APPLICATION OF BOUNDARY INTEGRAL EQUATION METHOD FOR THERMOELASTICITY PROBLEMS
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Vorona Yu.V.
2015-12-01
Full Text Available Boundary Integral Equation Method is used for solving analytically the problems of coupled thermoelastic spherical wave propagation. The resulting mathematical expressions coincide with the solutions obtained in a conventional manner.
Local defect correction for boundary integral equation methods
Kakuba, G.; Anthonissen, M.J.H.
2014-01-01
The aim in this paper is to develop a new local defect correction approach to gridding for problems with localised regions of high activity in the boundary element method. The technique of local defect correction has been studied for other methods as finite difference methods and finite volume
Local defect correction for boundary integral equation methods
Kakuba, G.; Anthonissen, M.J.H.
2013-01-01
This paper presents a new approach to gridding for problems with localised regions of high activity. The technique of local defect correction has been studied for other methods as ¿nite difference methods and ¿nite volume methods. In this paper we develop the technique for the boundary element
Application of Monte Carlo method to solving boundary value problem of differential equations
International Nuclear Information System (INIS)
Zuo Yinghong; Wang Jianguo
2012-01-01
This paper introduces the foundation of the Monte Carlo method and the way how to generate the random numbers. Based on the basic thought of the Monte Carlo method and finite differential method, the stochastic model for solving the boundary value problem of differential equations is built. To investigate the application of the Monte Carlo method to solving the boundary value problem of differential equations, the model is used to solve Laplace's equations with the first boundary condition and the unsteady heat transfer equation with initial values and boundary conditions. The results show that the boundary value problem of differential equations can be effectively solved with the Monte Carlo method, and the differential equations with initial condition can also be calculated by using a stochastic probability model which is based on the time-domain finite differential equations. Both the simulation results and theoretical analyses show that the errors of numerical results are lowered as the number of simulation particles is increased. (authors)
Implicit Boundary Integral Methods for the Helmholtz Equation in Exterior Domains
2016-06-01
solve the Helmholtz equation as ∂Ω goes through significant change in its shape and topology — applications for which implicit representation of the...boundary-value problems for the wave equation and maxwell’s equations. Russian Math . Surv., 1965. [16] S. Reutskiy. The method of fundamental
Alam Khan, Najeeb; Razzaq, Oyoon Abdul
2016-03-01
In the present work a wavelets approximation method is employed to solve fuzzy boundary value differential equations (FBVDEs). Essentially, a truncated Legendre wavelets series together with the Legendre wavelets operational matrix of derivative are utilized to convert FB- VDE into a simple computational problem by reducing it into a system of fuzzy algebraic linear equations. The capability of scheme is investigated on second order FB- VDE considered under generalized H-differentiability. Solutions are represented graphically showing competency and accuracy of this method.
Yousef, Hamood Mohammed; Ismail, Ahmad Izani
2017-11-01
In this paper, Laplace Adomian decomposition method (LADM) was applied to solve Delay differential equations with Boundary Value Problems. The solution is in the form of a convergent series which is easy to compute. This approach is tested on two test problem. The findings obtained exhibit the reliability and efficiency of the proposed method.
International Nuclear Information System (INIS)
Srivastava, A.C.; Hazarika, G.C.
1990-01-01
An algorithm based on the shooting method has been developed for the solution of a two-point boundary value problem consisting of a system of third order simultaneous ordinary differential equations. The Falkner-Skan equations for electrically conducting viscous fluid with applied magnetic field has been solved by using this algorithm for various values of the wedge angle and magnetic parameters. The shooting method seems to be well convergent for a system as the results are in good agreement with those obtained by other methods. It is observed that both viscous boundary layer and magnetic boundary layer decrease while velocity as well as magnetic field increase with the increase of the wedge angle. (author). 6 tabs., 7 refs
The Method of Subsuper Solutions for Weighted p(r-Laplacian Equation Boundary Value Problems
Directory of Open Access Journals (Sweden)
Zhimei Qiu
2008-10-01
Full Text Available This paper investigates the existence of solutions for weighted p(r-Laplacian ordinary boundary value problems. Our method is based on Leray-Schauder degree. As an application, we give the existence of weak solutions for p(x-Laplacian partial differential equations.
Boundary integral equation methods in eigenvalue problems of elastodynamics and thin plates
Kitahara, M
1985-01-01
The boundary integral equation (BIE) method has been used more and more in the last 20 years for solving various engineering problems. It has important advantages over other techniques for numerical treatment of a wide class of boundary value problems and is now regarded as an indispensable tool for potential problems, electromagnetism problems, heat transfer, fluid flow, elastostatics, stress concentration and fracture problems, geomechanical problems, and steady-state and transient electrodynamics.In this book, the author gives a complete, thorough and detailed survey of the method. It pro
Directory of Open Access Journals (Sweden)
Hongwu Zhang
2011-08-01
Full Text Available In this article, we study a Cauchy problem for an elliptic equation with variable coefficients. It is well-known that such a problem is severely ill-posed; i.e., the solution does not depend continuously on the Cauchy data. We propose a modified quasi-boundary value regularization method to solve it. Convergence estimates are established under two a priori assumptions on the exact solution. A numerical example is given to illustrate our proposed method.
Brehm, Christoph; Barad, Michael F.; Kiris, Cetin C.
2016-01-01
An immersed boundary method for the compressible Navier-Stokes equation and the additional infrastructure that is needed to solve moving boundary problems and fully coupled fluid-structure interaction is described. All the methods described in this paper were implemented in NASA's LAVA solver framework. The underlying immersed boundary method is based on the locally stabilized immersed boundary method that was previously introduced by the authors. In the present paper this method is extended to account for all aspects that are involved for fluid structure interaction simulations, such as fast geometry queries and stencil computations, the treatment of freshly cleared cells, and the coupling of the computational fluid dynamics solver with a linear structural finite element method. The current approach is validated for moving boundary problems with prescribed body motion and fully coupled fluid structure interaction problems in 2D and 3D. As part of the validation procedure, results from the second AIAA aeroelastic prediction workshop are also presented. The current paper is regarded as a proof of concept study, while more advanced methods for fluid structure interaction are currently being investigated, such as geometric and material nonlinearities, and advanced coupling approaches.
A method for the approximate solutions of the unsteady boundary layer equations
International Nuclear Information System (INIS)
Abdus Sattar, Md.
1990-12-01
The approximate integral method proposed by Bianchini et al. to solve the unsteady boundary layer equations is considered here with a simple modification to the scale function for the similarity variable. This is done by introducing a time dependent length scale. The closed form solutions, thus obtained, give satisfactory results for the velocity profile and the skin friction to a limiting case in comparison with the results of the past investigators. (author). 7 refs, 2 figs
International Nuclear Information System (INIS)
Kupka, F.
1997-11-01
This thesis deals with the extension of sparse grid techniques to spectral methods for the solution of partial differential equations with periodic boundary conditions. A review on boundary and initial-boundary value problems and a discussion on numerical resolution is used to motivate this research. Spectral methods are introduced by projection techniques, and by three model problems: the stationary and the transient Helmholtz equations, and the linear advection equation. The approximation theory on the hyperbolic cross is reviewed and its close relation to sparse grids is demonstrated. This approach extends to non-periodic problems. Various Sobolev spaces with dominant mixed derivative are introduced to provide error estimates for Fourier approximation and interpolation on the hyperbolic cross and on sparse grids by means of Sobolev norms. The theorems are immediately applicable to the stability and convergence analysis of sparse grid spectral methods. This is explicitly demonstrated for the three model problems. A variant of the von Neumann condition is introduced to simplify the stability analysis of the time-dependent model problems. The discrete Fourier transformation on sparse grids is discussed together with its software implementation. Results on numerical experiments are used to illustrate the performance of the new method with respect to the smoothness properties of each example. The potential of the method in mathematical modelling is estimated and generalizations to other sparse grid methods are suggested. The appendix includes a complete Fortran90 program to solve the linear advection equation by the sparse grid Fourier collocation method and a third-order Runge-Kutta routine for integration in time. (author)
A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains
Johansen, Hans; Colella, Phillip
1998-11-01
We present a numerical method for solving Poisson's equation, with variable coefficients and Dirichlet boundary conditions, on two-dimensional regions. The approach uses a finite-volume discretization, which embeds the domain in a regular Cartesian grid. We treat the solution as a cell-centered quantity, even when those centers are outside the domain. Cells that contain a portion of the domain boundary use conservative differencing of second-order accurate fluxes on each cell volume. The calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes. This allows us to use multigrid iterations with a simple point relaxation strategy. We have combined this with an adaptive mesh refinement (AMR) procedure. We provide evidence that the algorithm is second-order accurate on various exact solutions and compare the adaptive and nonadaptive calculations.
Young, D. P.; Woo, A. C.; Bussoletti, J. E.; Johnson, F. T.
1986-01-01
A general method is developed combining fast direct methods and boundary integral equation methods to solve Poisson's equation on irregular exterior regions. The method requires O(N log N) operations where N is the number of grid points. Error estimates are given that hold for regions with corners and other boundary irregularities. Computational results are given in the context of computational aerodynamics for a two-dimensional lifting airfoil. Solutions of boundary integral equations for lifting and nonlifting aerodynamic configurations using preconditioned conjugate gradient are examined for varying degrees of thinness.
Verschaeve, Joris C G
2011-06-13
By means of the continuity equation of the incompressible Navier-Stokes equations, additional physical arguments for the derivation of a formulation of the no-slip boundary condition for the lattice Boltzmann method for straight walls at rest are obtained. This leads to a boundary condition that is second-order accurate with respect to the grid spacing and conserves mass. In addition, the boundary condition is stable for relaxation frequencies close to two.
International Nuclear Information System (INIS)
Itagaki, Masafumi; Sahashi, Naoki.
1997-01-01
The multiple reciprocity boundary element method has been applied to three-dimensional two-group neutron diffusion problems. A matrix-type boundary integral equation has been derived to solve the first and the second group neutron diffusion equations simultaneously. The matrix-type fundamental solutions used here satisfy the equation which has a point source term and is adjoint to the neutron diffusion equations. A multiple reciprocity method has been employed to transform the matrix-type domain integral related to the fission source into an equivalent boundary one. The higher order fundamental solutions required for this formulation are composed of a series of two types of analytic functions. The eigenvalue itself is also calculated using only boundary integrals. Three-dimensional test calculations indicate that the present method provides stable and accurate solutions for criticality problems. (author)
Boundary integral equation methods and numerical solutions thin plates on an elastic foundation
Constanda, Christian; Hamill, William
2016-01-01
This book presents and explains a general, efficient, and elegant method for solving the Dirichlet, Neumann, and Robin boundary value problems for the extensional deformation of a thin plate on an elastic foundation. The solutions of these problems are obtained both analytically—by means of direct and indirect boundary integral equation methods (BIEMs)—and numerically, through the application of a boundary element technique. The text discusses the methodology for constructing a BIEM, deriving all the attending mathematical properties with full rigor. The model investigated in the book can serve as a template for the study of any linear elliptic two-dimensional problem with constant coefficients. The representation of the solution in terms of single-layer and double-layer potentials is pivotal in the development of a BIEM, which, in turn, forms the basis for the second part of the book, where approximate solutions are computed with a high degree of accuracy. The book is intended for graduate students and r...
A stable penalty method for the compressible Navier-Stokes equations: I. Open boundary conditions
DEFF Research Database (Denmark)
Hesthaven, Jan; Gottlieb, D.
1996-01-01
The purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible Navier-Stokes equations in three spatial dimensions. The treatment uses the conservation form of the Navier-Stokes equations and utilizes linearization...
Directory of Open Access Journals (Sweden)
Yuan Li
2013-01-01
Full Text Available This paper presents two-level iteration penalty finite element methods to approximate the solution of the Navier-Stokes equations with friction boundary conditions. The basic idea is to solve the Navier-Stokes type variational inequality problem on a coarse mesh with mesh size H in combining with solving a Stokes, Oseen, or linearized Navier-Stokes type variational inequality problem for Stokes, Oseen, or Newton iteration on a fine mesh with mesh size h. The error estimate obtained in this paper shows that if H, h, and ε can be chosen appropriately, then these two-level iteration penalty methods are of the same convergence orders as the usual one-level iteration penalty method.
Scalable smoothing strategies for a geometric multigrid method for the immersed boundary equations
Energy Technology Data Exchange (ETDEWEB)
Bhalla, Amneet Pal Singh [Univ. of North Carolina, Chapel Hill, NC (United States); Knepley, Matthew G. [Rice Univ., Houston, TX (United States); Adams, Mark F. [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Guy, Robert D. [Univ. of California, Davis, CA (United States); Griffith, Boyce E. [Univ. of North Carolina, Chapel Hill, NC (United States)
2016-12-20
The immersed boundary (IB) method is a widely used approach to simulating fluid-structure interaction (FSI). Although explicit versions of the IB method can suffer from severe time step size restrictions, these methods remain popular because of their simplicity and generality. In prior work (Guy et al., Adv Comput Math, 2015), some of us developed a geometric multigrid preconditioner for a stable semi-implicit IB method under Stokes flow conditions; however, this solver methodology used a Vanka-type smoother that presented limited opportunities for parallelization. This work extends this Stokes-IB solver methodology by developing smoothing techniques that are suitable for parallel implementation. Specifically, we demonstrate that an additive version of the Vanka smoother can yield an effective multigrid preconditioner for the Stokes-IB equations, and we introduce an efficient Schur complement-based smoother that is also shown to be effective for the Stokes-IB equations. We investigate the performance of these solvers for a broad range of material stiffnesses, both for Stokes flows and flows at nonzero Reynolds numbers, and for thick and thin structural models. We show here that linear solver performance degrades with increasing Reynolds number and material stiffness, especially for thin interface cases. Nonetheless, the proposed approaches promise to yield effective solution algorithms, especially at lower Reynolds numbers and at modest-to-high elastic stiffnesses.
Ge, Liang; Sotiropoulos, Fotis
2007-08-01
A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g. the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [A. Gilmanov, F. Sotiropoulos, A hybrid cartesian/immersed boundary method for simulating flows with 3d, geometrically complex, moving bodies, Journal of Computational Physics 207 (2005) 457-492.]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow
Directory of Open Access Journals (Sweden)
Dang Quang A
2013-02-01
Full Text Available In this paper we consider a mixed boundary value problem for biharmonic equation of the Airy stress function which models a crack problem of a solid elastic plate. An iterative method for reducing the problem to a sequence of mixed problems for Poisson equations is proposed and investigated. The convergence of the method is established theoretically and illustrated on many numerical experiments.
Azarnavid, Babak; Parand, Kourosh; Abbasbandy, Saeid
2018-06-01
This article discusses an iterative reproducing kernel method with respect to its effectiveness and capability of solving a fourth-order boundary value problem with nonlinear boundary conditions modeling beams on elastic foundations. Since there is no method of obtaining reproducing kernel which satisfies nonlinear boundary conditions, the standard reproducing kernel methods cannot be used directly to solve boundary value problems with nonlinear boundary conditions as there is no knowledge about the existence and uniqueness of the solution. The aim of this paper is, therefore, to construct an iterative method by the use of a combination of reproducing kernel Hilbert space method and a shooting-like technique to solve the mentioned problems. Error estimation for reproducing kernel Hilbert space methods for nonlinear boundary value problems have yet to be discussed in the literature. In this paper, we present error estimation for the reproducing kernel method to solve nonlinear boundary value problems probably for the first time. Some numerical results are given out to demonstrate the applicability of the method.
Vazquez-Leal, Hector; Benhammouda, Brahim; Filobello-Nino, Uriel Antonio; Sarmiento-Reyes, Arturo; Jimenez-Fernandez, Victor Manuel; Marin-Hernandez, Antonio; Herrera-May, Agustin Leobardo; Diaz-Sanchez, Alejandro; Huerta-Chua, Jesus
2014-01-01
In this article, we propose the application of a modified Taylor series method (MTSM) for the approximation of nonlinear problems described on finite intervals. The issue of Taylor series method with mixed boundary conditions is circumvented using shooting constants and extra derivatives of the problem. In order to show the benefits of this proposal, three different kinds of problems are solved: three-point boundary valued problem (BVP) of third-order with a hyperbolic sine nonlinearity, two-point BVP for a second-order nonlinear differential equation with an exponential nonlinearity, and a two-point BVP for a third-order nonlinear differential equation with a radical nonlinearity. The result shows that the MTSM method is capable to generate easily computable and highly accurate approximations for nonlinear equations. 34L30.
International Nuclear Information System (INIS)
Lu Jia; Zhou Huaichun
2016-01-01
To deal with the staircase approximation problem in the standard finite-difference time-domain (FDTD) simulation, the two-dimensional boundary condition equations (BCE) method is proposed in this paper. In the BCE method, the standard FDTD algorithm can be used as usual, and the curved surface is treated by adding the boundary condition equations. Thus, while maintaining the simplicity and computational efficiency of the standard FDTD algorithm, the BCE method can solve the staircase approximation problem. The BCE method is validated by analyzing near field and far field scattering properties of the PEC and dielectric cylinders. The results show that the BCE method can maintain a second-order accuracy by eliminating the staircase approximation errors. Moreover, the results of the BCE method show good accuracy for cylinder scattering cases with different permittivities. (paper)
The boundary integral equations method for analysis of high-frequency vibrations of an elastic layer
Czech Academy of Sciences Publication Activity Database
Sorokin, S.; Kolman, Radek; Kopačka, Ján
2017-01-01
Roč. 87, č. 4 (2017), s. 737-750 ISSN 0939-1533 R&D Projects: GA ČR(CZ) GA16-03823S; GA MŠk(CZ) EF15_003/0000493 Institutional support: RVO:61388998 Keywords : an elastic layer * symmetric and skew-symmetric waves * the Green’s matrix * boundary integral equations * eigen frequencies Subject RIV: BI - Acoustics OBOR OECD: Acoustics Impact factor: 1.490, year: 2016 https://link.springer.com/article/10.1007/s00419-016-1220-y
International Nuclear Information System (INIS)
Chiba, Gou; Tsuji, Masashi; Shimazu, Yoichiro
2001-01-01
A hierarchical domain decomposition boundary element method (HDD-BEM) that was developed to solve a two-dimensional neutron diffusion equation has been modified to deal with three-dimensional problems. In the HDD-BEM, the domain is decomposed into homogeneous regions. The boundary conditions on the common inner boundaries between decomposed regions and the neutron multiplication factor are initially assumed. With these assumptions, the neutron diffusion equations defined in decomposed homogeneous regions can be solved respectively by applying the boundary element method. This part corresponds to the 'lower level' calculations. At the 'higher level' calculations, the assumed values, the inner boundary conditions and the neutron multiplication factor, are modified so as to satisfy the continuity conditions for the neutron flux and the neutron currents on the inner boundaries. These procedures of the lower and higher levels are executed alternately and iteratively until the continuity conditions are satisfied within a convergence tolerance. With the hierarchical domain decomposition, it is possible to deal with problems composing a large number of regions, something that has been difficult with the conventional BEM. In this paper, it is showed that a three-dimensional problem even with 722 regions can be solved with a fine accuracy and an acceptable computation time. (author)
Directory of Open Access Journals (Sweden)
Taohua Liu
2017-01-01
Full Text Available Fractional advection-dispersion equations, as generalizations of classical integer-order advection-dispersion equations, are used to model the transport of passive tracers carried by fluid flow in a porous medium. In this paper, we develop an implicit finite difference method for fractional advection-dispersion equations with fractional derivative boundary conditions. First-order consistency, solvability, unconditional stability, and first-order convergence of the method are proven. Then, we present a fast iterative method for the implicit finite difference scheme, which only requires storage of O(K and computational cost of O(KlogK. Traditionally, the Gaussian elimination method requires storage of O(K2 and computational cost of O(K3. Finally, the accuracy and efficiency of the method are checked with a numerical example.
Yang, S A
2002-10-01
This paper presents an effective solution method for predicting acoustic radiation and scattering fields in two dimensions. The difficulty of the fictitious characteristic frequency is overcome by incorporating an auxiliary interior surface that satisfies certain boundary condition into the body surface. This process gives rise to a set of uniquely solvable boundary integral equations. Distributing monopoles with unknown strengths over the body and interior surfaces yields the simple source formulation. The modified boundary integral equations are further transformed to ordinary ones that contain nonsingular kernels only. This implementation allows direct application of standard quadrature formulas over the entire integration domain; that is, the collocation points are exactly the positions at which the integration points are located. Selecting the interior surface is an easy task. Moreover, only a few corresponding interior nodal points are sufficient for the computation. Numerical calculations consist of the acoustic radiation and scattering by acoustically hard elliptic and rectangular cylinders. Comparisons with analytical solutions are made. Numerical results demonstrate the efficiency and accuracy of the current solution method.
A highly accurate boundary integral equation method for surfactant-laden drops in 3D
Sorgentone, Chiara; Tornberg, Anna-Karin
2018-05-01
The presence of surfactants alters the dynamics of viscous drops immersed in an ambient viscous fluid. This is specifically true at small scales, such as in applications of droplet based microfluidics, where the interface dynamics become of increased importance. At such small scales, viscous forces dominate and inertial effects are often negligible. Considering Stokes flow, a numerical method based on a boundary integral formulation is presented for simulating 3D drops covered by an insoluble surfactant. The method is able to simulate drops with different viscosities and close interactions, automatically controlling the time step size and maintaining high accuracy also when substantial drop deformation appears. To achieve this, the drop surfaces as well as the surfactant concentration on each surface are represented by spherical harmonics expansions. A novel reparameterization method is introduced to ensure a high-quality representation of the drops also under deformation, specialized quadrature methods for singular and nearly singular integrals that appear in the formulation are evoked and the adaptive time stepping scheme for the coupled drop and surfactant evolution is designed with a preconditioned implicit treatment of the surfactant diffusion.
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Yurii M. Streliaiev
2016-06-01
Full Text Available Three-dimensional quasistatic contact problem of two linearly elastic bodies' interaction with Coulomb friction taken into account is considered. The boundary conditions of the problem have been simplified by the modification of the Coulomb's law of friction. This modification is based on the introducing of a delay in normal contact tractions that bound tangent contact tractions in the Coulomb's law of friction expressions. At this statement the problem is reduced to a sequence of similar systems of nonlinear integral equations describing bodies' interaction at each step of loading. A method for an approximate solution of the integral equations system corresponded to each step of loading is applied. This method consists of system regularization, discretization of regularized system and iterative process application for solving the discretized system. A numerical solution of a contact problem of an elastic sphere with an elastic half-space interaction under increasing and subsequently decreasing normal compressive force has been obtained.
The laminar boundary layer equations
Curle, N
2017-01-01
Thorough introduction to boundary layer problems offers an ordered, logical presentation accessible to undergraduates. The text's careful expositions of the limitations and accuracy of various methods will also benefit professionals. 1962 edition.
Variational methods for boundary value problems for systems of elliptic equations
Lavrent'ev, M A
2012-01-01
Famous monograph by a distinguished mathematician presents an innovative approach to classical boundary value problems. The treatment employs the basic scheme first suggested by Hilbert and developed by Tonnelli. 1963 edition.
Boundary value problems and partial differential equations
Powers, David L
2005-01-01
Boundary Value Problems is the leading text on boundary value problems and Fourier series. The author, David Powers, (Clarkson) has written a thorough, theoretical overview of solving boundary value problems involving partial differential equations by the methods of separation of variables. Professors and students agree that the author is a master at creating linear problems that adroitly illustrate the techniques of separation of variables used to solve science and engineering.* CD with animations and graphics of solutions, additional exercises and chapter review questions* Nearly 900 exercises ranging in difficulty* Many fully worked examples
International Nuclear Information System (INIS)
Ozgener, B.
1998-01-01
A boundary integral equation (BIE) is developed for the application of the boundary element method to the multigroup neutron diffusion equations. The developed BIE contains no explicit scattering term; the scattering effects are taken into account by redefining the unknowns. Boundary elements of the linear and constant variety are utilised for validation of the developed boundary integral formulation
Reaction diffusion equations with boundary degeneracy
Directory of Open Access Journals (Sweden)
Huashui Zhan
2016-03-01
Full Text Available In this article, we consider the reaction diffusion equation $$ \\frac{\\partial u}{\\partial t} = \\Delta A(u,\\quad (x,t\\in \\Omega \\times (0,T, $$ with the homogeneous boundary condition. Inspired by the Fichera-Oleinik theory, if the equation is not only strongly degenerate in the interior of $\\Omega$, but also degenerate on the boundary, we show that the solution of the equation is free from any limitation of the boundary condition.
The Boundary Function Method. Fundamentals
Kot, V. A.
2017-03-01
The boundary function method is proposed for solving applied problems of mathematical physics in the region defined by a partial differential equation of the general form involving constant or variable coefficients with a Dirichlet, Neumann, or Robin boundary condition. In this method, the desired function is defined by a power polynomial, and a boundary function represented in the form of the desired function or its derivative at one of the boundary points is introduced. Different sequences of boundary equations have been set up with the use of differential operators. Systems of linear algebraic equations constructed on the basis of these sequences allow one to determine the coefficients of a power polynomial. Constitutive equations have been derived for initial boundary-value problems of all the main types. With these equations, an initial boundary-value problem is transformed into the Cauchy problem for the boundary function. The determination of the boundary function by its derivative with respect to the time coordinate completes the solution of the problem.
Nonlinear streak computation using boundary region equations
Energy Technology Data Exchange (ETDEWEB)
Martin, J A; Martel, C, E-mail: juanangel.martin@upm.es, E-mail: carlos.martel@upm.es [Depto. de Fundamentos Matematicos, E.T.S.I Aeronauticos, Universidad Politecnica de Madrid, Plaza Cardenal Cisneros 3, 28040 Madrid (Spain)
2012-08-01
The boundary region equations (BREs) are applied for the simulation of the nonlinear evolution of a spanwise periodic array of streaks in a flat plate boundary layer. The well-known BRE formulation is obtained from the complete Navier-Stokes equations in the high Reynolds number limit, and provides the correct asymptotic description of three-dimensional boundary layer streaks. In this paper, a fast and robust streamwise marching scheme is introduced to perform their numerical integration. Typical streak computations present in the literature correspond to linear streaks or to small-amplitude nonlinear streaks computed using direct numerical simulation (DNS) or the nonlinear parabolized stability equations (PSEs). We use the BREs to numerically compute high-amplitude streaks, a method which requires much lower computational effort than DNS and does not have the consistency and convergence problems of the PSE. It is found that the flow configuration changes substantially as the amplitude of the streaks grows and the nonlinear effects come into play. The transversal motion (in the wall normal-streamwise plane) becomes more important and strongly distorts the streamwise velocity profiles, which end up being quite different from those of the linear case. We analyze in detail the resulting flow patterns for the nonlinearly saturated streaks and compare them with available experimental results. (paper)
Integrable boundary conditions and modified Lax equations
International Nuclear Information System (INIS)
Avan, Jean; Doikou, Anastasia
2008-01-01
We consider integrable boundary conditions for both discrete and continuum classical integrable models. Local integrals of motion generated by the corresponding 'transfer' matrices give rise to time evolution equations for the initial Lax operator. We systematically identify the modified Lax pairs for both discrete and continuum boundary integrable models, depending on the classical r-matrix and the boundary matrix
Discrete transparent boundary conditions for Schroedinger-type equations
International Nuclear Information System (INIS)
Schmidt, F.; Yevick, D.
1997-01-01
We present a general technique for constructing nonlocal transparent boundary conditions for one-dimensional Schroedinger-type equations. Our method supplies boundary conditions for the θ-family of implicit one-step discretizations of Schroedinger's equation in time. The use of Mikusinski's operator approach in time avoids direct and inverse transforms between time and frequency domains and thus implements the boundary conditions in a direct manner. 14 refs., 9 figs
Zhao, Ying; Schillinger, Dominik; Xu, Bai-Xiang
2017-07-01
The primal variational formulation of the fourth-order Cahn-Hilliard equation requires C1-continuous finite element discretizations, e.g., in the context of isogeometric analysis. In this paper, we explore the variational imposition of essential boundary conditions that arise from the thermodynamic derivation of the Cahn-Hilliard equation in primal variables. Our formulation is based on the symmetric variant of Nitsche's method, does not introduce additional degrees of freedom and is shown to be variationally consistent. In contrast to strong enforcement, the new boundary condition formulation can be naturally applied to any mapped isogeometric parametrization of any polynomial degree. In addition, it preserves full accuracy, including higher-order rates of convergence, which we illustrate for boundary-fitted discretizations of several benchmark tests in one, two and three dimensions. Unfitted Cartesian B-spline meshes constitute an effective alternative to boundary-fitted isogeometric parametrizations for constructing C1-continuous discretizations, in particular for complex geometries. We combine our variational boundary condition formulation with unfitted Cartesian B-spline meshes and the finite cell method to simulate chemical phase segregation in a composite electrode. This example, involving coupling of chemical fields with mechanical stresses on complex domains and coupling of different materials across complex interfaces, demonstrates the flexibility of variational boundary conditions in the context of higher-order unfitted isogeometric discretizations.
Energy Technology Data Exchange (ETDEWEB)
Matsushima, Toshiki; Ishioka, Keiichi, E-mail: matsushima@kugi.kyoto-u.ac.jp, E-mail: ishioka@gfd-dennou.org [Graduate School of Science, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502 (Japan)
2017-04-15
This paper presents a spectral method for numerically solving the Navier–Stokes equations in a semi-infinite domain bounded by a flat plane: the aim is to obtain high accuracy with flexible boundary conditions. The proposed use is for numerical simulations of small-scale atmospheric phenomena near the ground. We introduce basis functions that fit the semi-infinite domain, and an integral condition for vorticity is used to reduce the computational cost when solving the partial differential equations that appear when the viscosity term is treated implicitly. Furthermore, in order to ensure high accuracy, two iteration techniques are applied when solving the system of linear equations and in determining boundary values. This significantly reduces numerical errors, and the proposed method enables high-resolution numerical experiments. This is demonstrated by numerical experiments showing the collision of a vortex ring into a wall; these were performed using numerical models based on the proposed method. It is shown that the time evolution of the flow field is successfully obtained not only near the boundary, but also in a region far from the boundary. The applicability of the proposed method and the integral condition is discussed. (paper)
Lee, Byungjoon; Min, Chohong
2018-05-01
We introduce a stable method for solving the incompressible Navier-Stokes equations with variable density and viscosity. Our method is stable in the sense that it does not increase the total energy of dynamics that is the sum of kinetic energy and potential energy. Instead of velocity, a new state variable is taken so that the kinetic energy is formulated by the L2 norm of the new variable. Navier-Stokes equations are rephrased with respect to the new variable, and a stable time discretization for the rephrased equations is presented. Taking into consideration the incompressibility in the Marker-And-Cell (MAC) grid, we present a modified Lax-Friedrich method that is L2 stable. Utilizing the discrete integration-by-parts in MAC grid and the modified Lax-Friedrich method, the time discretization is fully discretized. An explicit CFL condition for the stability of the full discretization is given and mathematically proved.
Nakagawa, Y.
1981-01-01
The method described as the method of nearcharacteristics by Nakagawa (1980) is renamed the method of projected characteristics. Making full use of properties of the projected characteristics, a new and simpler formulation is developed. As a result, the formulation for the examination of the general three-dimensional problems is presented. It is noted that since in practice numerical solutions must be obtained, the final formulation is given in the form of difference equations. The possibility of including effects of viscous and ohmic dissipations in the formulation is considered, and the physical interpretation is discussed. A systematic manner is then presented for deriving physically self-consistent, time-dependent boundary equations for MHD initial boundary problems. It is demonstrated that the full use of the compatibility equations (differential equations relating variations at two spatial locations and times) is required in determining the time-dependent boundary conditions. In order to provide a clear physical picture as an example, the evolution of axisymmetric global magnetic field by photospheric differential rotation is considered.
Directory of Open Access Journals (Sweden)
A. Becker
2003-01-01
Full Text Available In this paper a hybrid method combining the FDTD/FIT with a Time Domain Boundary-Integral Marching-on-in-Time Algorithm (TD-BIM is presented. Inhomogeneous regions are modelled with the FIT-method, an alternative formulation of the FDTD. Homogeneous regions (which is in the presented numerical example the open space are modelled using a TD-BIM with equivalent electric and magnetic currents flowing on the boundary between the inhomogeneous and the homogeneous regions. The regions are coupled by the tangential magnetic fields just outside the inhomogeneous regions. These fields are calculated by making use of a Mixed Potential Integral Formulation for the magnetic field. The latter consists of equivalent electric and magnetic currents on the boundary plane between the homogeneous and the inhomogeneous region. The magnetic currents result directly from the electric fields of the Yee lattice. Electric currents in the same plane are calculated by making use of the TD-BIM and using the electric field of the Yee lattice as boundary condition. The presented hybrid method only needs the interpolations inherent in FIT and no additional interpolation. A numerical result is compared to a calculation that models both regions with FDTD.
Sirenko, Kostyantyn
2013-01-01
A scheme that discretizes exact absorbing boundary conditions (EACs) to incorporate them into a time-domain discontinuous Galerkin finite element method (TD-DG-FEM) is described. The proposed TD-DG-FEM with EACs is used for accurately characterizing transient electromagnetic wave interactions on two-dimensional waveguides. Numerical results demonstrate the proposed method\\'s superiority over the TD-DG-FEM that employs approximate boundary conditions and perfectly matched layers. Additionally, it is shown that the proposed method can produce the solution with ten-eleven digit accuracy when high-order spatial basis functions are used to discretize the Maxwell equations as well as the EACs. © 1963-2012 IEEE.
Optimal Wentzell Boundary Control of Parabolic Equations
International Nuclear Information System (INIS)
Luo, Yousong
2017-01-01
This paper deals with a class of optimal control problems governed by an initial-boundary value problem of a parabolic equation. The case of semi-linear boundary control is studied where the control is applied to the system via the Wentzell boundary condition. The differentiability of the state variable with respect to the control is established and hence a necessary condition is derived for the optimal solution in the case of both unconstrained and constrained problems. The condition is also sufficient for the unconstrained convex problems. A second order condition is also derived.
Optimal Wentzell Boundary Control of Parabolic Equations
Energy Technology Data Exchange (ETDEWEB)
Luo, Yousong, E-mail: yousong.luo@rmit.edu.au [RMIT University, School of Mathematical and Geospatial Sciences (Australia)
2017-04-15
This paper deals with a class of optimal control problems governed by an initial-boundary value problem of a parabolic equation. The case of semi-linear boundary control is studied where the control is applied to the system via the Wentzell boundary condition. The differentiability of the state variable with respect to the control is established and hence a necessary condition is derived for the optimal solution in the case of both unconstrained and constrained problems. The condition is also sufficient for the unconstrained convex problems. A second order condition is also derived.
Boundary value problems for multi-term fractional differential equations
Daftardar-Gejji, Varsha; Bhalekar, Sachin
2008-09-01
Multi-term fractional diffusion-wave equation along with the homogeneous/non-homogeneous boundary conditions has been solved using the method of separation of variables. It is observed that, unlike in the one term case, solution of multi-term fractional diffusion-wave equation is not necessarily non-negative, and hence does not represent anomalous diffusion of any kind.
A.R. Ansari; B. Hossain; B. Koren (Barry); G.I. Shishkin (Gregori)
2007-01-01
textabstractWe investigate the model problem of flow of a viscous incompressible fluid past a symmetric curved surface when the flow is parallel to its axis. This problem is known to exhibit boundary layers. Also the problem does not have solutions in closed form, it is modelled by boundary-layer
International Nuclear Information System (INIS)
Olschewski, J.; Stein, E.; Wagner, W.; Wetjen, D.
1981-01-01
This paper is a first step in the development of thermodynamically consistent material equations for inelastic materials, such as polycrystalline rock salt. In this context it is of particular importance to reduce the number and the structure of the internal variables, in order to allow for a fit with available experimental data. As an example this is demonstrated in detail in the case of the so-called dislocation model. As physical non-linearities and in addition also geometrical non-linearities lead to an inhomogeneous deformation - and stress state even in the case of simple samples, boundary value problems have to be studied, in order to test the material equations. For this purpose the finite element method has been used. (orig./HP) [de
Recent advances in boundary element methods
Manolis, GD
2009-01-01
Addresses the needs of the computational mechanics research community in terms of information on boundary integral equation-based methods and techniques applied to a variety of fields. This book collects both original and review articles on contemporary Boundary Element Methods (BEM) as well as on the Mesh Reduction Methods (MRM).
Numerical solution of boundary-integral equations for molecular electrostatics.
Bardhan, Jaydeep P
2009-03-07
Numerous molecular processes, such as ion permeation through channel proteins, are governed by relatively small changes in energetics. As a result, theoretical investigations of these processes require accurate numerical methods. In the present paper, we evaluate the accuracy of two approaches to simulating boundary-integral equations for continuum models of the electrostatics of solvation. The analysis emphasizes boundary-element method simulations of the integral-equation formulation known as the apparent-surface-charge (ASC) method or polarizable-continuum model (PCM). In many numerical implementations of the ASC/PCM model, one forces the integral equation to be satisfied exactly at a set of discrete points on the boundary. We demonstrate in this paper that this approach to discretization, known as point collocation, is significantly less accurate than an alternative approach known as qualocation. Furthermore, the qualocation method offers this improvement in accuracy without increasing simulation time. Numerical examples demonstrate that electrostatic part of the solvation free energy, when calculated using the collocation and qualocation methods, can differ significantly; for a polypeptide, the answers can differ by as much as 10 kcal/mol (approximately 4% of the total electrostatic contribution to solvation). The applicability of the qualocation discretization to other integral-equation formulations is also discussed, and two equivalences between integral-equation methods are derived.
A Coordinate Transformation for Unsteady Boundary Layer Equations
Directory of Open Access Journals (Sweden)
Paul G. A. CIZMAS
2011-12-01
Full Text Available This paper presents a new coordinate transformation for unsteady, incompressible boundary layer equations that applies to both laminar and turbulent flows. A generalization of this coordinate transformation is also proposed. The unsteady boundary layer equations are subsequently derived. In addition, the boundary layer equations are derived using a time linearization approach and assuming harmonically varying small disturbances.
Numerical Methods for Partial Differential Equations
Guo, Ben-yu
1987-01-01
These Proceedings of the first Chinese Conference on Numerical Methods for Partial Differential Equations covers topics such as difference methods, finite element methods, spectral methods, splitting methods, parallel algorithm etc., their theoretical foundation and applications to engineering. Numerical methods both for boundary value problems of elliptic equations and for initial-boundary value problems of evolution equations, such as hyperbolic systems and parabolic equations, are involved. The 16 papers of this volume present recent or new unpublished results and provide a good overview of current research being done in this field in China.
Asgharzadeh, Hafez; Borazjani, Iman
2017-02-15
The explicit and semi-implicit schemes in flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates. Implicit schemes can be used to overcome these restrictions, but implementing them to solve the Navier-Stokes equations is not straightforward due to their non-linearity. Among the implicit schemes for nonlinear equations, Newton-based techniques are preferred over fixed-point techniques because of their high convergence rate but each Newton iteration is more expensive than a fixed-point iteration. Krylov subspace methods are one of the most advanced iterative methods that can be combined with Newton methods, i.e., Newton-Krylov Methods (NKMs) to solve non-linear systems of equations. The success of NKMs vastly depends on the scheme for forming the Jacobian, e.g., automatic differentiation is very expensive, and matrix-free methods without a preconditioner slow down as the mesh is refined. A novel, computationally inexpensive analytical Jacobian for NKM is developed to solve unsteady incompressible Navier-Stokes momentum equations on staggered overset-curvilinear grids with immersed boundaries. Moreover, the analytical Jacobian is used to form preconditioner for matrix-free method in order to improve its performance. The NKM with the analytical Jacobian was validated and verified against Taylor-Green vortex, inline oscillations of a cylinder in a fluid initially at rest, and pulsatile flow in a 90 degree bend. The capability of the method in handling complex geometries with multiple overset grids and immersed boundaries is shown by simulating an intracranial aneurysm. It was shown that the NKM with an analytical Jacobian is 1.17 to 14.77 times faster than the fixed-point Runge-Kutta method, and 1.74 to 152.3 times (excluding an intensively stretched grid) faster than automatic differentiation depending on the grid (size) and the flow problem. In addition, it was shown that using only the
Asgharzadeh, Hafez; Borazjani, Iman
2016-01-01
The explicit and semi-implicit schemes in flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates. Implicit schemes can be used to overcome these restrictions, but implementing them to solve the Navier-Stokes equations is not straightforward due to their non-linearity. Among the implicit schemes for nonlinear equations, Newton-based techniques are preferred over fixed-point techniques because of their high convergence rate but each Newton iteration is more expensive than a fixed-point iteration. Krylov subspace methods are one of the most advanced iterative methods that can be combined with Newton methods, i.e., Newton-Krylov Methods (NKMs) to solve non-linear systems of equations. The success of NKMs vastly depends on the scheme for forming the Jacobian, e.g., automatic differentiation is very expensive, and matrix-free methods without a preconditioner slow down as the mesh is refined. A novel, computationally inexpensive analytical Jacobian for NKM is developed to solve unsteady incompressible Navier-Stokes momentum equations on staggered overset-curvilinear grids with immersed boundaries. Moreover, the analytical Jacobian is used to form preconditioner for matrix-free method in order to improve its performance. The NKM with the analytical Jacobian was validated and verified against Taylor-Green vortex, inline oscillations of a cylinder in a fluid initially at rest, and pulsatile flow in a 90 degree bend. The capability of the method in handling complex geometries with multiple overset grids and immersed boundaries is shown by simulating an intracranial aneurysm. It was shown that the NKM with an analytical Jacobian is 1.17 to 14.77 times faster than the fixed-point Runge-Kutta method, and 1.74 to 152.3 times (excluding an intensively stretched grid) faster than automatic differentiation depending on the grid (size) and the flow problem. In addition, it was shown that using only the
Kou, Jisheng
2015-03-01
In this paper, we consider multi-component dynamic two-phase interface models, which are formulated by the Cahn-Hilliard system with Peng-Robinson equation of state and various boundary conditions. These models can be derived from the minimum problems of Helmholtz free energy or grand potential in the realistic thermodynamic systems. The resulted Cahn-Hilliard systems with various boundary conditions are fully coupled and strongly nonlinear. A linear transformation is introduced to decouple the relations between different components, and as a result, the models are simplified. From this, we further propose a semi-implicit unconditionally stable time discretization scheme, which allows us to solve the Cahn-Hilliard system by a decoupled way, and thus, our method can significantly reduce the computational cost and memory requirements. The mixed finite element methods are employed for the spatial discretization, and the approximate errors are also analyzed for both space and time. Numerical examples are tested to demonstrate the efficiency of our proposed methods. © 2015 Elsevier B.V.
Controllability for a Wave Equation with Moving Boundary
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Lizhi Cui
2014-01-01
Full Text Available We investigate the controllability for a one-dimensional wave equation in domains with moving boundary. This model characterizes small vibrations of a stretched elastic string when one of the two endpoints varies. When the speed of the moving endpoint is less than 1-1/e, by Hilbert uniqueness method, sidewise energy estimates method, and multiplier method, we get partial Dirichlet boundary controllability. Moreover, we will give a sharper estimate on controllability time that only depends on the speed of the moving endpoint.
Transport equation solving methods
International Nuclear Information System (INIS)
Granjean, P.M.
1984-06-01
This work is mainly devoted to Csub(N) and Fsub(N) methods. CN method: starting from a lemma stated by Placzek, an equivalence is established between two problems: the first one is defined in a finite medium bounded by a surface S, the second one is defined in the whole space. In the first problem the angular flux on the surface S is shown to be the solution of an integral equation. This equation is solved by Galerkin's method. The Csub(N) method is applied here to one-velocity problems: in plane geometry, slab albedo and transmission with Rayleigh scattering, calculation of the extrapolation length; in cylindrical geometry, albedo and extrapolation length calculation with linear scattering. Fsub(N) method: the basic integral transport equation of the Csub(N) method is integrated on Case's elementary distributions; another integral transport equation is obtained: this equation is solved by a collocation method. The plane problems solved by the Csub(N) method are also solved by the Fsub(N) method. The Fsub(N) method is extended to any polynomial scattering law. Some simple spherical problems are also studied. Chandrasekhar's method, collision probability method, Case's method are presented for comparison with Csub(N) and Fsub(N) methods. This comparison shows the respective advantages of the two methods: a) fast convergence and possible extension to various geometries for Csub(N) method; b) easy calculations and easy extension to polynomial scattering for Fsub(N) method [fr
Microlocal methods in the analysis of the boundary element method
DEFF Research Database (Denmark)
Pedersen, Michael
1993-01-01
The application of the boundary element method in numerical analysis is based upon the use of boundary integral operators stemming from multiple layer potentials. The regularity properties of these operators are vital in the development of boundary integral equations and error estimates. We show...
International Nuclear Information System (INIS)
Tsuji, Masashi; Chiba, Gou
2000-01-01
A hierarchical domain decomposition boundary element method (HDD-BEM) for solving the multiregion neutron diffusion equation (NDE) has been fully parallelized, both for numerical computations and for data communications, to accomplish a high parallel efficiency on distributed memory message passing parallel computers. Data exchanges between node processors that are repeated during iteration processes of HDD-BEM are implemented, without any intervention of the host processor that was used to supervise parallel processing in the conventional parallelized HDD-BEM (P-HDD-BEM). Thus, the parallel processing can be executed with only cooperative operations of node processors. The communication overhead was even the dominant time consuming part in the conventional P-HDD-BEM, and the parallelization efficiency decreased steeply with the increase of the number of processors. With the parallel data communication, the efficiency is affected only by the number of boundary elements assigned to decomposed subregions, and the communication overhead can be drastically reduced. This feature can be particularly advantageous in the analysis of three-dimensional problems where a large number of processors are required. The proposed P-HDD-BEM offers a promising solution to the deterioration problem of parallel efficiency and opens a new path to parallel computations of NDEs on distributed memory message passing parallel computers. (author)
On the wave equation with semilinear porous acoustic boundary conditions
Graber, Philip Jameson; Said-Houari, Belkacem
2012-01-01
The goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary/interior sources and a nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. The main difficulty in proving the local existence result is that the Neumann boundary conditions experience loss of regularity due to boundary sources. Using an approximation method involving truncated sources and adapting the ideas in Lasiecka and Tataru (1993) [28], we show that the existence of solutions can still be obtained. Second, we prove that under some restrictions on the source terms, then the local solution can be extended to be global in time. In addition, it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ODE and depends on the behavior of the damping terms. In several situations, the obtained ODE can be easily solved and the decay rates can be given explicitly. Third, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution ceases to exists and blows up in finite time. Moreover, in either the absence of the interior source or the boundary source, then we prove that the solution is unbounded and grows as an exponential function. © 2012 Elsevier Inc.
On the wave equation with semilinear porous acoustic boundary conditions
Graber, Philip Jameson
2012-05-01
The goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary/interior sources and a nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. The main difficulty in proving the local existence result is that the Neumann boundary conditions experience loss of regularity due to boundary sources. Using an approximation method involving truncated sources and adapting the ideas in Lasiecka and Tataru (1993) [28], we show that the existence of solutions can still be obtained. Second, we prove that under some restrictions on the source terms, then the local solution can be extended to be global in time. In addition, it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ODE and depends on the behavior of the damping terms. In several situations, the obtained ODE can be easily solved and the decay rates can be given explicitly. Third, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution ceases to exists and blows up in finite time. Moreover, in either the absence of the interior source or the boundary source, then we prove that the solution is unbounded and grows as an exponential function. © 2012 Elsevier Inc.
Numerical methods for differential equations and applications
International Nuclear Information System (INIS)
Ixaru, L.G.
1984-01-01
This book is addressed to persons who, without being professionals in applied mathematics, are often faced with the problem of numerically solving differential equations. In each of the first three chapters a definite class of methods is discussed for the solution of the initial value problem for ordinary differential equations: multistep methods; one-step methods; and piecewise perturbation methods. The fourth chapter is mainly focussed on the boundary value problems for linear second-order equations, with a section devoted to the Schroedinger equation. In the fifth chapter the eigenvalue problem for the radial Schroedinger equation is solved in several ways, with computer programs included. (Auth.)
Numerical solution of the resistive magnetohydrodynamic boundary-layer equations
International Nuclear Information System (INIS)
Glasser, A.H.; Jardin, S.C.; Tesauro, G.
1983-10-01
Three different techniques are presented for numerical solution of the equations governing the boundary layer of resistive magnetohydrodynamic tearing and interchange instabilities in toroidal geometry. Excellent agreement among these methods and with analytical results provides confidence in the correctness of the results. Solutions obtained in regimes where analytical medthods fail indicate a new scaling for the tearing mode as well as the existence of a new regime of stability
Beshtokov, M. Kh.
2017-12-01
Boundary value problems for loaded third-order pseudo-parabolic equations with variable coefficients are considered. A priori estimates for the solutions of the problems in the differential and difference formulations are obtained. These a priori estimates imply the uniqueness and stability of the solution with respect to the initial data and the right-hand side on a layer, as well as the convergence of the solution of each difference problem to the solution of the corresponding differential problem.
Beshtokov, M. Kh.
2016-10-01
A nonlocal boundary value problem for a degenerating third-order pseudo-parabolic equation with variable coefficients is considered. For solving this problem, a priori estimates in the differential and difference forms are obtained. The a priori estimates imply the uniqueness and stability of the solution on a layer with respect to the initial data and the right-hand side and the convergence of the solution of the difference problem to the solution of the differential problem.
Tokamak plasma shape identification based on the boundary integral equations
International Nuclear Information System (INIS)
Kurihara, Kenichi; Kimura, Toyoaki
1992-05-01
A necessary condition for tokamak plasma shape identification is discussed and a new identification method is proposed in this article. This method is based on the boundary integral equations governing a vacuum region around a plasma with only the measurement of either magnetic fluxes or magnetic flux intensities. It can identify various plasmas with low to high ellipticities with the precision determined by the number of the magnetic sensors. This method is applicable to real-time control and visualization using a 'table-look-up' procedure. (author)
Discretization of the induced-charge boundary integral equation.
Bardhan, Jaydeep P; Eisenberg, Robert S; Gillespie, Dirk
2009-07-01
Boundary-element methods (BEMs) for solving integral equations numerically have been used in many fields to compute the induced charges at dielectric boundaries. In this paper, we consider a more accurate implementation of BEM in the context of ions in aqueous solution near proteins, but our results are applicable more generally. The ions that modulate protein function are often within a few angstroms of the protein, which leads to the significant accumulation of polarization charge at the protein-solvent interface. Computing the induced charge accurately and quickly poses a numerical challenge in solving a popular integral equation using BEM. In particular, the accuracy of simulations can depend strongly on seemingly minor details of how the entries of the BEM matrix are calculated. We demonstrate that when the dielectric interface is discretized into flat tiles, the qualocation method of Tausch [IEEE Trans Comput.-Comput.-Aided Des. 20, 1398 (2001)] to compute the BEM matrix elements is always more accurate than the traditional centroid-collocation method. Qualocation is not more expensive to implement than collocation and can save significant computational time by reducing the number of boundary elements needed to discretize the dielectric interfaces.
Discretization of the induced-charge boundary integral equation.
Energy Technology Data Exchange (ETDEWEB)
Bardhan, J. P.; Eisenberg, R. S.; Gillespie, D.; Rush Univ. Medical Center
2009-07-01
Boundary-element methods (BEMs) for solving integral equations numerically have been used in many fields to compute the induced charges at dielectric boundaries. In this paper, we consider a more accurate implementation of BEM in the context of ions in aqueous solution near proteins, but our results are applicable more generally. The ions that modulate protein function are often within a few angstroms of the protein, which leads to the significant accumulation of polarization charge at the protein-solvent interface. Computing the induced charge accurately and quickly poses a numerical challenge in solving a popular integral equation using BEM. In particular, the accuracy of simulations can depend strongly on seemingly minor details of how the entries of the BEM matrix are calculated. We demonstrate that when the dielectric interface is discretized into flat tiles, the qualocation method of Tausch et al. [IEEE Trans Comput.-Comput.-Aided Des. 20, 1398 (2001)] to compute the BEM matrix elements is always more accurate than the traditional centroid-collocation method. Qualocation is not more expensive to implement than collocation and can save significant computational time by reducing the number of boundary elements needed to discretize the dielectric interfaces.
Boundary conditions for the diffusion equation in radiative transfer
International Nuclear Information System (INIS)
Haskell, R.C.; Svaasand, L.O.; Tsay, T.; Feng, T.; McAdams, M.S.; Tromberg, B.J.
1994-01-01
Using the method of images, we examine the three boundary conditions commonly applied to the surface of a semi-infinite turbid medium. We find that the image-charge configurations of the partial-current and extrapolated-boundary conditions have the same dipole and quadrupole moments and that the two corresponding solutions to the diffusion equation are approximately equal. In the application of diffusion theory to frequency-domain photon-migration (FDPM) data, these two approaches yield values for the scattering and absorption coefficients that are equal to within 3%. Moreover, the two boundary conditions can be combined to yield a remarkably simple, accurate, and computationally fast method for extracting values for optical parameters from FDPM data. FDPM data were taken both at the surface and deep inside tissue phantoms, and the difference in data between the two geometries is striking. If one analyzes the surface data without accounting for the boundary, values deduced for the optical coefficients are in error by 50% or more. As expected, when aluminum foil was placed on the surface of a tissue phantom, phase and modulation data were closer to the results for an infinite-medium geometry. Raising the reflectivity of a tissue surface can, in principle, eliminate the effect of the boundary. However, we find that phase and modulation data are highly sensitive to the reflectivity in the range of 80--100%, and a minimum value of 98% is needed to mimic an infinite-medium geometry reliably. We conclude that noninvasive measurements of optically thick tissue require a rigorous treatment of the tissue boundary, and we suggest a unified partial-current--extrapolated boundary approach
Directory of Open Access Journals (Sweden)
Dandan Guo
2017-08-01
Full Text Available In this article we consider the boundary stabilization of a wave equation with variable coefficients. This equation has an acceleration term and a delayed velocity term on the boundary. Under suitable geometric conditions, we obtain the exponential decay for the solutions. Our proof relies on the geometric multiplier method and the Lyapunov approach.
Solution of Moving Boundary Space-Time Fractional Burger’s Equation
Directory of Open Access Journals (Sweden)
E. A.-B. Abdel-Salam
2014-01-01
Full Text Available The fractional Riccati expansion method is used to solve fractional differential equations with variable coefficients. To illustrate the effectiveness of the method, the moving boundary space-time fractional Burger’s equation is studied. The obtained solutions include generalized trigonometric and hyperbolic function solutions. Among these solutions, some are found for the first time. The linear and periodic moving boundaries for the kink solution of the Burger’s equation are presented graphically and discussed.
Boundary element method for modelling creep behaviour
International Nuclear Information System (INIS)
Zarina Masood; Shah Nor Basri; Abdel Majid Hamouda; Prithvi Raj Arora
2002-01-01
A two dimensional initial strain direct boundary element method is proposed to numerically model the creep behaviour. The boundary of the body is discretized into quadratic element and the domain into quadratic quadrilaterals. The variables are also assumed to have a quadratic variation over the elements. The boundary integral equation is solved for each boundary node and assembled into a matrix. This matrix is solved by Gauss elimination with partial pivoting to obtain the variables on the boundary and in the interior. Due to the time-dependent nature of creep, the solution has to be derived over increments of time. Automatic time incrementation technique and backward Euler method for updating the variables are implemented to assure stability and accuracy of results. A flowchart of the solution strategy is also presented. (Author)
Ene, Remus-Daniel; Marinca, Vasile; Marinca, Bogdan
2016-01-01
Analytic approximate solutions using Optimal Homotopy Perturbation Method (OHPM) are given for steady boundary layer flow over a nonlinearly stretching wall in presence of partial slip at the boundary. The governing equations are reduced to nonlinear ordinary differential equation by means of similarity transformations. Some examples are considered and the effects of different parameters are shown. OHPM is a very efficient procedure, ensuring a very rapid convergence of the solutions after only two iterations.
Methods for Equating Mental Tests.
1984-11-01
1983) compared conventional and IRT methods for equating the Test of English as a Foreign Language ( TOEFL ) after chaining. Three conventional and...three IRT equating methods were examined in this study; two sections of TOEFL were each (separately) equated. The IRT methods included the following: (a...group. A separate base form was established for each of the six equating methods. Instead of equating the base-form TOEFL to itself, the last (eighth
Sirenko, Kostyantyn; Liu, Meilin; Bagci, Hakan
2013-01-01
A scheme that discretizes exact absorbing boundary conditions (EACs) to incorporate them into a time-domain discontinuous Galerkin finite element method (TD-DG-FEM) is described. The proposed TD-DG-FEM with EACs is used for accurately characterizing
A One-Dimensional Wave Equation with White Noise Boundary Condition
International Nuclear Information System (INIS)
Kim, Jong Uhn
2006-01-01
We discuss the Cauchy problem for a one-dimensional wave equation with white noise boundary condition. We also establish the existence of an invariant measure when the noise is additive. Similar problems for parabolic equations were discussed by several authors. To our knowledge, there is only one work which investigated the initial-boundary value problem for a wave equation with random noise at the boundary. We handle a more general case by a different method. Our result on the existence of an invariant measure relies on the author's recent work on a certain class of stochastic evolution equations
Solution to random differential equations with boundary conditions
Directory of Open Access Journals (Sweden)
Fairouz Tchier
2017-04-01
Full Text Available We study a family of random differential equations with boundary conditions. Using a random fixed point theorem, we prove an existence theorem that yields a unique random solution.
Boundary-value problems with free boundaries for elliptic systems of equations
Monakhov, V N
1983-01-01
This book is concerned with certain classes of nonlinear problems for elliptic systems of partial differential equations: boundary-value problems with free boundaries. The first part has to do with the general theory of boundary-value problems for analytic functions and its applications to hydrodynamics. The second presents the theory of quasiconformal mappings, along with the theory of boundary-value problems for elliptic systems of equations and applications of it to problems in the mechanics of continuous media with free boundaries: problems in subsonic gas dynamics, filtration theory, and problems in elastico-plasticity.
Einstein boundary conditions for the 3+1 Einstein equations
International Nuclear Information System (INIS)
Frittelli, Simonetta; Gomez, Roberto
2003-01-01
In the 3+1 framework of the Einstein equations for the case of a vanishing shift vector and arbitrary lapse, we calculate explicitly the four boundary equations arising from the vanishing of the projection of the Einstein tensor along the normal to the boundary surface of the initial-boundary value problem. Such conditions take the form of evolution equations along (as opposed to across) the boundary for certain components of the extrinsic curvature and for certain space derivatives of the three-metric. We argue that, in general, such boundary conditions do not follow necessarily from the evolution equations and the initial data, but need to be imposed on the boundary values of the fundamental variables. Using the Einstein-Christoffel formulation, which is strongly hyperbolic, we show how three of the boundary equations up to linear combinations should be used to prescribe the values of some incoming characteristic fields. Additionally, we show that the fourth one imposes conditions on some outgoing fields
Vragov’s boundary value problem for an implicit equation of mixed type
Egorov, I. E.
2017-10-01
We study a Vragov boundary value problem for a third-order implicit equation of mixed type with an arbitrary manifold of type switch. These Sobolev-type equations arise in many important applied problems. Given certain constraints on the coefficients and the right-hand side of the equation, we demonstrate, using nonstationary Galerkin method and regularization method, the unique regular solvability of the boundary value problem. We also obtain an error estimate for approximate solutions of the boundary value problem in terms of the regularization parameter and the eigenvalues of the Dirichlet spectral problem for the Laplace operator.
Sinc-collocation method for solving the Blasius equation
International Nuclear Information System (INIS)
Parand, K.; Dehghan, Mehdi; Pirkhedri, A.
2009-01-01
Sinc-collocation method is applied for solving Blasius equation which comes from boundary layer equations. It is well known that sinc procedure converges to the solution at an exponential rate. Comparison with Howarth and Asaithambi's numerical solutions reveals that the proposed method is of high accuracy and reduces the solution of Blasius' equation to the solution of a system of algebraic equations.
Boundary value problem for Caputo-Hadamard fractional differential equations
Directory of Open Access Journals (Sweden)
Yacine Arioua
2017-09-01
Full Text Available The aim of this work is to study the existence and uniqueness solutions for boundary value problem of nonlinear fractional differential equations with Caputo-Hadamard derivative in bounded domain. We used the standard and Krasnoselskii's fixed point theorems. Some new results of existence and uniqueness solutions for Caputo-Hadamard fractional equations are obtained.
On a stochastic Burgers equation with Dirichlet boundary conditions
Directory of Open Access Journals (Sweden)
Ekaterina T. Kolkovska
2003-01-01
Full Text Available We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.
RECTC/RECTCF, 2. Order Elliptical Partial Differential Equation, Arbitrary Boundary Conditions
International Nuclear Information System (INIS)
Hackbusch, W.
1983-01-01
1 - Description of problem or function: A general linear elliptical second order partial differential equation on a rectangle with arbitrary boundary conditions is solved. 2 - Method of solution: Multi-grid iteration
Partial differential equations and boundary-value problems with applications
Pinsky, Mark A
2011-01-01
Building on the basic techniques of separation of variables and Fourier series, the book presents the solution of boundary-value problems for basic partial differential equations: the heat equation, wave equation, and Laplace equation, considered in various standard coordinate systems-rectangular, cylindrical, and spherical. Each of the equations is derived in the three-dimensional context; the solutions are organized according to the geometry of the coordinate system, which makes the mathematics especially transparent. Bessel and Legendre functions are studied and used whenever appropriate th
Test equating methods and practices
Kolen, Michael J
1995-01-01
In recent years, many researchers in the psychology and statistical communities have paid increasing attention to test equating as issues of using multiple test forms have arisen and in response to criticisms of traditional testing techniques This book provides a practically oriented introduction to test equating which both discusses the most frequently used equating methodologies and covers many of the practical issues involved The main themes are - the purpose of equating - distinguishing between equating and related methodologies - the importance of test equating to test development and quality control - the differences between equating properties, equating designs, and equating methods - equating error, and the underlying statistical assumptions for equating The authors are acknowledged experts in the field, and the book is based on numerous courses and seminars they have presented As a result, educators, psychometricians, professionals in measurement, statisticians, and students coming to the subject for...
Boundary conditions for the numerical solution of elliptic equations in exterior regions
International Nuclear Information System (INIS)
Bayliss, A.; Gunzburger, M.; Turkel, E.
1982-01-01
Elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the well-posedness of the problem. Examples of practical applications include the Helmholtz equation and Laplace's equation. Computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface. Direct imposition of the condition at infinity along the finite boundary results in large errors. A sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain. Estimates of the error due to the finite boundary are obtained for several cases. Computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions. The examples are based on a finite element formulation but finite difference methods can also be used
Positive Solutions of Two-Point Boundary Value Problems for Monge-Ampère Equations
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Baoqiang Yan
2015-01-01
Full Text Available This paper considers the following boundary value problem: ((-u'(tn'=ntn-1f(u(t, 01 is odd. We establish the method of lower and upper solutions for some boundary value problems which generalizes the above equations and using this method we present a necessary and sufficient condition for the existence of positive solutions to the above boundary value problem and some sufficient conditions for the existence of positive solutions.
Directory of Open Access Journals (Sweden)
Guotao Wang
2012-01-01
Full Text Available We study nonlinear impulsive differential equations of fractional order with irregular boundary conditions. Some existence and uniqueness results are obtained by applying standard fixed-point theorems. For illustration of the results, some examples are discussed.
The complex variable boundary element method: Applications in determining approximative boundaries
Hromadka, T.V.
1984-01-01
The complex variable boundary element method (CVBEM) is used to determine approximation functions for boundary value problems of the Laplace equation such as occurs in potential theory. By determining an approximative boundary upon which the CVBEM approximator matches the desired constant (level curves) boundary conditions, the CVBEM is found to provide the exact solution throughout the interior of the transformed problem domain. Thus, the acceptability of the CVBEM approximation is determined by the closeness-of-fit of the approximative boundary to the study problem boundary. ?? 1984.
Boundary value problemfor multidimensional fractional advection-dispersion equation
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Khasambiev Mokhammad Vakhaevich
2015-05-01
authors first considered the boundary value problem for stationary equation for mass transfer in super-diffusion conditions and abnormal advection. Then the solution of the problem is explicitly given. The solution is obtained by the Fourier’s method.The obtained results will be useful in liquid filtration theory in fractal medium and for modeling the temperature variations in the heated bar.
A simple and efficient outflow boundary condition for the incompressible Navier–Stokes equations
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Yibao Li
2017-01-01
Full Text Available Many researchers have proposed special treatments for outlet boundary conditions owing to lack of information at the outlet. Among them, the simplest method requires a large enough computational domain to prevent or reduce numerical errors at the boundaries. However, an efficient method generally requires special treatment to overcome the problems raised by the outlet boundary condition used. For example, mass flux is not conserved and the fluid field is not divergence-free at the outlet boundary. Overcoming these problems requires additional computational cost. In this paper, we present a simple and efficient outflow boundary condition for the incompressible Navier–Stokes equations, aiming to reduce the computational domain for simulating flow inside a long channel in the streamwise direction. The proposed outflow boundary condition is based on the transparent equation, where a weak formulation is used. The pressure boundary condition is derived by using the Navier–Stokes equations and the outlet flow boundary condition. In the numerical algorithm, a staggered marker-and-cell grid is used and temporal discretization is based on a projection method. The intermediate velocity boundary condition is consistently adopted to handle the velocity–pressure coupling. Characteristic numerical experiments are presented to demonstrate the robustness and accuracy of the proposed numerical scheme. Furthermore, the agreement of computational results from small and large domains suggests that our proposed outflow boundary condition can significantly reduce computational domain sizes.
On the Boussinesq-Burgers equations driven by dynamic boundary conditions
Zhu, Neng; Liu, Zhengrong; Zhao, Kun
2018-02-01
We study the qualitative behavior of the Boussinesq-Burgers equations on a finite interval subject to the Dirichlet type dynamic boundary conditions. Assuming H1 ×H2 initial data which are compatible with boundary conditions and utilizing energy methods, we show that under appropriate conditions on the dynamic boundary data, there exist unique global-in-time solutions to the initial-boundary value problem, and the solutions converge to the boundary data as time goes to infinity, regardless of the magnitude of the initial data.
Entropy viscosity method applied to Euler equations
International Nuclear Information System (INIS)
Delchini, M. O.; Ragusa, J. C.; Berry, R. A.
2013-01-01
The entropy viscosity method [4] has been successfully applied to hyperbolic systems of equations such as Burgers equation and Euler equations. The method consists in adding dissipative terms to the governing equations, where a viscosity coefficient modulates the amount of dissipation. The entropy viscosity method has been applied to the 1-D Euler equations with variable area using a continuous finite element discretization in the MOOSE framework and our results show that it has the ability to efficiently smooth out oscillations and accurately resolve shocks. Two equations of state are considered: Ideal Gas and Stiffened Gas Equations Of State. Results are provided for a second-order time implicit schemes (BDF2). Some typical Riemann problems are run with the entropy viscosity method to demonstrate some of its features. Then, a 1-D convergent-divergent nozzle is considered with open boundary conditions. The correct steady-state is reached for the liquid and gas phases with a time implicit scheme. The entropy viscosity method correctly behaves in every problem run. For each test problem, results are shown for both equations of state considered here. (authors)
Modified Differential Transform Method for Two Singular Boundary Values Problems
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Yinwei Lin
2014-01-01
Full Text Available This paper deals with the two singular boundary values problems of second order. Two singular points are both boundary values points of the differential equation. The numerical solutions are developed by modified differential transform method (DTM for expanded point. Linear and nonlinear models are solved by this method to get more reliable and efficient numerical results. It can also solve ordinary differential equations where the traditional one fails. Besides, we give the convergence of this new method.
Bessaih, Hakima
2015-04-01
The evolution Stokes equation in a domain containing periodically distributed obstacles subject to Fourier boundary condition on the boundaries is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the obstacles. We represent the solid obstacles by holes in the fluid domain. The macroscopic (homogenized) equation is derived as another stochastic partial differential equation, defined in the whole non perforated domain. Here, the initial stochastic perturbation on the boundary becomes part of the homogenized equation as another stochastic force. We use the twoscale convergence method after extending the solution with 0 in the holes to pass to the limit. By Itô stochastic calculus, we get uniform estimates on the solution in appropriate spaces. In order to pass to the limit on the boundary integrals, we rewrite them in terms of integrals in the whole domain. In particular, for the stochastic integral on the boundary, we combine the previous idea of rewriting it on the whole domain with the assumption that the Brownian motion is of trace class. Due to the particular boundary condition dealt with, we get that the solution of the stochastic homogenized equation is not divergence free. However, it is coupled with the cell problem that has a divergence free solution. This paper represents an extension of the results of Duan and Wang (Comm. Math. Phys. 275:1508-1527, 2007), where a reaction diffusion equation with a dynamical boundary condition with a noise source term on both the interior of the domain and on the boundary was studied, and through a tightness argument and a pointwise two scale convergence method the homogenized equation was derived. © American Institute of Mathematical Sciences.
Energy Technology Data Exchange (ETDEWEB)
Bal, G. [Departement MMN, Service IMA, Direction des Etudes et Recherches, Electricite de France (EDF), 92 - Clamart (France)
1995-10-01
Neutron transport in nuclear reactors is quite well modelled by the linear Boltzmann transport equation. Its solution is relatively easy, but unfortunately too expensive to achieve whole core computations. Thus, we have to simplify it, for example by homogenizing some physical characteristics. However, the solution may then be inaccurate. Moreover, in strongly homogeneous areas, the error may be too big. Then we would like to deal with such an inconvenient by solving the equation accurately on this area, but more coarsely away from it, so that the computation is not too expensive. This problem is the subject of a thesis. We present here some results obtained for slab geometry. The couplings between the fine and coarse discretization regions could be conceived in a number of approaches. Here, we only deal with the coupling at crossing the interface between two sub-domains. In the first section, we present the coupling of discrete ordinate methods for solving the homogeneous, isotropic and mono-kinetic equation. Coupling operators are defined and shown to be optimal. The second and the third sections are devoted to an extension of the previous results when the equation is non-homogeneous, anisotropic and multigroup (under some restrictive assumptions). Some numerical results are given in the case of isotropic and mono-kinetic equations. (author) 15 refs.
Exact solution of nonsteady thermal boundary layer equation
International Nuclear Information System (INIS)
Dorfman, A.S.
1995-01-01
There are only a few exact solutions of the thermal boundary layer equation. Most of them are derived for a specific surface temperature distribution. The first exact solution of the steady-state boundary layer equation was given for a plate with constant surface temperature and free-stream velocity. The same problem for a plate with polynomial surface temperature distribution was solved by Chapmen and Rubesin. Levy gave the exact solution for the case of a power law distribution of both surface temperature and free-stream velocity. The exact solution of the steady-state boundary layer equation for an arbitrary surface temperature and a power law free-stream velocity distribution was given by the author in two forms: of series and of the integral with an influence function of unheated zone. A similar solution of the nonsteady thermal boundary layer equation for an arbitrary surface temperature and a power law free-stream velocity distribution is presented here. In this case, the coefficients of series depend on time, and in the limit t → ∞ they become the constant coefficients of a similar solution published before. This solution, unlike the one presented here, does not satisfy the initial conditions at t = 0, and, hence, can be used only in time after the beginning of the process. The solution in the form of a series becomes a closed-form exact solution for polynomial surface temperature and a power law free-stream velocity distribution. 7 refs., 2 figs
Liouville equation with boundary conditions derived from classical strings
International Nuclear Information System (INIS)
Marnelius, R.
1983-01-01
It is shown in terms of the classical string theory that a breaking of the Weyl invariance necessarily requires the Liouville equation for the variable phi=1n rho, where rho is the variable that appears in the conformal gauge gsub(α#betta#)=rhoetasub(α#betta#). Appropriate boundary conditions on phi for open and closed strings are then derived. (orig.)
Partial differential equations & boundary value problems with Maple
Articolo, George A
2009-01-01
Partial Differential Equations and Boundary Value Problems with Maple presents all of the material normally covered in a standard course on partial differential equations, while focusing on the natural union between this material and the powerful computational software, Maple. The Maple commands are so intuitive and easy to learn, students can learn what they need to know about the software in a matter of hours- an investment that provides substantial returns. Maple''s animation capabilities allow students and practitioners to see real-time displays of the solutions of partial differential equations. Maple files can be found on the books website. Ancillary list: Maple files- http://www.elsevierdirect.com/companion.jsp?ISBN=9780123747327 Provides a quick overview of the software w/simple commands needed to get startedIncludes review material on linear algebra and Ordinary Differential equations, and their contribution in solving partial differential equationsIncorporates an early introduction to Sturm-L...
International Nuclear Information System (INIS)
Kotikov, A.V.
1993-01-01
A new method of massive Feynman diagrams calculation is presented. It provides a fairly simple procedure to obtain the result without the D-space integral calculation (for the dimensional regularization). Some diagrams are calculated as an illustration of this method capacities. (author). 7 refs
Differential equations methods and applications
Said-Houari, Belkacem
2015-01-01
This book presents a variety of techniques for solving ordinary differential equations analytically and features a wealth of examples. Focusing on the modeling of real-world phenomena, it begins with a basic introduction to differential equations, followed by linear and nonlinear first order equations and a detailed treatment of the second order linear equations. After presenting solution methods for the Laplace transform and power series, it lastly presents systems of equations and offers an introduction to the stability theory. To help readers practice the theory covered, two types of exercises are provided: those that illustrate the general theory, and others designed to expand on the text material. Detailed solutions to all the exercises are included. The book is excellently suited for use as a textbook for an undergraduate class (of all disciplines) in ordinary differential equations. .
Integral equation methods for electromagnetics
Volakis, John
2012-01-01
This text/reference is a detailed look at the development and use of integral equation methods for electromagnetic analysis, specifically for antennas and radar scattering. Developers and practitioners will appreciate the broad-based approach to understanding and utilizing integral equation methods and the unique coverage of historical developments that led to the current state-of-the-art. In contrast to existing books, Integral Equation Methods for Electromagnetics lays the groundwork in the initial chapters so students and basic users can solve simple problems and work their way up to the mo
ON A PARABOLIC FREE BOUNDARY EQUATION MODELING PRICE FORMATION
MARKOWICH, P. A.
2009-10-01
We discuss existence and uniqueness of solutions for a one-dimensional parabolic evolution equation with a free boundary. This problem was introduced by Lasry and Lions as description of the dynamical formation of the price of a trading good. Short time existence and uniqueness is established by a contraction argument. Then we discuss the issue of global-in-time-extension of the local solution which is closely related to the regularity of the free boundary. We also present numerical results. © 2009 World Scientific Publishing Company.
ON A PARABOLIC FREE BOUNDARY EQUATION MODELING PRICE FORMATION
MARKOWICH, P. A.; MATEVOSYAN, N.; PIETSCHMANN, J.-F.; WOLFRAM, M.-T.
2009-01-01
We discuss existence and uniqueness of solutions for a one-dimensional parabolic evolution equation with a free boundary. This problem was introduced by Lasry and Lions as description of the dynamical formation of the price of a trading good. Short time existence and uniqueness is established by a contraction argument. Then we discuss the issue of global-in-time-extension of the local solution which is closely related to the regularity of the free boundary. We also present numerical results. © 2009 World Scientific Publishing Company.
Zhu, C
2003-01-01
This paper is concerned with the existence and uniqueness of the entropy solution to the initial boundary value problem for the inviscid Burgers equation. To apply the method of vanishing viscosity to study the existence of the entropy solution, we first introduce the initial boundary value problem for the viscous Burgers equation, and as in Evans (1998 Partial Differential Equations (Providence, RI: American Mathematical Society) and Hopf (1950 Commun. Pure Appl. Math. 3 201-30), give the formula of the corresponding viscosity solutions by Hopf-Cole transformation. Secondly, we prove the convergence of the viscosity solution sequences and verify that the limiting function is an entropy solution. Finally, we give an example to show how our main result can be applied to solve the initial boundary value problem for the Burgers equation.
International Nuclear Information System (INIS)
Zhu, Changjiang; Duan, Renjun
2003-01-01
This paper is concerned with the existence and uniqueness of the entropy solution to the initial boundary value problem for the inviscid Burgers equation. To apply the method of vanishing viscosity to study the existence of the entropy solution, we first introduce the initial boundary value problem for the viscous Burgers equation, and as in Evans (1998 Partial Differential Equations (Providence, RI: American Mathematical Society) and Hopf (1950 Commun. Pure Appl. Math. 3 201-30), give the formula of the corresponding viscosity solutions by Hopf-Cole transformation. Secondly, we prove the convergence of the viscosity solution sequences and verify that the limiting function is an entropy solution. Finally, we give an example to show how our main result can be applied to solve the initial boundary value problem for the Burgers equation
Compressible stability of growing boundary layers using parabolized stability equations
Chang, Chau-Lyan; Malik, Mujeeb R.; Erlebacher, Gordon; Hussaini, M. Y.
1991-01-01
The parabolized stability equation (PSE) approach is employed to study linear and nonlinear compressible stability with an eye to providing a capability for boundary-layer transition prediction in both 'quiet' and 'disturbed' environments. The governing compressible stability equations are solved by a rational parabolizing approximation in the streamwise direction. Nonparallel flow effects are studied for both the first- and second-mode disturbances. For oblique waves of the first-mode type, the departure from the parallel results is more pronounced as compared to that for the two-dimensional waves. Results for the Mach 4.5 case show that flow nonparallelism has more influence on the first mode than on the second. The disturbance growth rate is shown to be a strong function of the wall-normal distance due to either flow nonparallelism or nonlinear interactions. The subharmonic and fundamental types of breakdown are found to be similar to the ones in incompressible boundary layers.
Comparison of Kernel Equating and Item Response Theory Equating Methods
Meng, Yu
2012-01-01
The kernel method of test equating is a unified approach to test equating with some advantages over traditional equating methods. Therefore, it is important to evaluate in a comprehensive way the usefulness and appropriateness of the Kernel equating (KE) method, as well as its advantages and disadvantages compared with several popular item…
A coupled boundary element-finite difference solution of the elliptic modified mild slope equation
DEFF Research Database (Denmark)
Naserizadeh, R.; Bingham, Harry B.; Noorzad, A.
2011-01-01
The modified mild slope equation of [5] is solved using a combination of the boundary element method (BEM) and the finite difference method (FDM). The exterior domain of constant depth and infinite horizontal extent is solved by a BEM using linear or quadratic elements. The interior domain...
Modeling mode interactions in boundary layer flows via the Parabolized Floquet Equations
Ran, Wei; Zare, Armin; Hack, M. J. Philipp; Jovanović, Mihailo R.
2017-01-01
In this paper, we develop a linear model to study interactions between different modes in slowly-growing boundary layer flows. Our method consists of two steps. First, we augment the Blasius boundary layer profile with a disturbance field resulting from the linear Parabolized Stability Equations (PSE) to obtain the modified base flow; and, second, we combine Floquet analysis with the linear PSE to capture the spatial evolution of flow fluctuations. This procedure yields the Parabolized Floque...
Variational linear algebraic equations method
International Nuclear Information System (INIS)
Moiseiwitsch, B.L.
1982-01-01
A modification of the linear algebraic equations method is described which ensures a variational bound on the phaseshifts for potentials having a definite sign at all points. The method is illustrated by the elastic scattering of s-wave electrons by the static field of atomic hydrogen. (author)
Cacio, Emanuela; Cohn, Stephen E.; Spigler, Renato
2011-01-01
A numerical method is devised to solve a class of linear boundary-value problems for one-dimensional parabolic equations degenerate at the boundaries. Feller theory, which classifies the nature of the boundary points, is used to decide whether boundary conditions are needed to ensure uniqueness, and, if so, which ones they are. The algorithm is based on a suitable preconditioned implicit finite-difference scheme, grid, and treatment of the boundary data. Second-order accuracy, unconditional stability, and unconditional convergence of solutions of the finite-difference scheme to a constant as the time-step index tends to infinity are further properties of the method. Several examples, pertaining to financial mathematics, physics, and genetics, are presented for the purpose of illustration.
The determination of an unknown boundary condition in a fractional diffusion equation
Rundell, William
2013-07-01
In this article we consider an inverse boundary problem, in which the unknown boundary function ∂u/∂v = f(u) is to be determined from overposed data in a time-fractional diffusion equation. Based upon the free space fundamental solution, we derive a representation for the solution f as a nonlinear Volterra integral equation of second kind with a weakly singular kernel. Uniqueness and reconstructibility by iteration is an immediate result of a priori assumption on f and applying the fixed point theorem. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method. © 2013 Copyright Taylor and Francis Group, LLC.
Directory of Open Access Journals (Sweden)
Nadjafikhah M.
2017-07-01
Full Text Available Lie group method is applicable to both linear and non-linear partial differential equations, which leads to find new solutions for partial differential equations. Lie symmetry group method is applied to study Newtonian incompressible fluid’s equations flow in turbulent boundary layers. The symmetry group and its optimal system are given, and group invariant solutions associated to the symmetries are obtained. Finally the structure of the Lie algebra such as Levi decomposition, radical subalgebra, solvability and simplicity of symmetries is given.
Ghil, M.; Balgovind, R.
1979-01-01
The inhomogeneous Cauchy-Riemann equations in a rectangle are discretized by a finite difference approximation. Several different boundary conditions are treated explicitly, leading to algorithms which have overall second-order accuracy. All boundary conditions with either u or v prescribed along a side of the rectangle can be treated by similar methods. The algorithms presented here have nearly minimal time and storage requirements and seem suitable for development into a general-purpose direct Cauchy-Riemann solver for arbitrary boundary conditions.
Stabilizing local boundary conditions for two-dimensional shallow water equations
Dia, Ben Mansour
2018-03-27
In this article, we present a sub-critical two-dimensional shallow water flow regulation. From the energy estimate of a set of one-dimensional boundary stabilization problems, we obtain a set of polynomial equations with respect to the boundary values as a requirement for the energy decrease. Using the Riemann invariant analysis, we build stabilizing local boundary conditions that guarantee the stability of the hydrodynamical state around a given steady state. Numerical results for the controller applied to the nonlinear problem demonstrate the performance of the method.
Stabilization of the Wave Equation with Boundary Time-Varying Delay
Directory of Open Access Journals (Sweden)
Hao Li
2014-01-01
Full Text Available We study the stabilization of the wave equation with variable coefficients in a bounded domain and a time-varying delay term in the time-varying, weakly nonlinear boundary feedbacks. By the Riemannian geometry methods and a suitable assumption of nonlinearity, we obtain the uniform decay of the energy of the closed loop system.
Boundary Observability and Stabilization for Westervelt Type Wave Equations without Interior Damping
International Nuclear Information System (INIS)
Kaltenbacher, Barbara
2010-01-01
In this paper we show boundary observability and boundary stabilizability by linear feedbacks for a class of nonlinear wave equations including the undamped Westervelt model used in nonlinear acoustics. We prove local existence for undamped generalized Westervelt equations with homogeneous Dirichlet boundary conditions as well as global existence and exponential decay with absorbing type boundary conditions.
Retarded potentials and time domain boundary integral equations a road map
Sayas, Francisco-Javier
2016-01-01
This book offers a thorough and self-contained exposition of the mathematics of time-domain boundary integral equations associated to the wave equation, including applications to scattering of acoustic and elastic waves. The book offers two different approaches for the analysis of these integral equations, including a systematic treatment of their numerical discretization using Galerkin (Boundary Element) methods in the space variables and Convolution Quadrature in the time variable. The first approach follows classical work started in the late eighties, based on Laplace transforms estimates. This approach has been refined and made more accessible by tailoring the necessary mathematical tools, avoiding an excess of generality. A second approach contains a novel point of view that the author and some of his collaborators have been developing in recent years, using the semigroup theory of evolution equations to obtain improved results. The extension to electromagnetic waves is explained in one of the appendices...
Partial differential equations with numerical methods
Larsson, Stig
2003-01-01
The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering. The main theme is the integration of the theory of linear PDEs and the numerical solution of such equations. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. As preparation, the two-point boundary value problem and the initial-value problem for ODEs are discussed in separate chapters. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. Some background on linear functional analysis and Sobolev spaces, and also on numerical linear algebra, is reviewed in two appendices.
Iterative Splitting Methods for Differential Equations
Geiser, Juergen
2011-01-01
Iterative Splitting Methods for Differential Equations explains how to solve evolution equations via novel iterative-based splitting methods that efficiently use computational and memory resources. It focuses on systems of parabolic and hyperbolic equations, including convection-diffusion-reaction equations, heat equations, and wave equations. In the theoretical part of the book, the author discusses the main theorems and results of the stability and consistency analysis for ordinary differential equations. He then presents extensions of the iterative splitting methods to partial differential
Boundary methods for mode estimation
Pierson, William E., Jr.; Ulug, Batuhan; Ahalt, Stanley C.
1999-08-01
This paper investigates the use of Boundary Methods (BMs), a collection of tools used for distribution analysis, as a method for estimating the number of modes associated with a given data set. Model order information of this type is required by several pattern recognition applications. The BM technique provides a novel approach to this parameter estimation problem and is comparable in terms of both accuracy and computations to other popular mode estimation techniques currently found in the literature and automatic target recognition applications. This paper explains the methodology used in the BM approach to mode estimation. Also, this paper quickly reviews other common mode estimation techniques and describes the empirical investigation used to explore the relationship of the BM technique to other mode estimation techniques. Specifically, the accuracy and computational efficiency of the BM technique are compared quantitatively to the a mixture of Gaussian (MOG) approach and a k-means approach to model order estimation. The stopping criteria of the MOG and k-means techniques is the Akaike Information Criteria (AIC).
International Nuclear Information System (INIS)
Semenova, V.N.
2016-01-01
A boundary value problem for a nonlinear second order differential equation has been considered. A numerical method has been proposed to solve this problem using power series. Results of numerical experiments have been presented in the paper [ru
Auxiliary equation method for solving nonlinear partial differential equations
International Nuclear Information System (INIS)
Sirendaoreji,; Jiong, Sun
2003-01-01
By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation
Straight velocity boundaries in the lattice Boltzmann method
Latt, Jonas; Chopard, Bastien; Malaspinas, Orestis; Deville, Michel; Michler, Andreas
2008-05-01
Various ways of implementing boundary conditions for the numerical solution of the Navier-Stokes equations by a lattice Boltzmann method are discussed. Five commonly adopted approaches are reviewed, analyzed, and compared, including local and nonlocal methods. The discussion is restricted to velocity Dirichlet boundary conditions, and to straight on-lattice boundaries which are aligned with the horizontal and vertical lattice directions. The boundary conditions are first inspected analytically by applying systematically the results of a multiscale analysis to boundary nodes. This procedure makes it possible to compare boundary conditions on an equal footing, although they were originally derived from very different principles. It is concluded that all five boundary conditions exhibit second-order accuracy, consistent with the accuracy of the lattice Boltzmann method. The five methods are then compared numerically for accuracy and stability through benchmarks of two-dimensional and three-dimensional flows. None of the methods is found to be throughout superior to the others. Instead, the choice of a best boundary condition depends on the flow geometry, and on the desired trade-off between accuracy and stability. From the findings of the benchmarks, the boundary conditions can be classified into two major groups. The first group comprehends boundary conditions that preserve the information streaming from the bulk into boundary nodes and complete the missing information through closure relations. Boundary conditions in this group are found to be exceptionally accurate at low Reynolds number. Boundary conditions of the second group replace all variables on boundary nodes by new values. They exhibit generally much better numerical stability and are therefore dedicated for use in high Reynolds number flows.
Directory of Open Access Journals (Sweden)
Boričić Zoran
2009-01-01
Full Text Available This paper concerns with unsteady two-dimensional temperature laminar magnetohydrodynamic (MHD boundary layer of incompressible fluid. It is assumed that induction of outer magnetic field is function of longitudinal coordinate with force lines perpendicular to the body surface on which boundary layer forms. Outer electric filed is neglected and magnetic Reynolds number is significantly lower then one i.e. considered problem is in inductionless approximation. Characteristic properties of fluid are constant because velocity of flow is much lower than speed of light and temperature difference is small enough (under 50ºC . Introduced assumptions simplify considered problem in sake of mathematical solving, but adopted physical model is interesting from practical point of view, because its relation with large number of technically significant MHD flows. Obtained partial differential equations can be solved with modern numerical methods for every particular problem. Conclusions based on these solutions are related only with specific temperature MHD boundary layer problem. In this paper, quite different approach is used. First new variables are introduced and then sets of similarity parameters which transform equations on the form which don't contain inside and in corresponding boundary conditions characteristics of particular problems and in that sense equations are considered as universal. Obtained universal equations in appropriate approximation can be solved numerically once for all. So-called universal solutions of equations can be used to carry out general conclusions about temperature MHD boundary layer and for calculation of arbitrary particular problems. To calculate any particular problem it is necessary also to solve corresponding momentum integral equation.
Hilbert space methods in partial differential equations
Showalter, Ralph E
1994-01-01
This graduate-level text opens with an elementary presentation of Hilbert space theory sufficient for understanding the rest of the book. Additional topics include boundary value problems, evolution equations, optimization, and approximation.1979 edition.
(Environmental and geophysical modeling, fracture mechanics, and boundary element methods)
Energy Technology Data Exchange (ETDEWEB)
Gray, L.J.
1990-11-09
Technical discussions at the various sites visited centered on application of boundary integral methods for environmental modeling, seismic analysis, and computational fracture mechanics in composite and smart'' materials. The traveler also attended the International Association for Boundary Element Methods Conference at Rome, Italy. While many aspects of boundary element theory and applications were discussed in the papers, the dominant topic was the analysis and application of hypersingular equations. This has been the focus of recent work by the author, and thus the conference was highly relevant to research at ORNL.
On the wall-normal velocity of the compressible boundary-layer equations
Pruett, C. David
1991-01-01
Numerical methods for the compressible boundary-layer equations are facilitated by transformation from the physical (x,y) plane to a computational (xi,eta) plane in which the evolution of the flow is 'slow' in the time-like xi direction. The commonly used Levy-Lees transformation results in a computationally well-behaved problem for a wide class of non-similar boundary-layer flows, but it complicates interpretation of the solution in physical space. Specifically, the transformation is inherently nonlinear, and the physical wall-normal velocity is transformed out of the problem and is not readily recovered. In light of recent research which shows mean-flow non-parallelism to significantly influence the stability of high-speed compressible flows, the contribution of the wall-normal velocity in the analysis of stability should not be routinely neglected. Conventional methods extract the wall-normal velocity in physical space from the continuity equation, using finite-difference techniques and interpolation procedures. The present spectrally-accurate method extracts the wall-normal velocity directly from the transformation itself, without interpolation, leaving the continuity equation free as a check on the quality of the solution. The present method for recovering wall-normal velocity, when used in conjunction with a highly-accurate spectral collocation method for solving the compressible boundary-layer equations, results in a discrete solution which is extraordinarily smooth and accurate, and which satisfies the continuity equation nearly to machine precision. These qualities make the method well suited to the computation of the non-parallel mean flows needed by spatial direct numerical simulations (DNS) and parabolized stability equation (PSE) approaches to the analysis of stability.
Biala, T A; Jator, S N
2015-01-01
In this article, the boundary value method is applied to solve three dimensional elliptic and hyperbolic partial differential equations. The partial derivatives with respect to two of the spatial variables (y, z) are discretized using finite difference approximations to obtain a large system of ordinary differential equations (ODEs) in the third spatial variable (x). Using interpolation and collocation techniques, a continuous scheme is developed and used to obtain discrete methods which are applied via the Block unification approach to obtain approximations to the resulting large system of ODEs. Several test problems are investigated to elucidate the solution process.
Existence of solutions to boundary value problem of fractional differential equations with impulsive
Directory of Open Access Journals (Sweden)
Weihua JIANG
2016-12-01
Full Text Available In order to solve the boundary value problem of fractional impulsive differential equations with countable impulses and integral boundary conditions on the half line, the existence of solutions to the boundary problem is specifically studied. By defining suitable Banach spaces, norms and operators, using the properties of fractional calculus and applying the contraction mapping principle and Krasnoselskii's fixed point theorem, the existence of solutions for the boundary value problem of fractional impulsive differential equations with countable impulses and integral boundary conditions on the half line is proved, and examples are given to illustrate the existence of solutions to this kind of equation boundary value problems.
Abstract methods in partial differential equations
Carroll, Robert W
2012-01-01
Detailed, self-contained treatment examines modern abstract methods in partial differential equations, especially abstract evolution equations. Suitable for graduate students with some previous exposure to classical partial differential equations. 1969 edition.
On Solutions of the Integrable Boundary Value Problem for KdV Equation on the Semi-Axis
International Nuclear Information System (INIS)
Ignatyev, M. Yu.
2013-01-01
This paper is concerned with the Korteweg–de Vries (KdV) equation on the semi-axis. The boundary value problem with inhomogeneous integrable boundary conditions is studied. We establish some characteristic properties of solutions of the problem. Also we construct a wide class of solutions of the problem using the inverse spectral method.
Active flow control insight gained from a modified integral boundary layer equation
Seifert, Avraham
2016-11-01
Active Flow Control (AFC) can alter the development of boundary layers with applications (e.g., reducing drag by separation delay or separating the boundary layers and enhancing vortex shedding to increase drag). Historically, significant effects of steady AFC methods were observed. Unsteady actuation is significantly more efficient than steady. Full-scale AFC tests were conducted with varying levels of success. While clearly relevant to industry, AFC implementation relies on expert knowledge with proven intuition and or costly and lengthy computational efforts. This situation hinders the use of AFC while simple, quick and reliable design method is absent. An updated form of the unsteady integral boundary layer (UIBL) equations, that include AFC terms (unsteady wall transpiration and body forces) can be used to assist in AFC analysis and design. With these equations and given a family of suitable velocity profiles, the momentum thickness can be calculated and matched with an outer, potential flow solution in 2D and 3D manner to create an AFC design tool, parallel to proven tools for airfoil design. Limiting cases of the UIBL equation can be used to analyze candidate AFC concepts in terms of their capability to modify the boundary layers development and system performance.
About potential of double layer and boundary value problems for Laplace equation
International Nuclear Information System (INIS)
Aleshin, M.V.
1991-01-01
An integral operator raisen by a kernel of the double layer's potential is investigated. The kernel is defined on S (S - two-digit variety of C 2 class presented by a boundary of the finite domain in R 3 ). The operator is considered on C(S). Following results are received: the operator's spectrum belongs to [-1,1]; it's eigenvalues and eigenfunctions may be found by Kellog's method; knowledge of the operator's spectrum is enough to construct it's resolvent. These properties permit to point out the determined interation processes, solving boundary value problems for Laplace equation. One of such processes - solving of Roben problem - is generalized on electrostatic problems. 6 refs
Gerbi, Sté phane; Said-Houari, Belkacem
2013-01-01
The goal of this work is to study a model of the wave equation with dynamic boundary conditions and a viscoelastic term. First, applying the Faedo-Galerkin method combined with the fixed point theorem, we show the existence and uniqueness of a local in time solution. Second, we show that under some restrictions on the initial data, the solution continues to exist globally in time. On the other hand, if the interior source dominates the boundary damping, then the solution is unbounded and grows as an exponential function. In addition, in the absence of the strong damping, then the solution ceases to exist and blows up in finite time.
Gerbi, Stéphane
2013-01-15
The goal of this work is to study a model of the wave equation with dynamic boundary conditions and a viscoelastic term. First, applying the Faedo-Galerkin method combined with the fixed point theorem, we show the existence and uniqueness of a local in time solution. Second, we show that under some restrictions on the initial data, the solution continues to exist globally in time. On the other hand, if the interior source dominates the boundary damping, then the solution is unbounded and grows as an exponential function. In addition, in the absence of the strong damping, then the solution ceases to exist and blows up in finite time.
Chebyshev Finite Difference Method for Fractional Boundary Value Problems
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Boundary
2015-09-01
Full Text Available This paper presents a numerical method for fractional differential equations using Chebyshev finite difference method. The fractional derivatives are described in the Caputo sense. Numerical results show that this method is of high accuracy and is more convenient and efficient for solving boundary value problems involving fractional ordinary differential equations. AMS Subject Classification: 34A08 Keywords and Phrases: Chebyshev polynomials, Gauss-Lobatto points, fractional differential equation, finite difference 1. Introduction The idea of a derivative which interpolates between the familiar integer order derivatives was introduced many years ago and has gained increasing importance only in recent years due to the development of mathematical models of a certain situations in engineering, materials science, control theory, polymer modelling etc. For example see [20, 22, 25, 26]. Most fractional order differential equations describing real life situations, in general do not have exact analytical solutions. Several numerical and approximate analytical methods for ordinary differential equation Received: December 2014; Accepted: March 2015 57 Journal of Mathematical Extension Vol. 9, No. 3, (2015, 57-71 ISSN: 1735-8299 URL: http://www.ijmex.com Chebyshev Finite Difference Method for Fractional Boundary Value Problems H. Azizi Taft Branch, Islamic Azad University Abstract. This paper presents a numerical method for fractional differential equations using Chebyshev finite difference method. The fractional derivative
Iterative observer based method for source localization problem for Poisson equation in 3D
Majeed, Muhammad Usman; Laleg-Kirati, Taous-Meriem
2017-01-01
A state-observer based method is developed to solve point source localization problem for Poisson equation in a 3D rectangular prism with available boundary data. The technique requires a weighted sum of solutions of multiple boundary data
Bessaih, Hakima; Efendiev, Yalchin; Maris, Florin
2015-01-01
The evolution Stokes equation in a domain containing periodically distributed obstacles subject to Fourier boundary condition on the boundaries is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior
Ding, Xiao-Li; Nieto, Juan J.
2017-11-01
In this paper, we consider the analytical solutions of coupling fractional partial differential equations (FPDEs) with Dirichlet boundary conditions on a finite domain. Firstly, the method of successive approximations is used to obtain the analytical solutions of coupling multi-term time fractional ordinary differential equations. Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the coupling FPDEs to the coupling multi-term time fractional ordinary differential equations. By applying the obtained analytical solutions to the resulting multi-term time fractional ordinary differential equations, the desired analytical solutions of the coupling FPDEs are given. Our results are applied to derive the analytical solutions of some special cases to demonstrate their applicability.
Liu, Ping; Shi, Junping
2018-01-01
The bifurcation of non-trivial steady state solutions of a scalar reaction-diffusion equation with nonlinear boundary conditions is considered using several new abstract bifurcation theorems. The existence and stability of positive steady state solutions are proved using a unified approach. The general results are applied to a Laplace equation with nonlinear boundary condition and bistable nonlinearity, and an elliptic equation with superlinear nonlinearity and sublinear boundary conditions.
On problems with displacement in boundary conditions for hyperbolic equation
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Elena A. Utkina
2016-03-01
Full Text Available We consider three problems for hyperbolic equation on a plane in the characteristic domain. In these problems at least one of the conditions of the Goursat problem is replaced by nonlocal condition on the relevant characteristic. Non-local conditions are the linear combinations of the normal derivatives at points on opposite characteristics. In case of replacement of one condition we solve the problem by reduction to the Goursat problem for which it exists and is unique. To find the unknown Goursat condition author receives the integral equation, rewrite it in operational form and finds its unique solvability cases. To prove the unique solvability of the equation, the author shows the continuous linear operator and the fact, that some degree of the resulting operator is a contraction mapping. It is known that in this case the required Goursat condition can be written as Neumann series. We considered in detail only one of the tasks, but for both the unique solvability theorems are formulated. If the two conditions are changed, the uniqueness of the solution on the assumption that it exists, is proved by the method of a priori estimates. For this purpose, the inner product and the norm in $L_2$ are used. As a result, the conditions were obtained for the coefficients of a hyperbolic equation that ensure the uniqueness of the solution. An example is given, confirming that these conditions are essential. Namely, constructed an equation whose coefficients do not satisfy the conditions of the last theorem, given the conditions on the characteristics and nontrivial solution is built.
International Nuclear Information System (INIS)
Boisseau, Bruno; Forgacs, Peter; Giacomini, Hector
2007-01-01
A new (algebraic) approximation scheme to find global solutions of two-point boundary value problems of ordinary differential equations (ODEs) is presented. The method is applicable for both linear and nonlinear (coupled) ODEs whose solutions are analytic near one of the boundary points. It is based on replacing the original ODEs by a sequence of auxiliary first-order polynomial ODEs with constant coefficients. The coefficients in the auxiliary ODEs are uniquely determined from the local behaviour of the solution in the neighbourhood of one of the boundary points. The problem of obtaining the parameters of the global (connecting) solutions, analytic at one of the boundary points, reduces to find the appropriate zeros of algebraic equations. The power of the method is illustrated by computing the approximate values of the 'connecting parameters' for a number of nonlinear ODEs arising in various problems in field theory. We treat in particular the static and rotationally symmetric global vortex, the skyrmion, the Abrikosov-Nielsen-Olesen vortex, as well as the 't Hooft-Polyakov magnetic monopole. The total energy of the skyrmion and of the monopole is also computed by the new method. We also consider some ODEs coming from the exact renormalization group. The ground-state energy level of the anharmonic oscillator is also computed for arbitrary coupling strengths with good precision. (fast track communication)
Convergence of a random walk method for the Burgers equation
International Nuclear Information System (INIS)
Roberts, S.
1985-10-01
In this paper we consider a random walk algorithm for the solution of Burgers' equation. The algorithm uses the method of fractional steps. The non-linear advection term of the equation is solved by advecting ''fluid'' particles in a velocity field induced by the particles. The diffusion term of the equation is approximated by adding an appropriate random perturbation to the positions of the particles. Though the algorithm is inefficient as a method for solving Burgers' equation, it does model a similar method, the random vortex method, which has been used extensively to solve the incompressible Navier-Stokes equations. The purpose of this paper is to demonstrate the strong convergence of our random walk method and so provide a model for the proof of convergence for more complex random walk algorithms; for instance, the random vortex method without boundaries
Absorbing boundary conditions for Einstein's field equations
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Sarbach, Olivier [Instituto de Fisica y Matematicas, Universidad Michoacana de San Nicolas de Hidalgo, Edificio C-3, Cd. Universitaria. C. P. 58040 Morelia, Michoacan (Mexico)
2007-11-15
A common approach for the numerical simulation of wave propagation on a spatially unbounded domain is to truncate the domain via an artificial boundary, thus forming a finite computational domain with an outer boundary. Absorbing boundary conditions must then be specified at the boundary such that the resulting initial-boundary value problem is well posed and such that the amount of spurious reflection is minimized. In this article, we review recent results on the construction of absorbing boundary conditions in General Relativity and their application to numerical relativity.
The SMM Model as a Boundary Value Problem Using the Discrete Diffusion Equation
Campbell, Joel
2007-01-01
A generalized single step stepwise mutation model (SMM) is developed that takes into account an arbitrary initial state to a certain partial difference equation. This is solved in both the approximate continuum limit and the more exact discrete form. A time evolution model is developed for Y DNA or mtDNA that takes into account the reflective boundary modeling minimum microsatellite length and the original difference equation. A comparison is made between the more widely known continuum Gaussian model and a discrete model, which is based on modified Bessel functions of the first kind. A correction is made to the SMM model for the probability that two individuals are related that takes into account a reflecting boundary modeling minimum microsatellite length. This method is generalized to take into account the general n-step model and exact solutions are found. A new model is proposed for the step distribution.
Directory of Open Access Journals (Sweden)
Guo Chun Wen
2009-05-01
Full Text Available This article concerns the oblique derivative problems for second-order quasilinear degenerate equations of mixed type with several characteristic boundaries, which include the Tricomi problem as a special case. First we formulate the problem and obtain estimates of its solutions, then we show the existence of solutions by the successive iterations and the Leray-Schauder theorem. We use a complex analytic method: elliptic complex functions are used in the elliptic domain, and hyperbolic complex functions in the hyperbolic domain, such that second-order equations of mixed type with degenerate curve are reduced to the first order mixed complex equations with singular coefficients. An application of the complex analytic method, solves (1.1 below with $m=n=1$, $a=b=0$, which was posed as an open problem by Rassias.
International Nuclear Information System (INIS)
Werby, M.F.; Broadhead, M.K.; Strayer, M.R.; Bottcher, C.
1992-01-01
The Helmholtz-Poincarf Wave Equation (H-PWE) arises in many areas of classical wave scattering theory. In particular it can be found for the cases of acoustical scattering from submerged bounded objects and electromagnetic scattering from objects. The extended boundary integral equations (EBIE) method is derived from considering both the exterior and interior solutions of the H-PWECs. This coupled set of expressions has the advantage of not only offering a prescription for obtaining a solution for the exterior scattering problem, but it also obviates the problem of irregular values corresponding to fictitious interior eigenvalues. Once the coupled equations are derived, they can be obtained in matrix form by expanding all relevant terms in partial wave expansions, including a bi-orthogonal expansion of the Green's function. However some freedom in the choice of the surface expansion is available since the unknown surface quantities may be expanded in a variety of ways so long as closure is obtained. Out of many possible choices, we develop an optimal method to obtain such expansions which is based on the optimum eigenfunctions related to the surface of the object. In effect, we convert part of the problem (that associated with the Fredholms integral equation of the first kind) an eigenvalue problem of a related Hermitian operator. The methodology will be explained in detail and examples will be presented
International Nuclear Information System (INIS)
Zeng, Huihui
2015-01-01
In this paper we establish the global existence of smooth solutions to vacuum free boundary problems of the one-dimensional compressible isentropic Navier–Stokes equations for which the smoothness extends all the way to the boundaries. The results obtained in this work include the physical vacuum for which the sound speed is C 1/2 -Hölder continuous near the vacuum boundaries when 1 < γ < 3. The novelty of this result is its global-in-time regularity which is in contrast to the previous main results of global weak solutions in the literature. Moreover, in previous studies of the one-dimensional free boundary problems of compressible Navier–Stokes equations, the Lagrangian mass coordinates method has often been used, but in the present work the particle path (flow trajectory) method is adopted, which has the advantage that the particle paths and, in particular, the free boundaries can be traced. (paper)
Lubricated immersed boundary method in two dimensions
Fai, Thomas G.; Rycroft, Chris H.
2018-03-01
Many biological examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen and the intracellular trafficking of vesicles into dendritic spines, involve the near-contact of elastic structures separated by thin layers of fluid. Motivated by such problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We demonstrate 2nd-order accurate convergence for simple two-dimensional flows with known exact solutions to showcase the increased accuracy of this method compared to the standard immersed boundary method. Motivated by the phenomenon of wall-induced migration, we apply the lubricated immersed boundary method to simulate an elastic vesicle near a wall in shear flow. We also simulate the dynamics of a vesicle traveling through a narrow channel and observe the ability of the lubricated method to capture the vesicle motion on relatively coarse fluid grids.
Boundary integral methods for unsaturated flow
International Nuclear Information System (INIS)
Martinez, M.J.; McTigue, D.F.
1990-01-01
Many large simulations may be required to assess the performance of Yucca Mountain as a possible site for the nations first high level nuclear waste repository. A boundary integral equation method (BIEM) is described for numerical analysis of quasilinear steady unsaturated flow in homogeneous material. The applicability of the exponential model for the dependence of hydraulic conductivity on pressure head is discussed briefly. This constitutive assumption is at the heart of the quasilinear transformation. Materials which display a wide distribution in pore-size are described reasonably well by the exponential. For materials with a narrow range in pore-size, the exponential is suitable over more limited ranges in pressure head. The numerical implementation of the BIEM is used to investigate the infiltration from a strip source to a water table. The net infiltration of moisture into a finite-depth layer is well-described by results for a semi-infinite layer if αD > 4, where α is the sorptive number and D is the depth to the water table. the distribution of moisture exhibits a similar dependence on αD. 11 refs., 4 figs.,
Solving equations by topological methods
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Lech Górniewicz
2005-01-01
Full Text Available In this paper we survey most important results from topological fixed point theory which can be directly applied to differential equations. Some new formulations are presented. We believe that our article will be useful for analysts applying topological fixed point theory in nonlinear analysis and in differential equations.
Lattice Boltzmann methods for moving boundary flows
International Nuclear Information System (INIS)
Inamuro, Takaji
2012-01-01
The lattice Boltzmann methods (LBMs) for moving boundary flows are presented. The LBM for two-phase fluid flows with the same density and the LBM combined with the immersed boundary method are described. In addition, the LBM on a moving multi-block grid is explained. Three numerical examples (a droplet moving in a constricted tube, the lift generation of a flapping wing and the sedimentation of an elliptical cylinder) are shown in order to demonstrate the applicability of the LBMs to moving boundary problems. (invited review)
Lattice Boltzmann methods for moving boundary flows
Energy Technology Data Exchange (ETDEWEB)
Inamuro, Takaji, E-mail: inamuro@kuaero.kyoto-u.ac.jp [Department of Aeronautics and Astronautics, and Advanced Research Institute of Fluid Science and Engineering, Graduate School of Engineering, Kyoto University, Kyoto 606-8501 (Japan)
2012-04-01
The lattice Boltzmann methods (LBMs) for moving boundary flows are presented. The LBM for two-phase fluid flows with the same density and the LBM combined with the immersed boundary method are described. In addition, the LBM on a moving multi-block grid is explained. Three numerical examples (a droplet moving in a constricted tube, the lift generation of a flapping wing and the sedimentation of an elliptical cylinder) are shown in order to demonstrate the applicability of the LBMs to moving boundary problems. (invited review)
Continuum and Discrete Initial-Boundary Value Problems and Einstein's Field Equations
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Olivier Sarbach
2012-08-01
Full Text Available Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black-hole problem within Einstein's theory of gravitation, in which one computes the gravitational radiation emitted from the inspiral of the two black holes, merger and ringdown. Powerful mathematical tools can be used to establish qualitative statements about the solutions, such as their existence, uniqueness, continuous dependence on the initial data, or their asymptotic behavior over large time scales. However, one is often interested in computing the solution itself, and unless the partial differential equation is very simple, or the initial data possesses a high degree of symmetry, this computation requires approximation by numerical discretization. When solving such discrete problems on a machine, one is faced with a finite limit to computational resources, which leads to the replacement of the infinite continuum domain with a finite computer grid. This, in turn, leads to a discrete initial-boundary value problem. The hope is to recover, with high accuracy, the exact solution in the limit where the grid spacing converges to zero with the boundary being pushed to infinity. The goal of this article is to review some of the theory necessary to understand the continuum and discrete initial boundary-value problems arising from hyperbolic partial differential equations and to discuss its applications to numerical relativity; in particular, we present well-posed initial and initial-boundary value formulations of Einstein's equations, and we discuss multi-domain high-order finite difference and spectral methods to solve them.
Continuum and Discrete Initial-Boundary Value Problems and Einstein's Field Equations.
Sarbach, Olivier; Tiglio, Manuel
2012-01-01
Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black-hole problem within Einstein's theory of gravitation, in which one computes the gravitational radiation emitted from the inspiral of the two black holes, merger and ringdown. Powerful mathematical tools can be used to establish qualitative statements about the solutions, such as their existence, uniqueness, continuous dependence on the initial data, or their asymptotic behavior over large time scales. However, one is often interested in computing the solution itself, and unless the partial differential equation is very simple, or the initial data possesses a high degree of symmetry, this computation requires approximation by numerical discretization. When solving such discrete problems on a machine, one is faced with a finite limit to computational resources, which leads to the replacement of the infinite continuum domain with a finite computer grid. This, in turn, leads to a discrete initial-boundary value problem. The hope is to recover, with high accuracy, the exact solution in the limit where the grid spacing converges to zero with the boundary being pushed to infinity. The goal of this article is to review some of the theory necessary to understand the continuum and discrete initial boundary-value problems arising from hyperbolic partial differential equations and to discuss its applications to numerical relativity; in particular, we present well-posed initial and initial-boundary value formulations of Einstein's equations, and we discuss multi-domain high-order finite difference and spectral methods to solve them.
DEFF Research Database (Denmark)
Johannessen, Kim
2014-01-01
The exact solution to the one-dimensional Poisson–Boltzmann equation with asymmetric boundary conditions can be expressed in terms of the Jacobi elliptic functions. The boundary conditions determine the modulus of the Jacobi elliptic functions. The boundary conditions can not be solved analytically...
Constraint-preserving boundary treatment for a harmonic formulation of the Einstein equations
Energy Technology Data Exchange (ETDEWEB)
Seiler, Jennifer; Szilagyi, Bela; Pollney, Denis; Rezzolla, Luciano [Max-Planck-Institut fuer Gravitationsphysik, Albert-Einstein-Institut, Golm (Germany)
2008-09-07
We present a set of well-posed constraint-preserving boundary conditions for a first-order in time, second-order in space, harmonic formulation of the Einstein equations. The boundary conditions are tested using robust stability, linear and nonlinear waves, and are found to be both less reflective and constraint preserving than standard Sommerfeld-type boundary conditions.
Constraint-preserving boundary treatment for a harmonic formulation of the Einstein equations
International Nuclear Information System (INIS)
Seiler, Jennifer; Szilagyi, Bela; Pollney, Denis; Rezzolla, Luciano
2008-01-01
We present a set of well-posed constraint-preserving boundary conditions for a first-order in time, second-order in space, harmonic formulation of the Einstein equations. The boundary conditions are tested using robust stability, linear and nonlinear waves, and are found to be both less reflective and constraint preserving than standard Sommerfeld-type boundary conditions
Miyake, Y.; Noda, H.
2017-12-01
Earthquake sequences involve many processes in a wide range of time scales, from quasistatic loading to dynamic rupture. At a depth of brittle-plastic transitional and deeper, rock behaves as a viscous fluid in a long timescale, but as an elastic material in a short timescale. Viscoelastic stress relaxation may be important in the interseismic periods at the depth, near the deeper limit of the seismogenic layer or the region of slow slip events (SSEs) [Namiki et al., 2014 and references therein]. In the present study, we implemented the viscoelastic effect (Maxwell material) in fully-dynamic earthquake sequence simulations using a spectral boundary integral equation method (SBIEM) [e.g., Lapusta et al., 2000]. SBIEM is efficient in calculation of convolutional terms for dynamic stress transfer, and the problem size is limited by the amount of memory available. Linear viscoelasticity could be implemented by convolution of slip rate history and Green's function, but this method requires additional memory and thus not suitable for the implementation to the present code. Instead, we integrated the evolution of "effective slip" distribution, which gives static stress distribution when convolved with static elastic Green's function. This method works only for simple viscoelastic property distributions, but such models are suitable for numerical experiments aiming basic understanding of the system behavior because of the virtue of SBIEM, the ability of fine on-fault spatial resolution and efficient computation utilizing the fast Fourier transformation. In the present study, we examined the effect of viscoelasticity on earthquake sequences of a fault with a rate-weakening patch. A series of simulations with various relaxation time tc revealed that as decreasing tc, recurrence intervals of earthquakes increases and seismicity ultimately disappears. As long as studied, this transition to aseismic behavior is NOT associated with SSEs. In a case where the rate-weakening patch
Directory of Open Access Journals (Sweden)
Nahed S. Hussein
2014-01-01
Full Text Available A numerical boundary integral scheme is proposed for the solution to the system of eld equations of plane. The stresses are prescribed on one-half of the circle, while the displacements are given. The considered problem with mixed boundary conditions in the circle is replaced by two problems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way, and the problem at this stage is reduced to the solution to a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution and the unknown boundary values of stresses or displacements on proper parts of the boundary. On the basis of the obtained results, it is inferred that a stress component has a singularity at each of the two separation points, thought to be of logarithmic type. The results are discussed and boundary plots are given. We have also calculated the unknown functions in the bulk directly from the given boundary conditions using the boundary collocation method. The obtained results in the bulk are discussed and three-dimensional plots are given. A tentative form for the singular solution is proposed and the corresponding singular stresses and displacements are plotted in the bulk. The form of the singular tangential stress is seen to be compatible with the boundary values obtained earlier. The efficiency of the used numerical schemes is discussed.
Introducing the Boundary Element Method with MATLAB
Ang, Keng-Cheng
2008-01-01
The boundary element method provides an excellent platform for learning and teaching a computational method for solving problems in physical and engineering science. However, it is often left out in many undergraduate courses as its implementation is deemed to be difficult. This is partly due to the perception that coding the method requires…
Sarna, Neeraj; Torrilhon, Manuel
2018-01-01
We define certain criteria, using the characteristic decomposition of the boundary conditions and energy estimates, which a set of stable boundary conditions for a linear initial boundary value problem, involving a symmetric hyperbolic system, must satisfy. We first use these stability criteria to show the instability of the Maxwell boundary conditions proposed by Grad (Commun Pure Appl Math 2(4):331-407, 1949). We then recognise a special block structure of the moment equations which arises due to the recursion relations and the orthogonality of the Hermite polynomials; the block structure will help us in formulating stable boundary conditions for an arbitrary order Hermite discretization of the Boltzmann equation. The formulation of stable boundary conditions relies upon an Onsager matrix which will be constructed such that the newly proposed boundary conditions stay close to the Maxwell boundary conditions at least in the lower order moments.
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Ruzanna Kh. Makaova
2017-12-01
Full Text Available In this paper we study the boundary value problem for a degenerating third order equation of hyperbolic type in a mixed domain. The equation under consideration in the positive part of the domain coincides with the Hallaire equation, which is a pseudoparabolic type equation. Moreover, in the negative part of the domain it coincides with a degenerating hyperbolic equation of the first kind, the particular case of the Bitsadze–Lykov equation. The existence and uniqueness theorem for the solution is proved. The uniqueness of the solution to the problem is proved with the Tricomi method. Using the functional relationships of the positive and negative parts of the domain on the degeneration line, we arrive at the convolution type Volterra integral equation of the 2nd kind with respect to the desired solution by a derivative trace. With the Laplace transform method, we obtain the solution of the integral equation in its explicit form. At last, the solution to the problem under study is written out explicitly as the solution of the second boundary-value problem in the positive part of the domain for the Hallaire equation and as the solution to the Cauchy problem in the negative part of the domain for a degenerate hyperbolic equation of the first kind.
Hyers-Ulam stability for second-order linear differential equations with boundary conditions
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Pasc Gavruta
2011-06-01
Full Text Available We prove the Hyers-Ulam stability of linear differential equations of second-order with boundary conditions or with initial conditions. That is, if y is an approximate solution of the differential equation $y''+ eta (x y = 0$ with $y(a = y(b =0$, then there exists an exact solution of the differential equation, near y.
Relaxation methods for gauge field equilibrium equations
International Nuclear Information System (INIS)
Adler, S.L.; Piran, T.
1984-01-01
This article gives a pedagogical introduction to relaxation methods for the numerical solution of elliptic partial differential equations, with particular emphasis on treating nonlinear problems with delta-function source terms and axial symmetry, which arise in the context of effective Lagrangian approximations to the dynamics of quantized gauge fields. The authors present a detailed theoretical analysis of three models which are used as numerical examples: the classical Abelian Higgs model (illustrating charge screening), the semiclassical leading logarithm model (illustrating flux confinement within a free boundary or ''bag''), and the axially symmetric Bogomol'nyi-Prasad-Sommerfield monopoles (illustrating the occurrence of p topological quantum numbers in non-Abelian gauge fields). They then proceed to a self-contained introduction to the theory of relaxation methods and allied iterative numerical methods and to the practical aspects of their implementation, with attention to general issues which arise in the three examples. The authors conclude with a brief discussion of details of the numerical solution of the models, presenting sample numerical results
International Nuclear Information System (INIS)
Frittelli, Simonetta; Gomez, Roberto
2004-01-01
We show how the use of the normal projection of the Einstein tensor as a set of boundary conditions relates to the propagation of the constraints, for two representations of the Einstein equations with vanishing shift vector: the Arnowitt-Deser-Misner formulation, which is ill posed, and the Einstein-Christoffel formulation, which is symmetric hyperbolic. Essentially, the components of the normal projection of the Einstein tensor that act as nontrivial boundary conditions are linear combinations of the evolution equations with the constraints that are not preserved at the boundary, in both cases. In the process, the relationship of the normal projection of the Einstein tensor to the recently introduced 'constraint-preserving' boundary conditions becomes apparent
Shooting method for solution of boundary-layer flows with massive blowing
Liu, T.-M.; Nachtsheim, P. R.
1973-01-01
A modified, bidirectional shooting method is presented for solving boundary-layer equations under conditions of massive blowing. Unlike the conventional shooting method, which is unstable when the blowing rate increases, the proposed method avoids the unstable direction and is capable of solving complex boundary-layer problems involving mass and energy balance on the surface.
Method of interior boundaries in a mixed problem of acoustic scattering
Directory of Open Access Journals (Sweden)
P. A. Krutitskii
1999-01-01
Full Text Available The mixed problem for the Helmholtz equation in the exterior of several bodies (obstacles is studied in 2 and 3 dimensions. The Dirichlet boundary condition is given on some obstacles and the impedance boundary condition is specified on the rest. The problem is investigated by a special modification of the boundary integral equation method. This modification can be called ‘Method of interior boundaries’, because additional boundaries are introduced inside scattering bodies, where impedance boundary condition is given. The solution of the problem is obtained in the form of potentials on the whole boundary. The density in the potentials satisfies the uniquely solvable Fredholm equation of the second kind and can be computed by standard codes. In fact our method holds for any positive wave numbers. The Neumann, Dirichlet, impedance problems and mixed Dirichlet–Neumann problem are particular cases of our problem.
Directory of Open Access Journals (Sweden)
Zhigang Hu
2014-01-01
Full Text Available In this paper, we apply the method of the Nehari manifold to study the fractional differential equation (d/dt((1/2 0Dt-β(u′(t+(1/2 tDT-β(u′(t= f(t,u(t, a.e. t∈[0,T], and u0=uT=0, where 0Dt-β, tDT-β are the left and right Riemann-Liouville fractional integrals of order 0≤β<1, respectively. We prove the existence of a ground state solution of the boundary value problem.
Boundary element method for internal axisymmetric flow
Directory of Open Access Journals (Sweden)
Gokhman Alexander
1999-01-01
Full Text Available We present an accurate fast method for the computation of potential internal axisymmetric flow based on the boundary element technique. We prove that the computed velocity field asymptotically satisfies reasonable boundary conditions at infinity for various types of inlet/exit. Computation of internal axisymmetric potential flow is an essential ingredient in the three-dimensional problem of computation of velocity fields in turbomachines. We include the results of a practical application of the method to the computation of flow in turbomachines of Kaplan and Francis types.
Boundary Shape Control of the Navier-Stokes Equations and Applications
Institute of Scientific and Technical Information of China (English)
Kaitai LI; Jian SU; Aixiang HUANG
2010-01-01
In this paper,the geometrical design for the blade's surface(s)in an impeller or for the profile of an aircraft,is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations.The objective function is the sum of a global dissipative function and the power of the fluid.The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations.The Euler-Lagrange equations of the optimal control problem are derived,which are an elliptic boundary value system of fourth order,coupled with the Navier-Stokes equations.The authors also prove the existence of the solution of the optimal control problem,the existence of the solution of the Navier-Stokes equations with mixed boundary conditions,the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface and the existence of solutions of the equations for the G(a)teaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.
Sorokin, Sergey V
2011-03-01
Helical springs serve as vibration isolators in virtually any suspension system. Various exact and approximate methods may be employed to determine the eigenfrequencies of vibrations of these structural elements and their dynamic transfer functions. The method of boundary integral equations is a meaningful alternative to obtain exact solutions of problems of the time-harmonic dynamics of elastic springs in the framework of Bernoulli-Euler beam theory. In this paper, the derivations of the Green's matrix, of the Somigliana's identities, and of the boundary integral equations are presented. The vibrational power transmission in an infinitely long spring is analyzed by means of the Green's matrix. The eigenfrequencies and the dynamic transfer functions are found by solving the boundary integral equations. In the course of analysis, the essential features and advantages of the method of boundary integral equations are highlighted. The reported analytical results may be used to study the time-harmonic motion in any wave guide governed by a system of linear differential equations in a single spatial coordinate along its axis. © 2011 Acoustical Society of America
Spline Collocation Method for Nonlinear Multi-Term Fractional Differential Equation
Choe, Hui-Chol; Kang, Yong-Suk
2013-01-01
We study an approximation method to solve nonlinear multi-term fractional differential equations with initial conditions or boundary conditions. First, we transform the nonlinear multi-term fractional differential equations with initial conditions and boundary conditions to nonlinear fractional integral equations and consider the relations between them. We present a Spline Collocation Method and prove the existence, uniqueness and convergence of approximate solution as well as error estimatio...
Approximate analytical solution to the Boussinesq equation with a sloping water-land boundary
Tang, Yuehao; Jiang, Qinghui; Zhou, Chuangbing
2016-04-01
An approximate solution is presented to the 1-D Boussinesq equation (BEQ) characterizing transient groundwater flow in an unconfined aquifer subject to a constant water variation at the sloping water-land boundary. The flow equation is decomposed to a linearized BEQ and a head correction equation. The linearized BEQ is solved using a Laplace transform. By means of the frozen-coefficient technique and Gauss function method, the approximate solution for the head correction equation can be obtained, which is further simplified to a closed-form expression under the condition of local energy equilibrium. The solutions of the linearized and head correction equations are discussed from physical concepts. Especially for the head correction equation, the well posedness of the approximate solution obtained by the frozen-coefficient method is verified to demonstrate its boundedness, which can be further embodied as the upper and lower error bounds to the exact solution of the head correction by statistical analysis. The advantage of this approximate solution is in its simplicity while preserving the inherent nonlinearity of the physical phenomenon. Comparisons between the analytical and numerical solutions of the BEQ validate that the approximation method can achieve desirable precisions, even in the cases with strong nonlinearity. The proposed approximate solution is applied to various hydrological problems, in which the algebraic expressions that quantify the water flow processes are derived from its basic solutions. The results are useful for the quantification of stream-aquifer exchange flow rates, aquifer response due to the sudden reservoir release, bank storage and depletion, and front position and propagation speed.
Optimal analytic method for the nonlinear Hasegawa-Mima equation
Baxter, Mathew; Van Gorder, Robert A.; Vajravelu, Kuppalapalle
2014-05-01
The Hasegawa-Mima equation is a nonlinear partial differential equation that describes the electric potential due to a drift wave in a plasma. In the present paper, we apply the method of homotopy analysis to a slightly more general Hasegawa-Mima equation, which accounts for hyper-viscous damping or viscous dissipation. First, we outline the method for the general initial/boundary value problem over a compact rectangular spatial domain. We use a two-stage method, where both the convergence control parameter and the auxiliary linear operator are optimally selected to minimize the residual error due to the approximation. To do the latter, we consider a family of operators parameterized by a constant which gives the decay rate of the solutions. After outlining the general method, we consider a number of concrete examples in order to demonstrate the utility of this approach. The results enable us to study properties of the initial/boundary value problem for the generalized Hasegawa-Mima equation. In several cases considered, we are able to obtain solutions with extremely small residual errors after relatively few iterations are computed (residual errors on the order of 10-15 are found in multiple cases after only three iterations). The results demonstrate that selecting a parameterized auxiliary linear operator can be extremely useful for minimizing residual errors when used concurrently with the optimal homotopy analysis method, suggesting that this approach can prove useful for a number of nonlinear partial differential equations arising in physics and nonlinear mechanics.
The integral equation method applied to eddy currents
International Nuclear Information System (INIS)
Biddlecombe, C.S.; Collie, C.J.; Simkin, J.; Trowbridge, C.W.
1976-04-01
An algorithm for the numerical solution of eddy current problems is described, based on the direct solution of the integral equation for the potentials. In this method only the conducting and iron regions need to be divided into elements, and there are no boundary conditions. Results from two computer programs using this method for iron free problems for various two-dimensional geometries are presented and compared with analytic solutions. (author)
Solution of the Stokes system by boundary integral equations and fixed point iterative schemes
International Nuclear Information System (INIS)
Chidume, C.E.; Lubuma, M.S.
1990-01-01
The solution to the exterior three dimensional Stokes problem is sought in the form of a single layer potential of unknown density. This reduces the problem to a boundary integral equation of the first kind whose operator is the velocity component of the single layer potential. It is shown that this component is an isomorphism between two appropriate Sobolev spaces containing the unknown densities and the data respectively. The isomorphism corresponds to a variational problem with coercive bilinear form. The latter property allows us to consider various fixed point iterative schemes that converge to the unique solution of the integral equation. Explicit error estimates are also obtained. The successive approximations are also considered in a more computable form by using the product integration method of Atkinson. (author). 47 refs
Antiperiodic Boundary Value Problems for Second-Order Impulsive Ordinary Differential Equations
Directory of Open Access Journals (Sweden)
2009-02-01
Full Text Available We consider a second-order ordinary differential equation with antiperiodic boundary conditions and impulses. By using Schaefer's fixed-point theorem, some existence results are obtained.
Directory of Open Access Journals (Sweden)
Weifeng Wang
2014-01-01
Full Text Available We study an optimal control problem governed by a semilinear parabolic equation, whose control variable is contained only in the boundary condition. An existence theorem for the optimal control is obtained.
Directory of Open Access Journals (Sweden)
Giai Giang Vo
2015-01-01
Full Text Available This paper is devoted to the study of a wave equation with a boundary condition of many-point type. The existence of weak solutions is proved by using the Galerkin method. Also, the uniqueness and the stability of solutions are established.
Directory of Open Access Journals (Sweden)
Mitsuhiro Nakao
2014-01-01
Full Text Available We prove the existence and uniqueness of a global decaying solution to the initial boundary value problem for the quasilinear wave equation with Kelvin-Voigt dissipation and a derivative nonlinearity. To derive the required estimates of the solutions we employ a 'loan' method and use a difference inequality on the energy.
Statistical Methods for Stochastic Differential Equations
Kessler, Mathieu; Sorensen, Michael
2012-01-01
The seventh volume in the SemStat series, Statistical Methods for Stochastic Differential Equations presents current research trends and recent developments in statistical methods for stochastic differential equations. Written to be accessible to both new students and seasoned researchers, each self-contained chapter starts with introductions to the topic at hand and builds gradually towards discussing recent research. The book covers Wiener-driven equations as well as stochastic differential equations with jumps, including continuous-time ARMA processes and COGARCH processes. It presents a sp
Method of controlling chaos in laser equations
International Nuclear Information System (INIS)
Duong-van, M.
1993-01-01
A method of controlling chaotic to laminar flows in the Lorenz equations using fixed points dictated by minimizing the Lyapunov functional was proposed by Singer, Wang, and Bau [Phys. Rev. Lett. 66, 1123 (1991)]. Using different fixed points, we find that the solutions in a chaotic regime can also be periodic. Since the laser equations are isomorphic to the Lorenz equations we use this method to control chaos when the laser is operated over the pump threshold. Furthermore, by solving the laser equations with an occasional proportional feedback mechanism, we recover the essential laser controlling features experimentally discovered by Roy, Murphy, Jr., Maier, Gills, and Hunt [Phys. Rev. Lett. 68, 1259 (1992)
Method of controlling chaos in laser equations
Duong-van, Minh
1993-01-01
A method of controlling chaotic to laminar flows in the Lorenz equations using fixed points dictated by minimizing the Lyapunov functional was proposed by Singer, Wang, and Bau [Phys. Rev. Lett. 66, 1123 (1991)]. Using different fixed points, we find that the solutions in a chaotic regime can also be periodic. Since the laser equations are isomorphic to the Lorenz equations we use this method to control chaos when the laser is operated over the pump threshold. Furthermore, by solving the laser equations with an occasional proportional feedback mechanism, we recover the essential laser controlling features experimentally discovered by Roy, Murphy, Jr., Maier, Gills, and Hunt [Phys. Rev. Lett. 68, 1259 (1992)].
Nonlinear $q$-fractional differential equations with nonlocal and sub-strip type boundary conditions
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Bashir Ahmad
2014-06-01
Full Text Available This paper is concerned with new boundary value problems of nonlinear $q$-fractional differential equations with nonlocal and sub-strip type boundary conditions. Our results are new in the present setting and rely on the contraction mapping principle and a fixed point theorem due to O'Regan. Some illustrative examples are also presented.
m-POINT BOUNDARY VALUE PROBLEM FOR SECOND ORDER IMPULSIVE DIFFERENTIAL EQUATION AT RESONANCE
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
In his paper,we obtain a general theorem concerning the existence of solutions to an m-point boundary value problem for the second-order differential equation with impulses.Moreover,the result can also be applied to study the usual m-point boundary value problem at resonance without impulses.
Stabilizing local boundary conditions for two-dimensional shallow water equations
Dia, Ben Mansour; Oppelstrup, Jesper
2018-01-01
In this article, we present a sub-critical two-dimensional shallow water flow regulation. From the energy estimate of a set of one-dimensional boundary stabilization problems, we obtain a set of polynomial equations with respect to the boundary
On nonseparated three-point boundary value problems for linear functional differential equations
Czech Academy of Sciences Publication Activity Database
Rontó, András; Rontó, M.
2011-01-01
Roč. 2011, - (2011), s. 326052 ISSN 1085-3375 Institutional research plan: CEZ:AV0Z10190503 Keywords : functional-differential equation * three-point boundary value problem * nonseparated boundary condition Subject RIV: BA - General Mathematics Impact factor: 1.318, year: 2011 http://www.hindawi.com/journals/ aaa /2011/326052/
Brito, Irene; Mena, Filipe C
2017-08-01
We prove that, for a given spherically symmetric fluid distribution with tangential pressure on an initial space-like hypersurface with a time-like boundary, there exists a unique, local in time solution to the Einstein equations in a neighbourhood of the boundary. As an application, we consider a particular elastic fluid interior matched to a vacuum exterior.
How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems
Cortazar, C.; Elgueta, M.; Rossi, J. D.; Wolanski, N.
2006-01-01
We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation with Neumann boundary conditions.
Homogenization of the stochastic Navier–Stokes equation with a stochastic slip boundary condition
Bessaih, Hakima
2015-11-02
The two-dimensional Navier–Stokes equation in a perforated domain with a dynamical slip boundary condition is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the holes. We consider a scaling (ᵋ for the viscosity and 1 for the density) that will lead to a time-dependent limit problem. However, the noncritical scaling (ᵋ, β > 1) is considered in front of the nonlinear term. The homogenized system in the limit is obtained as a Darcy’s law with memory with two permeabilities and an extra term that is due to the stochastic perturbation on the boundary of the holes. The nonhomogeneity on the boundary contains a stochastic part that yields in the limit an additional term in the Darcy’s law. We use the two-scale convergence method after extending the solution with 0 inside the holes to pass to the limit. By Itô stochastic calculus, we get uniform estimates on the solution in appropriate spaces. Due to the stochastic integral, the pressure that appears in the variational formulation does not have enough regularity in time. This fact made us rely only on the variational formulation for the passage to the limit on the solution. We obtain a variational formulation for the limit that is solution of a Stokes system with two pressures. This two-scale limit gives rise to three cell problems, two of them give the permeabilities while the third one gives an extra term in the Darcy’s law due to the stochastic perturbation on the boundary of the holes.
Directory of Open Access Journals (Sweden)
Javier A. Dottori
2015-01-01
Full Text Available A method for modeling outflow boundary conditions in the lattice Boltzmann method (LBM based on the maximization of the local entropy is presented. The maximization procedure is constrained by macroscopic values and downstream components. The method is applied to fully developed boundary conditions of the Navier-Stokes equations in rectangular channels. Comparisons are made with other alternative methods. In addition, the new downstream-conditioned entropy is studied and it was found that there is a correlation with the velocity gradient during the flow development.
A New Spectral Local Linearization Method for Nonlinear Boundary Layer Flow Problems
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S. S. Motsa
2013-01-01
Full Text Available We propose a simple and efficient method for solving highly nonlinear systems of boundary layer flow problems with exponentially decaying profiles. The algorithm of the proposed method is based on an innovative idea of linearizing and decoupling the governing systems of equations and reducing them into a sequence of subsystems of differential equations which are solved using spectral collocation methods. The applicability of the proposed method, hereinafter referred to as the spectral local linearization method (SLLM, is tested on some well-known boundary layer flow equations. The numerical results presented in this investigation indicate that the proposed method, despite being easy to develop and numerically implement, is very robust in that it converges rapidly to yield accurate results and is more efficient in solving very large systems of nonlinear boundary value problems of the similarity variable boundary layer type. The accuracy and numerical stability of the SLLM can further be improved by using successive overrelaxation techniques.
Ji, Songsong; Yang, Yibo; Pang, Gang; Antoine, Xavier
2018-01-01
The aim of this paper is to design some accurate artificial boundary conditions for the semi-discretized linear Schrödinger and heat equations in rectangular domains. The Laplace transform in time and discrete Fourier transform in space are applied to get Green's functions of the semi-discretized equations in unbounded domains with single-source. An algorithm is given to compute these Green's functions accurately through some recurrence relations. Furthermore, the finite-difference method is used to discretize the reduced problem with accurate boundary conditions. Numerical simulations are presented to illustrate the accuracy of our method in the case of the linear Schrödinger and heat equations. It is shown that the reflection at the corners is correctly eliminated.
Inverse Boundary Value Problem for Non-linear Hyperbolic Partial Differential Equations
Nakamura, Gen; Vashisth, Manmohan
2017-01-01
In this article we are concerned with an inverse boundary value problem for a non-linear wave equation of divergence form with space dimension $n\\geq 3$. This non-linear wave equation has a trivial solution, i.e. zero solution. By linearizing this equation at the trivial solution, we have the usual linear isotropic wave equation with the speed $\\sqrt{\\gamma(x)}$ at each point $x$ in a given spacial domain. For any small solution $u=u(t,x)$ of this non-linear equation, we have the linear isotr...
Advances in iterative methods for nonlinear equations
Busquier, Sonia
2016-01-01
This book focuses on the approximation of nonlinear equations using iterative methods. Nine contributions are presented on the construction and analysis of these methods, the coverage encompassing convergence, efficiency, robustness, dynamics, and applications. Many problems are stated in the form of nonlinear equations, using mathematical modeling. In particular, a wide range of problems in Applied Mathematics and in Engineering can be solved by finding the solutions to these equations. The book reveals the importance of studying convergence aspects in iterative methods and shows that selection of the most efficient and robust iterative method for a given problem is crucial to guaranteeing a good approximation. A number of sample criteria for selecting the optimal method are presented, including those regarding the order of convergence, the computational cost, and the stability, including the dynamics. This book will appeal to researchers whose field of interest is related to nonlinear problems and equations...
Gazzola, Filippo; Sweers, Guido
2010-01-01
This monograph covers higher order linear and nonlinear elliptic boundary value problems in bounded domains, mainly with the biharmonic or poly-harmonic operator as leading principal part. Underlying models and, in particular, the role of different boundary conditions are explained in detail. As for linear problems, after a brief summary of the existence theory and Lp and Schauder estimates, the focus is on positivity or - since, in contrast to second order equations, a general form of a comparison principle does not exist - on “near positivity.” The required kernel estimates are also presented in detail. As for nonlinear problems, several techniques well-known from second order equations cannot be utilized and have to be replaced by new and different methods. Subcritical, critical and supercritical nonlinearities are discussed and various existence and nonexistence results are proved. The interplay with the positivity topic from the ﬁrst part is emphasized and, moreover, a far-reaching Gidas-Ni-Nirenbe...
Immersed Boundary-Lattice Boltzmann Method Using Two Relaxation Times
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Kosuke Hayashi
2012-06-01
Full Text Available An immersed boundary-lattice Boltzmann method (IB-LBM using a two-relaxation time model (TRT is proposed. The collision operator in the lattice Boltzmann equation is modeled using two relaxation times. One of them is used to set the fluid viscosity and the other is for numerical stability and accuracy. A direct-forcing method is utilized for treatment of immersed boundary. A multi-direct forcing method is also implemented to precisely satisfy the boundary conditions at the immersed boundary. Circular Couette flows between a stationary cylinder and a rotating cylinder are simulated for validation of the proposed method. The method is also validated through simulations of circular and spherical falling particles. Effects of the functional forms of the direct-forcing term and the smoothed-delta function, which interpolates the fluid velocity to the immersed boundary and distributes the forcing term to fixed Eulerian grid points, are also examined. As a result, the following conclusions are obtained: (1 the proposed method does not cause non-physical velocity distribution in circular Couette flows even at high relaxation times, whereas the single-relaxation time (SRT model causes a large non-physical velocity distortion at a high relaxation time, (2 the multi-direct forcing reduces the errors in the velocity profile of a circular Couette flow at a high relaxation time, (3 the two-point delta function is better than the four-point delta function at low relaxation times, but worse at high relaxation times, (4 the functional form of the direct-forcing term does not affect predictions, and (5 circular and spherical particles falling in liquids are well predicted by using the proposed method both for two-dimensional and three-dimensional cases.
Hybrid finite difference/finite element immersed boundary method.
E Griffith, Boyce; Luo, Xiaoyu
2017-12-01
The immersed boundary method is an approach to fluid-structure interaction that uses a Lagrangian description of the structural deformations, stresses, and forces along with an Eulerian description of the momentum, viscosity, and incompressibility of the fluid-structure system. The original immersed boundary methods described immersed elastic structures using systems of flexible fibers, and even now, most immersed boundary methods still require Lagrangian meshes that are finer than the Eulerian grid. This work introduces a coupling scheme for the immersed boundary method to link the Lagrangian and Eulerian variables that facilitates independent spatial discretizations for the structure and background grid. This approach uses a finite element discretization of the structure while retaining a finite difference scheme for the Eulerian variables. We apply this method to benchmark problems involving elastic, rigid, and actively contracting structures, including an idealized model of the left ventricle of the heart. Our tests include cases in which, for a fixed Eulerian grid spacing, coarser Lagrangian structural meshes yield discretization errors that are as much as several orders of magnitude smaller than errors obtained using finer structural meshes. The Lagrangian-Eulerian coupling approach developed in this work enables the effective use of these coarse structural meshes with the immersed boundary method. This work also contrasts two different weak forms of the equations, one of which is demonstrated to be more effective for the coarse structural discretizations facilitated by our coupling approach. © 2017 The Authors International Journal for Numerical Methods in Biomedical Engineering Published by John Wiley & Sons Ltd.
Modeling of Airfoil Trailing Edge Flap with Immersed Boundary Method
DEFF Research Database (Denmark)
Zhu, Wei Jun; Shen, Wen Zhong; Sørensen, Jens Nørkær
2011-01-01
The present work considers incompressible flow over a 2D airfoil with a deformable trailing edge. The aerodynamic characteristics of an airfoil with a trailing edge flap is numerically investigated using computational fluid dynamics. A novel hybrid immersed boundary (IB) technique is applied...... to simulate the moving part of the trailing edge. Over the main fixed part of the airfoil the Navier-Stokes (NS) equations are solved using a standard body-fitted finite volume technique whereas the moving trailing edge flap is simulated with the immersed boundary method on a curvilinear mesh. The obtained...... results show that the hybrid approach is an efficient and accurate method for solving turbulent flows past airfoils with a trailing edge flap and flow control using trailing edge flap is an efficient way to regulate the aerodynamic loading on airfoils....
Multiscale molecular dynamics using the matched interface and boundary method
International Nuclear Information System (INIS)
Geng Weihua; Wei, G.W.
2011-01-01
The Poisson-Boltzmann (PB) equation is an established multiscale model for electrostatic analysis of biomolecules and other dielectric systems. PB based molecular dynamics (MD) approach has a potential to tackle large biological systems. Obstacles that hinder the current development of PB based MD methods are concerns in accuracy, stability, efficiency and reliability. The presence of complex solvent-solute interface, geometric singularities and charge singularities leads to challenges in the numerical solution of the PB equation and electrostatic force evaluation in PB based MD methods. Recently, the matched interface and boundary (MIB) method has been utilized to develop the first second order accurate PB solver that is numerically stable in dealing with discontinuous dielectric coefficients, complex geometric singularities and singular source charges. The present work develops the PB based MD approach using the MIB method. New formulation of electrostatic forces is derived to allow the use of sharp molecular surfaces. Accurate reaction field forces are obtained by directly differentiating the electrostatic potential. Dielectric boundary forces are evaluated at the solvent-solute interface using an accurate Cartesian-grid surface integration method. The electrostatic forces located at reentrant surfaces are appropriately assigned to related atoms. Extensive numerical tests are carried out to validate the accuracy and stability of the present electrostatic force calculation. The new PB based MD method is implemented in conjunction with the AMBER package. MIB based MD simulations of biomolecules are demonstrated via a few example systems.
Spline methods for conversation equations
International Nuclear Information System (INIS)
Bottcher, C.; Strayer, M.R.
1991-01-01
The consider the numerical solution of physical theories, in particular hydrodynamics, which can be formulated as systems of conservation laws. To this end we briefly describe the Basis Spline and collocation methods, paying particular attention to representation theory, which provides discrete analogues of the continuum conservation and dispersion relations, and hence a rigorous understanding of errors and instabilities. On this foundation we propose an algorithm for hydrodynamic problems in which most linear and nonlinear instabilities are brought under control. Numerical examples are presented from one-dimensional relativistic hydrodynamics. 9 refs., 10 figs
Ozdemir, Burhanettin
2017-01-01
The purpose of this study is to equate Trends in International Mathematics and Science Study (TIMSS) mathematics subtest scores obtained from TIMSS 2011 to scores obtained from TIMSS 2007 form with different nonlinear observed score equating methods under Non-Equivalent Anchor Test (NEAT) design where common items are used to link two or more test…
Boundary-integral equation formulation for time-dependent inelastic deformation in metals
Energy Technology Data Exchange (ETDEWEB)
Kumar, V; Mukherjee, S
1977-01-01
The mathematical structure of various constitutive relations proposed in recent years for representing time-dependent inelastic deformation behavior of metals at elevated temperatues has certain features which permit a simple formulation of the three-dimensional inelasticity problem in terms of real time rates. A direct formulation of the boundary-integral equation method in terms of rates is discussed for the analysis of time-dependent inelastic deformation of arbitrarily shaped three-dimensional metallic bodies subjected to arbitrary mechanical and thermal loading histories and obeying constitutive relations of the kind mentioned above. The formulation is based on the assumption of infinitesimal deformations. Several illustrative examples involving creep of thick-walled spheres, long thick-walled cylinders, and rotating discs are discussed. The implementation of the method appears to be far easier than analogous BIE formulations that have been suggested for elastoplastic problems.
Boundary element methods for electrical engineers
POLJAK, D
2005-01-01
In the last couple of decades the Boundary Element Method (BEM) has become a well-established technique that is widely used for solving various problems in electrical engineering and electromagnetics. Although there are many excellent research papers published in the relevant literature that describe various BEM applications in electrical engineering and electromagnetics, there has been a lack of suitable textbooks and monographs on the subject. This book presents BEM in a simple fashion in order to help the beginner to understand the very basic principles of the method. It initially derives B
Partial differential equations methods, applications and theories
Hattori, Harumi
2013-01-01
This volume is an introductory level textbook for partial differential equations (PDE's) and suitable for a one-semester undergraduate level or two-semester graduate level course in PDE's or applied mathematics. Chapters One to Five are organized according to the equations and the basic PDE's are introduced in an easy to understand manner. They include the first-order equations and the three fundamental second-order equations, i.e. the heat, wave and Laplace equations. Through these equations we learn the types of problems, how we pose the problems, and the methods of solutions such as the separation of variables and the method of characteristics. The modeling aspects are explained as well. The methods introduced in earlier chapters are developed further in Chapters Six to Twelve. They include the Fourier series, the Fourier and the Laplace transforms, and the Green's functions. The equations in higher dimensions are also discussed in detail. This volume is application-oriented and rich in examples. Going thr...
Integral equation methods for vesicle electrohydrodynamics in three dimensions
Veerapaneni, Shravan
2016-12-01
In this paper, we develop a new boundary integral equation formulation that describes the coupled electro- and hydro-dynamics of a vesicle suspended in a viscous fluid and subjected to external flow and electric fields. The dynamics of the vesicle are characterized by a competition between the elastic, electric and viscous forces on its membrane. The classical Taylor-Melcher leaky-dielectric model is employed for the electric response of the vesicle and the Helfrich energy model combined with local inextensibility is employed for its elastic response. The coupled governing equations for the vesicle position and its transmembrane electric potential are solved using a numerical method that is spectrally accurate in space and first-order in time. The method uses a semi-implicit time-stepping scheme to overcome the numerical stiffness associated with the governing equations.
A stochastic collocation method for the second order wave equation with a discontinuous random speed
Motamed, Mohammad; Nobile, Fabio; Tempone, Raul
2012-01-01
In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical
A device adaptive inflow boundary condition for Wigner equations of quantum transport
International Nuclear Information System (INIS)
Jiang, Haiyan; Lu, Tiao; Cai, Wei
2014-01-01
In this paper, an improved inflow boundary condition is proposed for Wigner equations in simulating a resonant tunneling diode (RTD), which takes into consideration the band structure of the device. The original Frensley inflow boundary condition prescribes the Wigner distribution function at the device boundary to be the semi-classical Fermi–Dirac distribution for free electrons in the device contacts without considering the effect of the quantum interaction inside the quantum device. The proposed device adaptive inflow boundary condition includes this effect by assigning the Wigner distribution to the value obtained from the Wigner transform of wave functions inside the device at zero external bias voltage, thus including the dominant effect on the electron distribution in the contacts due to the device internal band energy profile. Numerical results on computing the electron density inside the RTD under various incident waves and non-zero bias conditions show much improvement by the new boundary condition over the traditional Frensley inflow boundary condition
Parsani, Matteo; Carpenter, Mark H.; Nielsen, Eric J.
2015-01-01
Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneous-approximation-term penalty technique. Although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, discontinuous Galerkin, and flux reconstruction/correction procedure via reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions.
Entropy Stable Wall Boundary Conditions for the Compressible Navier-Stokes Equations
Parsani, Matteo; Carpenter, Mark H.; Nielsen, Eric J.
2014-01-01
Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the compressible Navier-Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneous-approximation-term penalty technique. Although discontinuous spectral collocation operators are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite volume, finite difference, discontinuous Galerkin, and flux reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions.
Boundary Equations and Regularity Theory for Geometric Variational Systems with Neumann Data
Schikorra, Armin
2018-02-01
We study boundary regularity of maps from two-dimensional domains into manifolds which are critical with respect to a generic conformally invariant variational functional and which, at the boundary, intersect perpendicularly with a support manifold. For example, harmonic maps, or H-surfaces, with a partially free boundary condition. In the interior it is known, by the celebrated work of Rivière, that these maps satisfy a system with an antisymmetric potential, from which one can derive the interior regularity of the solution. Avoiding a reflection argument, we show that these maps satisfy along the boundary a system of equations which also exhibits a (nonlocal) antisymmetric potential that combines information from the interior potential and the geometric Neumann boundary condition. We then proceed to show boundary regularity for solutions to such systems.
Numerical Methods for Free Boundary Problems
1991-01-01
About 80 participants from 16 countries attended the Conference on Numerical Methods for Free Boundary Problems, held at the University of Jyviiskylii, Finland, July 23-27, 1990. The main purpose of this conference was to provide up-to-date information on important directions of research in the field of free boundary problems and their numerical solutions. The contributions contained in this volume cover the lectures given in the conference. The invited lectures were given by H.W. Alt, V. Barbu, K-H. Hoffmann, H. Mittelmann and V. Rivkind. In his lecture H.W. Alt considered a mathematical model and existence theory for non-isothermal phase separations in binary systems. The lecture of V. Barbu was on the approximate solvability of the inverse one phase Stefan problem. K-H. Hoff mann gave an up-to-date survey of several directions in free boundary problems and listed several applications, but the material of his lecture is not included in this proceedings. H.D. Mittelmann handled the stability of thermo capi...
Kinetic equation solution by inverse kinetic method
International Nuclear Information System (INIS)
Salas, G.
1983-01-01
We propose a computer program (CAMU) which permits to solve the inverse kinetic equation. The CAMU code is written in HPL language for a HP 982 A microcomputer with a peripheral interface HP 9876 A ''thermal graphic printer''. The CAMU code solves the inverse kinetic equation by taking as data entry the output of the ionization chambers and integrating the equation with the help of the Simpson method. With this program we calculate the evolution of the reactivity in time for a given disturbance
Explicit Finite Difference Methods for the Delay Pseudoparabolic Equations
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I. Amirali
2014-01-01
Full Text Available Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown.
van Horssen, Wim T.; Wang, Yandong; Cao, Guohua
2018-06-01
In this paper, it is shown how characteristic coordinates, or equivalently how the well-known formula of d'Alembert, can be used to solve initial-boundary value problems for wave equations on fixed, bounded intervals involving Robin type of boundary conditions with time-dependent coefficients. A Robin boundary condition is a condition that specifies a linear combination of the dependent variable and its first order space-derivative on a boundary of the interval. Analytical methods, such as the method of separation of variables (SOV) or the Laplace transform method, are not applicable to those types of problems. The obtained analytical results by applying the proposed method, are in complete agreement with those obtained by using the numerical, finite difference method. For problems with time-independent coefficients in the Robin boundary condition(s), the results of the proposed method also completely agree with those as for instance obtained by the method of separation of variables, or by the finite difference method.
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M. W. Dunlop
Full Text Available Magnetic field measurements, taken by the magnetometer experiment (MAM on board the German Equator-S spacecraft, have been used to identify and categorise 131 crossings of the dawn-side magnetopause at low latitude, providing unusual, long duration coverage of the adjacent magnetospheric regions and near magnetosheath. The crossings occurred on 31 orbits, providing unbiased coverage over the full range of local magnetic shear from 06:00 to 10:40 LT. Apogee extent places the spacecraft in conditions associated with intermediate, rather than low, solar wind dynamic pressure, as it processes into the flank region. The apogee of the spacecraft remains close to the magnetopause for mean solar wind pressure. The occurrence of the magnetopause encounters are summarised and are found to compare well with predicted boundary location, where solar wind conditions are known. Most scale with solar wind pressure. Magnetopause shape is also documented and we find that the magnetopause orientation is consistently sunward of a model boundary and is not accounted for by IMF or local magnetic shear conditions. A number of well-established crossings, particularly those at high magnetic shear, or exhibiting unusually high-pressure states, were observed and have been analysed for their boundary characteristics and some details of their boundary and near magnetosheath properties are discussed. Of particular note are the occurrence of mirror-like signatures in the adjacent magnetosheath during a significant fraction of the encounters and a high number of multiple crossings over a long time period. The latter is facilitated by the spacecraft orbit which is designed to remain in the near magnetosheath for average solar wind pressure. For most encounters, a well-ordered, tangential (draped magnetosheath field is observed and there is little evidence of large deviations in local boundary orientations. Two passes corresponding to close conjunctions of the Geotail spacecraft
International Nuclear Information System (INIS)
Li Xicheng; Xu Mingyu; Wang Shaowei
2008-01-01
In this paper, we give similarity solutions of partial differential equations of fractional order with a moving boundary condition. The solutions are given in terms of a generalized Wright function. The time-fractional Caputo derivative and two types of space-fractional derivatives are considered. The scale-invariant variable and the form of the solution of the moving boundary are obtained by the Lie group analysis. A comparison between the solutions corresponding to two types of fractional derivative is also given
Gerbi, Sté phane; Said-Houari, Belkacem
2011-01-01
In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the KelvinVoigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained. © 2011 Elsevier Ltd. All rights reserved.
Gerbi, Stéphane
2011-12-01
In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the KelvinVoigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained. © 2011 Elsevier Ltd. All rights reserved.
Waveform relaxation methods for implicit differential equations
P.J. van der Houwen; W.A. van der Veen
1996-01-01
textabstractWe apply a Runge-Kutta-based waveform relaxation method to initial-value problems for implicit differential equations. In the implementation of such methods, a sequence of nonlinear systems has to be solved iteratively in each step of the integration process. The size of these systems
Crossing of the cosmological constant boundary-an equation of state description
International Nuclear Information System (INIS)
Stefancic, Hrvoje
2006-01-01
The phenomenon of the dark energy transition between the quintessence regime (w > -1) and the phantom regime (w < -1), also known as the cosmological constant boundary crossing, is analysed in terms of the dark energy equation of state. It is found that the dark energy equation of state in the dark energy models which exhibit the transition is implicitly defined. The generalizations of the models explicitly constructed to exhibit the transition are studied to gain insight into the mechanism of the transition. It is found that the cancellation of the terms corresponding to the cosmological constant boundary makes the transition possible
Hybrid immersed boundary method for airfoils with a trailing-edge flap
DEFF Research Database (Denmark)
Zhu, Wei Jun; Behrens, Tim; Shen, Wen Zhong
2013-01-01
In this paper, a hybrid immersed boundary technique has been developed for simulating turbulent flows past airfoils with moving trailing-edge flaps. Over the main fixed part of the airfoil, the equations are solved using a standard body-fitted finite volume technique, whereas the moving trailing......-edge flap is simulated using the immersed boundary method on a curvilinear mesh. An existing in-house-developed flow solver is employed to solve the incompressible Reynolds-Averaged Navier-Stokes equations together with the k-ω turbulence model. To achieve consistent wall boundary conditions at the immersed...... boundaries the k-ωturbulence model is modified and adapted to the local conditions associated with the immersed boundary method. The obtained results show that the hybrid approach is an efficient and accurate method for solving turbulent flows past airfoils with a trailing-edge flap and that flow control...
A new RBF-Trefftz meshless method for partial differential equations
International Nuclear Information System (INIS)
Cao Leilei; Zhao Ning; Qin Qinghua
2010-01-01
Based on the radial basis functions (RBF) and T-Trefftz solution, this paper presents a new meshless method for numerically solving various partial differential equation systems. First, the analog equation method (AEM) is used to convert the original patial differential equation to an equivalent Poisson's equation. Then, the radial basis functions (RBF) are employed to approxiamate the inhomogeneous term, while the homogeneous solution is obtained by linear combination of a set of T-Trefftz solutions. The present scheme, named RBF-Trefftz has the advantage over the fundamental solution (MFS) method due to the use of nonsingular T-Trefftz solution rather than singular fundamental solutions, so it does not require the artificial boundary. The application and efficiency of the proposed method are validated through several examples which include different type of differential equations, such as Laplace equation, Hellmholtz equation, convectin-diffusion equation and time-dependent equation.
Boundary layer phenomena for differential-delay equations with state-dependent time lags, I.
Mallet-Paret, John; Nussbaum, Roger D.
1992-11-01
In this paper we begin a study of the differential-delay equation \\varepsilon x'(t) = - x(t) + f(x(t - r)), r = r(x(t)) . We prove the existence of periodic solutions for 0equations. In a companion paper these results will be used to investigate the limiting profile and corresponding boundary layer phenomena for periodic solutions as ɛ approaches zero.
Directory of Open Access Journals (Sweden)
G. K. Parks
1999-12-01
Full Text Available An electrostatic analyser (ESA onboard the Equator-S spacecraft operating in coordination with a potential control device (PCD has obtained the first accurate electron energy spectrum with energies ≈7 eV–100 eV in the vicinity of the magnetopause. On 8 January, 1998, a solar wind pressure increase pushed the magnetopause inward, leaving the Equator-S spacecraft in the magnetosheath. On the return into the magnetosphere approximately 80 min later, the magnetopause was observed by the ESA and the solid state telescopes (the SSTs detected electrons and ions with energies ≈20–300 keV. The high time resolution (3 s data from ESA and SST show the boundary region contains of multiple plasma sources that appear to evolve in space and time. We show that electrons with energies ≈7 eV–100 eV permeate the outer regions of the magnetosphere, from the magnetopause to ≈6Re. Pitch-angle distributions of ≈20–300 keV electrons show the electrons travel in both directions along the magnetic field with a peak at 90° indicating a trapped configuration. The IMF during this interval was dominated by Bx and By components with a small Bz.Key words. Magnetospheric physics (magnetopause · cusp · and boundary layers; magnetospheric configuration and dynamics; solar wind · magnetosphere interactions
International Nuclear Information System (INIS)
Goncalez, Tifani T.; Segatto, Cynthia F.; Vilhena, Marco Tullio
2011-01-01
In this work, we report an analytical solution for the set of S N equations for the angular flux, in a rectangle, using the double Laplace transform technique. Its main idea comprehends the steps: application of the Laplace transform in one space variable, solution of the resulting equation by the LTS N method and reconstruction of the double Laplace transformed angular flux using the inversion theorem of the Laplace transform. We must emphasize that we perform the Laplace inversion by the LTS N method in the x direction, meanwhile we evaluate the inversion in the y direction performing the calculation of the corresponding line integral solution by the Stefest method. We have also to figure out that the application of Laplace transform to this type of boundary value problem introduces additional unknown functions associated to the partial derivatives of the angular flux at boundary. Based on the good results attained by the nodal LTS N method, we assume that the angular flux at boundary is also approximated by an exponential function. By analytical we mean that no approximation is done along the solution derivation except for the exponential hypothesis for the exiting angular flux at boundary. For sake of completeness, we report numerical comparisons of the obtained results against the ones of the literature. (author)
Existence and asymptotic behavior of the wave equation with dynamic boundary conditions
Graber, Philip Jameson; Said-Houari, Belkacem
2012-01-01
The goal of this work is to study a model of the strongly damped wave equation with dynamic boundary conditions and nonlinear boundary/interior sources and nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. In addition, we show that in the strongly damped case solutions gain additional regularity for positive times t>0. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution grows as an exponential function. Moreover, in the absence of the strong damping term, we prove that the solution ceases to exists and blows up in finite time. © 2012 Springer Science+Business Media, LLC.
Existence and asymptotic behavior of the wave equation with dynamic boundary conditions
Graber, Philip Jameson
2012-03-07
The goal of this work is to study a model of the strongly damped wave equation with dynamic boundary conditions and nonlinear boundary/interior sources and nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. In addition, we show that in the strongly damped case solutions gain additional regularity for positive times t>0. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution grows as an exponential function. Moreover, in the absence of the strong damping term, we prove that the solution ceases to exists and blows up in finite time. © 2012 Springer Science+Business Media, LLC.
Zuo, Chao; Chen, Qian; Li, Hongru; Qu, Weijuan; Asundi, Anand
2014-07-28
Boundary conditions play a crucial role in the solution of the transport of intensity equation (TIE). If not appropriately handled, they can create significant boundary artifacts across the reconstruction result. In a previous paper [Opt. Express 22, 9220 (2014)], we presented a new boundary-artifact-free TIE phase retrieval method with use of discrete cosine transform (DCT). Here we report its experimental investigations with applications to the micro-optics characterization. The experimental setup is based on a tunable lens based 4f system attached to a non-modified inverted bright-field microscope. We establish inhomogeneous Neumann boundary values by placing a rectangular aperture in the intermediate image plane of the microscope. Then the boundary values are applied to solve the TIE with our DCT-based TIE solver. Experimental results on microlenses highlight the importance of boundary conditions that often overlooked in simplified models, and confirm that our approach effectively avoid the boundary error even when objects are located at the image borders. It is further demonstrated that our technique is non-interferometric, accurate, fast, full-field, and flexible, rendering it a promising metrological tool for the micro-optics inspection.
An Adaptive Pseudospectral Method for Fractional Order Boundary Value Problems
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Mohammad Maleki
2012-01-01
Full Text Available An adaptive pseudospectral method is presented for solving a class of multiterm fractional boundary value problems (FBVP which involve Caputo-type fractional derivatives. The multiterm FBVP is first converted into a singular Volterra integrodifferential equation (SVIDE. By dividing the interval of the problem to subintervals, the unknown function is approximated using a piecewise interpolation polynomial with unknown coefficients which is based on shifted Legendre-Gauss (ShLG collocation points. Then the problem is reduced to a system of algebraic equations, thus greatly simplifying the problem. Further, some additional conditions are considered to maintain the continuity of the approximate solution and its derivatives at the interface of subintervals. In order to convert the singular integrals of SVIDE into nonsingular ones, integration by parts is utilized. In the method developed in this paper, the accuracy can be improved either by increasing the number of subintervals or by increasing the degree of the polynomial on each subinterval. Using several examples including Bagley-Torvik equation the proposed method is shown to be efficient and accurate.
Boundary value problems in time for wave equations on RN
Directory of Open Access Journals (Sweden)
M. W. Smiley
1990-01-01
Full Text Available Let Lλ denote the linear operator associated with the radially symmetric form of the wave operator ∂t2−Δ+λ together with the side conditions of decay to zero as r=‖x‖→+∞ and T-periodicity in time. Thus Lλω=ωtt−(ωrr+N−1rωr+λω, when there are N space variables. For δ,R,T>0 let DT,R=(0,T×(R,+∞ and Lδ2(D denote the weighted L2 space with weight function exp(δr. It is shown that Lλ is a Fredholm operator from dom(Lλ⊂L2(D onto Lδ2(D with non-negative index depending on λ. If [2πj/T]2<λ≤[2π(j+1/T]2 then the index is 2j+1. In addition it is shown that Lλ has a bounded partial inverse Kλ:Lδ2(D→Hδ1(D⋂Lδ∞(D, with all spaces weighted by the function exp(δr. This provides a key ingredient for the analysis of nonlinear problems via the method of alternative problems.
Initial-Boundary Value Problem Solution of the Nonlinear Shallow-water Wave Equations
Kanoglu, U.; Aydin, B.
2014-12-01
The hodograph transformation solutions of the one-dimensional nonlinear shallow-water wave (NSW) equations are usually obtained through integral transform techniques such as Fourier-Bessel transforms. However, the original formulation of Carrier and Greenspan (1958 J Fluid Mech) and its variant Carrier et al. (2003 J Fluid Mech) involve evaluation integrals. Since elliptic integrals are highly singular as discussed in Carrier et al. (2003), this solution methodology requires either approximation of the associated integrands by smooth functions or selection of regular initial/boundary data. It should be noted that Kanoglu (2004 J Fluid Mech) partly resolves this issue by simplifying the resulting integrals in closed form. Here, the hodograph transform approach is coupled with the classical eigenfunction expansion method rather than integral transform techniques and a new analytical model for nonlinear long wave propagation over a plane beach is derived. This approach is based on the solution methodology used in Aydın & Kanoglu (2007 CMES-Comp Model Eng) for wind set-down relaxation problem. In contrast to classical initial- or boundary-value problem solutions, here, the NSW equations are formulated to yield an initial-boundary value problem (IBVP) solution. In general, initial wave profile with nonzero initial velocity distribution is assumed and the flow variables are given in the form of Fourier-Bessel series. The results reveal that the developed method allows accurate estimation of the spatial and temporal variation of the flow quantities, i.e., free-surface height and depth-averaged velocity, with much less computational effort compared to the integral transform techniques such as Carrier et al. (2003), Kanoglu (2004), Tinti & Tonini (2005 J Fluid Mech), and Kanoglu & Synolakis (2006 Phys Rev Lett). Acknowledgments: This work is funded by project ASTARTE- Assessment, STrategy And Risk Reduction for Tsunamis in Europe. Grant 603839, 7th FP (ENV.2013.6.4-3 ENV
A modified SOR method for the Poisson equation in unsteady free-surface flow calculations.
Botta, E.F.F.; Ellenbroek, Marcellinus Hermannus Maria
1985-01-01
Convergence difficulties that sometimes occur if the successive overrelaxation (SOR) method is applied to the Poisson equation on a region with irregular free boundaries are analyzed. It is shown that these difficulties are related to the treatment of the free boundaries and caused by the appearance
A asymptotic numerical method for the steady-state convection diffusion equation
International Nuclear Information System (INIS)
Wu Qiguang
1988-01-01
In this paper, A asymptotic numerical method for the steady-state Convection diffusion equation is proposed, which need not take very fine mesh size in the neighbourhood of the boundary layer. Numerical computation for model problem show that we can obtain the numerical solution in the boundary layer with moderate step size
Savoye, Philippe
2009-01-01
In recent years, I started covering difference equations and z transform methods in my introductory differential equations course. This allowed my students to extend the "classical" methods for (ordinary differential equation) ODE's to discrete time problems arising in many applications.
Directory of Open Access Journals (Sweden)
Lingju Kong
2013-04-01
Full Text Available We study the existence of multiple solutions to the boundary value problem $$displaylines{ frac{d}{dt}Big(frac12{}_0D_t^{-eta}(u'(t+frac12{}_tD_T^{-eta}(u'(t Big+lambda abla F(t,u(t=0,quad tin [0,T],cr u(0=u(T=0, }$$ where $T>0$, $lambda>0$ is a parameter, $0leqeta<1$, ${}_0D_t^{-eta}$ and ${}_tD_T^{-eta}$ are, respectively, the left and right Riemann-Liouville fractional integrals of order $eta$, $F: [0,T]imesmathbb{R}^Nomathbb{R}$ is a given function. Our interest in the above system arises from studying the steady fractional advection dispersion equation. By applying variational methods, we obtain sufficient conditions under which the above equation has at least three solutions. Our results are new even for the special case when $eta=0$. Examples are provided to illustrate the applicability of our results.
Analytic solution of boundary-value problems for nonstationary model kinetic equations
International Nuclear Information System (INIS)
Latyshev, A.V.; Yushkanov, A.A.
1993-01-01
A theory for constructing the solutions of boundary-value problems for non-stationary model kinetic equations is constructed. This theory was incorrectly presented equation, separation of the variables is used, this leading to a characteristic equation. Eigenfunctions are found in the space of generalized functions, and the eigenvalue spectrum is investigated. An existence and uniqueness theorem for the expansion of the Laplace transform of the solution with respect to the eigenfunctions is proved. The proof is constructive and gives explicit expressions for the expansion coefficients. An application to the Rayleigh problem is obtained, and the corresponding result of Cercignani is corrected
Czech Academy of Sciences Publication Activity Database
Mukhigulashvili, Sulkhan
-, č. 35 (2015), s. 23-50 ISSN 1126-8042 Institutional support: RVO:67985840 Keywords : higher order functional differential equations * Dirichlet boundary value problem * strong singularity Subject RIV: BA - General Mathematics http://ijpam.uniud.it/online_issue/201535/03-Mukhigulashvili.pdf
Directory of Open Access Journals (Sweden)
Tengfei Shen
2015-12-01
Full Text Available This paper deals with the multiplicity of solutions for Dirichlet boundary conditions of second-order quasilinear equations with impulsive effects. By using critical point theory, a new result is obtained. An example is given to illustrate the main result.
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Araz R. Aliev
2013-10-01
Full Text Available We study a third-order operator-differential equation on the semi-axis with a discontinuous coefficient and boundary conditions which include an abstract linear operator. Sufficient conditions for the well-posed and unique solvability are found by means of properties of the operator coefficients in a Sobolev-type space.
BOUNDARY VALUE PROBLEM FOR A LOADED EQUATION ELLIPTIC-HYPERBOLIC TYPE IN A DOUBLY CONNECTED DOMAIN
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O.Kh. Abdullaev
2014-06-01
Full Text Available We study the existence and uniqueness of the solution of one boundary value problem for the loaded elliptic-hyperbolic equation of the second order with two lines of change of type in double-connected domain. Similar results have been received by D.M.Kuryhazov, when investigated domain is one-connected.
Czech Academy of Sciences Publication Activity Database
Lomtatidze, Alexander; Vodstrčil, Petr
2005-01-01
Roč. 84, č. 2 (2005), s. 197-209 ISSN 0003-6811 Institutional research plan: CEZ:AV0Z10190503 Keywords : second order linear functional differential equations * nonnegative solution * two-point boundary value problem Subject RIV: BA - General Mathematics http://www.tandfonline.com/doi/full/10.1080/00036810410001724427
M. Denche; A. L. Marhoune
2003-01-01
In this paper, we study a mixed problem with integral boundary conditions for a high order partial differential equation of mixed type. We prove the existence and uniqueness of the solution. The proof is based on energy inequality, and on the density of the range of the operator generated by the considered problem.
Initial boundary value problems of nonlinear wave equations in an exterior domain
International Nuclear Information System (INIS)
Chen Yunmei.
1987-06-01
In this paper, we investigate the existence and uniqueness of the global solutions to the initial boundary value problems of nonlinear wave equations in an exterior domain. When the space dimension n >= 3, the unique global solution of the above problem is obtained for small initial data, even if the nonlinear term is fully nonlinear and contains the unknown function itself. (author). 10 refs
DEFF Research Database (Denmark)
Backi, Christoph Josef; Bendtsen, Jan Dimon; Leth, John
2015-01-01
In this work the stability properties of a partial differential equation (PDE) with state-dependent parameters and asymmetric boundary conditions are investigated. The PDE describes the temperature distribution inside foodstuff, but can also hold for other applications and phenomena. We show...
Czech Academy of Sciences Publication Activity Database
Lomtatidze, Alexander
2016-01-01
Roč. 67, č. 1 (2016), s. 1-129 ISSN 1512-0015 Institutional support: RVO:67985840 Keywords : periodic boundary value problem * positive solution * singular equation Subject RIV: BA - General Mathematics http://rmi.tsu.ge/jeomj/memoirs/vol67/abs67-1.htm
Czech Academy of Sciences Publication Activity Database
Mukhigulashvili, Sulkhan; Půža, B.
2015-01-01
Roč. 2015, January (2015), s. 17 ISSN 1687-2770 Institutional support: RVO:67985840 Keywords : higher order nonlinear functional-differential equations * two-point right-focal boundary value problem * strong singularity Subject RIV: BA - General Mathematics Impact factor: 0.642, year: 2015 http://link.springer.com/article/10.1186%2Fs13661-014-0277-1
Remark on periodic boundary-value problem for second-order linear ordinary differential equations
Czech Academy of Sciences Publication Activity Database
Dosoudilová, M.; Lomtatidze, Alexander
2018-01-01
Roč. 2018, č. 13 (2018), s. 1-7 ISSN 1072-6691 Institutional support: RVO:67985840 Keywords : second-order linear equation * periodic boundary value problem * unique solvability Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics Impact factor: 0.954, year: 2016 https://ejde.math.txstate.edu/Volumes/2018/13/abstr.html
Energy Technology Data Exchange (ETDEWEB)
Chen, Ke [Univ. of Liverpool (United Kingdom)
1996-12-31
We study various preconditioning techniques for the iterative solution of boundary integral equations, and aim to provide a theory for a class of sparse preconditioners. Two related ideas are explored here: singularity separation and inverse approximation. Our preliminary conclusion is that singularity separation based preconditioners perform better than approximate inverse based while it is desirable to have both features.
Hybrid immersed interface-immersed boundary methods for AC dielectrophoresis
International Nuclear Information System (INIS)
Hossan, Mohammad Robiul; Dillon, Robert; Dutta, Prashanta
2014-01-01
Dielectrophoresis, a nonlinear electrokinetic transport mechanism, has become popular in many engineering applications including manipulation, characterization and actuation of biomaterials, particles and biological cells. In this paper, we present a hybrid immersed interface–immersed boundary method to study AC dielectrophoresis where an algorithm is developed to solve the complex Poisson equation using a real variable formulation. An immersed interface method is employed to obtain the AC electric field in a fluid media with suspended particles and an immersed boundary method is used for the fluid equations and particle transport. The convergence of the proposed algorithm as well as validation of the hybrid scheme with experimental results is presented. In this paper, the Maxwell stress tensor is used to calculate the dielectrophoretic force acting on particles by considering the physical effect of particles in the computational domain. Thus, this study eliminates the approximations used in point dipole methods for calculating dielectrophoretic force. A comparative study between Maxwell stress tensor and point dipole methods for computing dielectrophoretic forces are presented. The hybrid method is used to investigate the physics of dielectrophoresis in microfluidic devices using an AC electric field. The numerical results show that with proper design and appropriate selection of applied potential and frequency, global electric field minima can be obtained to facilitate multiple particle trapping by exploiting the mechanism of negative dielectrophoresis. Our numerical results also show that electrically neutral particles form a chain parallel to the applied electric field irrespective of their initial orientation when an AC electric field is applied. This proposed hybrid numerical scheme will help to better understand dielectrophoresis and to design and optimize microfluidic devices
Attractor of Beam Equation with Structural Damping under Nonlinear Boundary Conditions
Directory of Open Access Journals (Sweden)
Danxia Wang
2015-01-01
Full Text Available Simultaneously, considering the viscous effect of material, damping of medium, and rotational inertia, we study a kind of more general Kirchhoff-type extensible beam equation utt-uxxtt+uxxxx-σ(∫0l(ux2dxuxx-ϕ(∫0l(ux2dxuxxt=q(x, in [0,L]×R+ with the structural damping and the rotational inertia term. Little attention is paid to the longtime behavior of the beam equation under nonlinear boundary conditions. In this paper, under nonlinear boundary conditions, we prove not only the existence and uniqueness of global solutions by prior estimates combined with some inequality skills, but also the existence of a global attractor by the existence of an absorbing set and asymptotic compactness of corresponding solution semigroup. In addition, the same results also can be proved under the other nonlinear boundary conditions.
The Dirichlet problem with L2-boundary data for elliptic linear equations
Chabrowski, Jan
1991-01-01
The Dirichlet problem has a very long history in mathematics and its importance in partial differential equations, harmonic analysis, potential theory and the applied sciences is well-known. In the last decade the Dirichlet problem with L2-boundary data has attracted the attention of several mathematicians. The significant features of this recent research are the use of weighted Sobolev spaces, existence results for elliptic equations under very weak regularity assumptions on coefficients, energy estimates involving L2-norm of a boundary data and the construction of a space larger than the usual Sobolev space W1,2 such that every L2-function on the boundary of a given set is the trace of a suitable element of this space. The book gives a concise account of main aspects of these recent developments and is intended for researchers and graduate students. Some basic knowledge of Sobolev spaces and measure theory is required.
Structural equation modeling methods and applications
Wang, Jichuan
2012-01-01
A reference guide for applications of SEM using Mplus Structural Equation Modeling: Applications Using Mplus is intended as both a teaching resource and a reference guide. Written in non-mathematical terms, this book focuses on the conceptual and practical aspects of Structural Equation Modeling (SEM). Basic concepts and examples of various SEM models are demonstrated along with recently developed advanced methods, such as mixture modeling and model-based power analysis and sample size estimate for SEM. The statistical modeling program, Mplus, is also featured and provides researchers with a
Explicitly represented polygon wall boundary model for the explicit MPS method
Mitsume, Naoto; Yoshimura, Shinobu; Murotani, Kohei; Yamada, Tomonori
2015-05-01
This study presents an accurate and robust boundary model, the explicitly represented polygon (ERP) wall boundary model, to treat arbitrarily shaped wall boundaries in the explicit moving particle simulation (E-MPS) method, which is a mesh-free particle method for strong form partial differential equations. The ERP model expresses wall boundaries as polygons, which are explicitly represented without using the distance function. These are derived so that for viscous fluids, and with less computational cost, they satisfy the Neumann boundary condition for the pressure and the slip/no-slip condition on the wall surface. The proposed model is verified and validated by comparing computed results with the theoretical solution, results obtained by other models, and experimental results. Two simulations with complex boundary movements are conducted to demonstrate the applicability of the E-MPS method to the ERP model.
Simple equation method for nonlinear partial differential equations and its applications
Directory of Open Access Journals (Sweden)
Taher A. Nofal
2016-04-01
Full Text Available In this article, we focus on the exact solution of the some nonlinear partial differential equations (NLPDEs such as, Kodomtsev–Petviashvili (KP equation, the (2 + 1-dimensional breaking soliton equation and the modified generalized Vakhnenko equation by using the simple equation method. In the simple equation method the trial condition is the Bernoulli equation or the Riccati equation. It has been shown that the method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering problems.
Variational formulation and projectional methods for the second order transport equation
International Nuclear Information System (INIS)
Borysiewicz, M.; Stankiewicz, R.
1979-01-01
Herein the variational problem for a second-order boundary value problem for the neutron transport equation is formulated. The projectional methods solving the problem are examined. The approach is compared with that based on the original untransformed form of the neutron transport equation
Integral Method of Boundary Characteristics: Neumann Condition
Kot, V. A.
2018-05-01
A new algorithm, based on systems of identical equalities with integral and differential boundary characteristics, is proposed for solving boundary-value problems on the heat conduction in bodies canonical in shape at a Neumann boundary condition. Results of a numerical analysis of the accuracy of solving heat-conduction problems with variable boundary conditions with the use of this algorithm are presented. The solutions obtained with it can be considered as exact because their errors comprise hundredths and ten-thousandths of a persent for a wide range of change in the parameters of a problem.
Parallel Fast Multipole Boundary Element Method for crustal dynamics
International Nuclear Information System (INIS)
Quevedo, Leonardo; Morra, Gabriele; Mueller, R Dietmar
2010-01-01
Crustal faults and sharp material transitions in the crust are usually represented as triangulated surfaces in structural geological models. The complex range of volumes separating such surfaces is typically three-dimensionally meshed in order to solve equations that describe crustal deformation with the finite-difference (FD) or finite-element (FEM) methods. We show here how the Boundary Element Method, combined with the Multipole approach, can revolutionise the calculation of stress and strain, solving the problem of computational scalability from reservoir to basin scales. The Fast Multipole Boundary Element Method (Fast BEM) tackles the difficulty of handling the intricate volume meshes and high resolution of crustal data that has put classical Finite 3D approaches in a performance crisis. The two main performance enhancements of this method: the reduction of required mesh elements from cubic to quadratic with linear size and linear-logarithmic runtime; achieve a reduction of memory and runtime requirements allowing the treatment of a new scale of geodynamic models. This approach was recently tested and applied in a series of papers by [1, 2, 3] for regional and global geodynamics, using KD trees for fast identification of near and far-field interacting elements, and MPI parallelised code on distributed memory architectures, and is now in active development for crustal dynamics. As the method is based on a free-surface, it allows easy data transfer to geological visualisation tools where only changes in boundaries and material properties are required as input parameters. In addition, easy volume mesh sampling of physical quantities enables direct integration with existing FD/FEM code.
A Hamiltonian-based derivation of Scaled Boundary Finite Element Method for elasticity problems
International Nuclear Information System (INIS)
Hu Zhiqiang; Lin Gao; Wang Yi; Liu Jun
2010-01-01
The Scaled Boundary Finite Method (SBFEM) is a semi-analytical solution approach for solving partial differential equation. For problem in elasticity, the governing equations can be obtained by mechanically based formulation, Scaled-boundary-transformation-based formulation and principle of virtual work. The governing equations are described in the frame of Lagrange system and the unknowns are displacements. But in the solution procedure, the auxiliary variables are introduced and the equations are solved in the state space. Based on the observation that the duality system to solve elastic problem proposed by W.X. Zhong is similar to the above solution approach, the discretization of the SBFEM and the duality system are combined to derive the governing equations in the Hamilton system by introducing the dual variables in this paper. The Precise Integration Method (PIM) used in Duality system is also an efficient method for the solution of the governing equations of SBFEM in displacement and boundary stiffness matrix especially for the case which results some numerical difficulties in the usually uses the eigenvalue method. Numerical examples are used to demonstrate the validity and effectiveness of the PIM for solution of boundary static stiffness.
Hejranfar, Kazem; Parseh, Kaveh
2017-09-01
The preconditioned characteristic boundary conditions based on the artificial compressibility (AC) method are implemented at artificial boundaries for the solution of two- and three-dimensional incompressible viscous flows in the generalized curvilinear coordinates. The compatibility equations and the corresponding characteristic variables (or the Riemann invariants) are mathematically derived and then applied as suitable boundary conditions in a high-order accurate incompressible flow solver. The spatial discretization of the resulting system of equations is carried out by the fourth-order compact finite-difference (FD) scheme. In the preconditioning applied here, the value of AC parameter in the flow field and also at the far-field boundary is automatically calculated based on the local flow conditions to enhance the robustness and performance of the solution algorithm. The code is fully parallelized using the Concurrency Runtime standard and Parallel Patterns Library (PPL) and its performance on a multi-core CPU is analyzed. The incompressible viscous flows around a 2-D circular cylinder, a 2-D NACA0012 airfoil and also a 3-D wavy cylinder are simulated and the accuracy and performance of the preconditioned characteristic boundary conditions applied at the far-field boundaries are evaluated in comparison to the simplified boundary conditions and the non-preconditioned characteristic boundary conditions. It is indicated that the preconditioned characteristic boundary conditions considerably improve the convergence rate of the solution of incompressible flows compared to the other boundary conditions and the computational costs are significantly decreased.
Method of ATMS operators in the formalism of Faddeev equations
International Nuclear Information System (INIS)
Zubarev, D.A.
1991-01-01
The method of ATMS operators is generalized for the case of Faddeev equations. The method to construct effective equations for both elastic scattering and scattering with rearrangement is presented. Properties to obtained equations are considered
International Nuclear Information System (INIS)
Choi, C. Y.
1997-01-01
A geometrical inverse heat conduction problem is solved for the infrared scanning cavity detection by the boundary element method using minimal energy technique. By minimizing the kinetic energy of temperature field, boundary element equations are converted to the quadratic programming problem. A hypothetical inner boundary is defined such that the actual cavity is located interior to the domain. Temperatures at hypothetical inner boundary are determined to meet the constraints of measurement error of surface temperature obtained by infrared scanning, and then boundary element analysis is performed for the position of an unknown boundary (cavity). Cavity detection algorithm is provided, and the effects of minimal energy technique on the inverse solution method are investigated by means of numerical analysis
An efficient strongly coupled immersed boundary method for deforming bodies
Goza, Andres; Colonius, Tim
2016-11-01
Immersed boundary methods treat the fluid and immersed solid with separate domains. As a result, a nonlinear interface constraint must be satisfied when these methods are applied to flow-structure interaction problems. This typically results in a large nonlinear system of equations that is difficult to solve efficiently. Often, this system is solved with a block Gauss-Seidel procedure, which is easy to implement but can require many iterations to converge for small solid-to-fluid mass ratios. Alternatively, a Newton-Raphson procedure can be used to solve the nonlinear system. This typically leads to convergence in a small number of iterations for arbitrary mass ratios, but involves the use of large Jacobian matrices. We present an immersed boundary formulation that, like the Newton-Raphson approach, uses a linearization of the system to perform iterations. It therefore inherits the same favorable convergence behavior. However, we avoid large Jacobian matrices by using a block LU factorization of the linearized system. We derive our method for general deforming surfaces and perform verification on 2D test problems of flow past beams. These test problems involve large amplitude flapping and a wide range of mass ratios. This work was partially supported by the Jet Propulsion Laboratory and Air Force Office of Scientific Research.
Transmission problem for the Laplace equation and the integral equation method
Czech Academy of Sciences Publication Activity Database
Medková, Dagmar
2012-01-01
Roč. 387, č. 2 (2012), s. 837-843 ISSN 0022-247X Institutional research plan: CEZ:AV0Z10190503 Keywords : transmission problem * Laplace equation * boundary integral equation Subject RIV: BA - General Mathematics Impact factor: 1.050, year: 2012 http://www.sciencedirect.com/science/article/pii/S0022247X11008985
Energy Technology Data Exchange (ETDEWEB)
Kaikina, Elena I., E-mail: ekaikina@matmor.unam.mx [Centro de Ciencias Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán (Mexico)
2013-11-15
We consider the inhomogeneous Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation, formulated on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.
International Nuclear Information System (INIS)
Kaikina, Elena I.
2013-01-01
We consider the inhomogeneous Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation, formulated on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time
Discretization errors at free boundaries of the Grad-Schlueter-Shafranov equation
International Nuclear Information System (INIS)
Meyer-Spasche, R.; Fornberg, B.
1990-10-01
The numerical error of standard finite-difference schemes is analyzed at free boundaries of the Grad-Schlueter-Shafranov equation of plasma physics. A simple correction strategy is devised to eliminate (to leading order) the errors which arise as the free boundary crosses the rectangular grid at irregular locations. The resulting scheme can be solved by Gauss-Newton or Inverse iterations, or by multigrid iterations. Extrapolation (from 2nd to 3rd order of accuracy) is possible for the new scheme. (orig.)
On the physical solutions to the heat equation subjected to nonlinear boundary conditions
International Nuclear Information System (INIS)
Gama, R.M.S. da.
1990-01-01
This work consists of a discussion on the physical solutions to the steady-state heat transfer equation, when it is subjected to nonlinear boundary conditions. It will be presented a functional, whose minimum occurs for the (unique) physical solution to the condidered heat transfer problem, suitable for a large class of typical (nonlinear) boundary conditions (representing the radiative/convective loss from the body to the environment). It will be demonstrated that these problems admit-always one, and only one, physical solution (which represents the absolute temperature). (author)
Said-Houari, Belkacem
2012-09-01
The goal of this work is to study a model of the viscoelastic wave equation with nonlinear boundary/interior sources and a nonlinear interior damping. First, applying the Faedo-Galerkin approximations combined with the compactness method to obtain existence of regular global solutions to an auxiliary problem with globally Lipschitz source terms and with initial data in the potential well. It is important to emphasize that it is not possible to consider density arguments to pass from regular to weak solutions if one considers regular solutions of our problem where the source terms are locally Lipschitz functions. To overcome this difficulty, we use an approximation method involving truncated sources and adapting the ideas in [13] to show that the existence of weak solutions can still be obtained for our problem. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term, then the solution ceases to exist and blows up in finite time provided that the initial data are large enough.
Said-Houari, Belkacem; Nascimento, Flá vio A Falcã o
2012-01-01
The goal of this work is to study a model of the viscoelastic wave equation with nonlinear boundary/interior sources and a nonlinear interior damping. First, applying the Faedo-Galerkin approximations combined with the compactness method to obtain existence of regular global solutions to an auxiliary problem with globally Lipschitz source terms and with initial data in the potential well. It is important to emphasize that it is not possible to consider density arguments to pass from regular to weak solutions if one considers regular solutions of our problem where the source terms are locally Lipschitz functions. To overcome this difficulty, we use an approximation method involving truncated sources and adapting the ideas in [13] to show that the existence of weak solutions can still be obtained for our problem. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term, then the solution ceases to exist and blows up in finite time provided that the initial data are large enough.
Bardhan, Jaydeep P
2008-10-14
The importance of molecular electrostatic interactions in aqueous solution has motivated extensive research into physical models and numerical methods for their estimation. The computational costs associated with simulations that include many explicit water molecules have driven the development of implicit-solvent models, with generalized-Born (GB) models among the most popular of these. In this paper, we analyze a boundary-integral equation interpretation for the Coulomb-field approximation (CFA), which plays a central role in most GB models. This interpretation offers new insights into the nature of the CFA, which traditionally has been assessed using only a single point charge in the solute. The boundary-integral interpretation of the CFA allows the use of multiple point charges, or even continuous charge distributions, leading naturally to methods that eliminate the interpolation inaccuracies associated with the Still equation. This approach, which we call boundary-integral-based electrostatic estimation by the CFA (BIBEE/CFA), is most accurate when the molecular charge distribution generates a smooth normal displacement field at the solute-solvent boundary, and CFA-based GB methods perform similarly. Conversely, both methods are least accurate for charge distributions that give rise to rapidly varying or highly localized normal displacement fields. Supporting this analysis are comparisons of the reaction-potential matrices calculated using GB methods and boundary-element-method (BEM) simulations. An approximation similar to BIBEE/CFA exhibits complementary behavior, with superior accuracy for charge distributions that generate rapidly varying normal fields and poorer accuracy for distributions that produce smooth fields. This approximation, BIBEE by preconditioning (BIBEE/P), essentially generates initial guesses for preconditioned Krylov-subspace iterative BEMs. Thus, iterative refinement of the BIBEE/P results recovers the BEM solution; excellent agreement
A boundary-fitted staggered difference method for incompressible flow using Riemann geometry
International Nuclear Information System (INIS)
Koshizuka, Seiichi; Kondo, Shunsuke; Oka, Yoshiaki.
1990-01-01
A boundary-fitted staggered difference method (BFSDM) is investigated for incompressible flow in nuclear plants. BFSDM employs control cells for scalars, staggered location of velocity components, and integrated formulation of div=0. Governing equations are written as coordinate-free forms using Riemann geometry. Flow velocity is represented with contravariant physical components in the present method. Connection terms emerge as source terms in the coordinate-free governing equations. These terms are studied from the viewpoints of physical meaning, numerical stability, and conservative property. Some flows on a round or slant boundary are solved using boundary-fitted curvilinear (BFC) grids and rectangular grids to compare the present method and the rectangular-type (R-type) staggered difference method (SDM). Supercomputing of the present method, including vector processing, is also discussed compared with the R-type method. (author)
Application of the B-Determining Equations Method to One Problem of Free Turbulence
Directory of Open Access Journals (Sweden)
Oleg V. Kaptsov
2012-10-01
Full Text Available A three-dimensional model of the far turbulent wake behind a self-propelled body in a passively stratified medium is considered. The model is reduced to a system of ordinary differential equations by a similarity reduction and the B-determining equations method. The system of ordinary differential equations satisfying natural boundary conditions is solved numerically. The solutions obtained here are in close agreement with experimental data.
Conference on Boundary and Interior Layers : Computational and Asymptotic Methods
Stynes, Martin; Zhang, Zhimin
2017-01-01
This volume collects papers associated with lectures that were presented at the BAIL 2016 conference, which was held from 14 to 19 August 2016 at Beijing Computational Science Research Center and Tsinghua University in Beijing, China. It showcases the variety and quality of current research into numerical and asymptotic methods for theoretical and practical problems whose solutions involve layer phenomena. The BAIL (Boundary And Interior Layers) conferences, held usually in even-numbered years, bring together mathematicians and engineers/physicists whose research involves layer phenomena, with the aim of promoting interaction between these often-separate disciplines. These layers appear as solutions of singularly perturbed differential equations of various types, and are common in physical problems, most notably in fluid dynamics. This book is of interest for current researchers from mathematics, engineering and physics whose work involves the accurate app roximation of solutions of singularly perturbed diffe...
Variational derivation of the simplified P2 equations with boundary and interface conditions
International Nuclear Information System (INIS)
Tomasevic, D.I.; Larsen, E.W.
1995-01-01
The Simplified P 2 (SP 2 ) approximation to the transport equation is derived using a variational principle. The variational analysis yields the SP 2 equations, together with interface and Marshak-like boundary conditions. Numerical calculations show that for problems in which the P 1 solution is a reasonably accurate approximation to the transport solution, the corresponding SP 2 Solution is generally more accurate than the P 1 solution, for calculating integral quantities and detailed flux distributions, except in the close vicinity of material interfaces, where the SP 2 solution is discontinuous
Muskhelishvili, N I
2011-01-01
Singular integral equations play important roles in physics and theoretical mechanics, particularly in the areas of elasticity, aerodynamics, and unsteady aerofoil theory. They are highly effective in solving boundary problems occurring in the theory of functions of a complex variable, potential theory, the theory of elasticity, and the theory of fluid mechanics.This high-level treatment by a noted mathematician considers one-dimensional singular integral equations involving Cauchy principal values. Its coverage includes such topics as the Hölder condition, Hilbert and Riemann-Hilbert problem
Iterative observer based method for source localization problem for Poisson equation in 3D
Majeed, Muhammad Usman
2017-07-10
A state-observer based method is developed to solve point source localization problem for Poisson equation in a 3D rectangular prism with available boundary data. The technique requires a weighted sum of solutions of multiple boundary data estimation problems for Laplace equation over the 3D domain. The solution of each of these boundary estimation problems involves writing down the mathematical problem in state-space-like representation using one of the space variables as time-like. First, system observability result for 3D boundary estimation problem is recalled in an infinite dimensional setting. Then, based on the observability result, the boundary estimation problem is decomposed into a set of independent 2D sub-problems. These 2D problems are then solved using an iterative observer to obtain the solution. Theoretical results are provided. The method is implemented numerically using finite difference discretization schemes. Numerical illustrations along with simulation results are provided.
International Nuclear Information System (INIS)
Tao Ganqiang; Yu Qing; Xiao Xiao
2011-01-01
Viscous and incompressible fluid flow is important for numerous engineering mechanics problems. Because of high non linear and incompressibility for Navier-Stokes equation, it is very difficult to solve Navier-Stokes equation by numerical method. According to its characters of Navier-Stokes equation, quartic derivation controlling equation of the two dimensional incompressible Navier-Stokes equation is set up firstly. The method solves the problem for dealing with vorticity boundary and automatically meets incompressibility condition. Then Finite Element equation for Navier-Stokes equation is proposed by using quadratic quadrilateral unit with 8 nodes in which the unit function is quadratic and non linear.-Based on it, the Finite Element program of quadratic quadrilateral unit with 8 nodes is developed. Lastly, numerical experiment proves the accuracy and dependability of the method and also shows the method has good application prospect in computational fluid mechanics. (authors)
High-order finite-difference methods for Poisson's equation
van Linde, Hendrik Jan
1971-01-01
In this thesis finite-difference approximations to the three boundary value problems for Poisson’s equation are given, with discretization errors of O(H^3) for the mixed boundary value problem, O(H^3 |ln(h)| for the Neumann problem and O(H^4)for the Dirichlet problem respectively . First an operator
Holst, Michael; Meier, Caleb; Tsogtgerel, G.
2018-01-01
In this article we continue our effort to do a systematic development of the solution theory for conformal formulations of the Einstein constraint equations on compact manifolds with boundary. By building in a natural way on our recent work in Holst and Tsogtgerel (Class Quantum Gravity 30:205011, 2013), and Holst et al. (Phys Rev Lett 100(16):161101, 2008, Commun Math Phys 288(2):547-613, 2009), and also on the work of Maxwell (J Hyperbolic Differ Eqs 2(2):521-546, 2005a, Commun Math Phys 253(3):561-583, 2005b, Math Res Lett 16(4):627-645, 2009) and Dain (Class Quantum Gravity 21(2):555-573, 2004), under reasonable assumptions on the data we prove existence of both near- and far-from-constant mean curvature (CMC) solutions for a class of Robin boundary conditions commonly used in the literature for modeling black holes, with a third existence result for CMC appearing as a special case. Dain and Maxwell addressed initial data engineering for space-times that evolve to contain black holes, determining solutions to the conformal formulation on an asymptotically Euclidean manifold in the CMC setting, with interior boundary conditions representing excised interior black hole regions. Holst and Tsogtgerel compiled the interior boundary results covered by Dain and Maxwell, and then developed general interior conditions to model the apparent horizon boundary conditions of Dainand Maxwell for compact manifolds with boundary, and subsequently proved existence of solutions to the Lichnerowicz equation on compact manifolds with such boundary conditions. This paper picks up where Holst and Tsogtgerel left off, addressing the general non-CMC case for compact manifolds with boundary. As in our previous articles, our focus here is again on low regularity data and on the interaction between different types of boundary conditions. While our work here serves primarily to extend the solution theory for the compact with boundary case, we also develop several technical tools that have
Energy Technology Data Exchange (ETDEWEB)
Aarao, J; Bradshaw-Hajek, B H; Miklavcic, S J; Ward, D A, E-mail: Stan.Miklavcic@unisa.edu.a [School of Mathematics and Statistics, University of South Australia, Mawson Lakes, SA 5095 (Australia)
2010-05-07
Standard analytical solutions to elliptic boundary value problems on asymmetric domains are rarely, if ever, obtainable. In this paper, we propose a solution technique wherein we embed the original domain into one with simple boundaries where the classical eigenfunction solution approach can be used. The solution in the larger domain, when restricted to the original domain, is then the solution of the original boundary value problem. We call this the extended-domain-eigenfunction method. To illustrate the method's strength and scope, we apply it to Laplace's equation on an annular-like domain.
Efimova, Olga Yu.
2010-01-01
The modification of simplest equation method to look for exact solutions of nonlinear partial differential equations is presented. Using this method we obtain exact solutions of generalized Korteweg-de Vries equation with cubic source and exact solutions of third-order Kudryashov-Sinelshchikov equation describing nonlinear waves in liquids with gas bubbles.
Transient Growth Analysis of Compressible Boundary Layers with Parabolized Stability Equations
Paredes, Pedro; Choudhari, Meelan M.; Li, Fei; Chang, Chau-Lyan
2016-01-01
The linear form of parabolized linear stability equations (PSE) is used in a variational approach to extend the previous body of results for the optimal, non-modal disturbance growth in boundary layer flows. This methodology includes the non-parallel effects associated with the spatial development of boundary layer flows. As noted in literature, the optimal initial disturbances correspond to steady counter-rotating stream-wise vortices, which subsequently lead to the formation of stream-wise-elongated structures, i.e., streaks, via a lift-up effect. The parameter space for optimal growth is extended to the hypersonic Mach number regime without any high enthalpy effects, and the effect of wall cooling is studied with particular emphasis on the role of the initial disturbance location and the value of the span-wise wavenumber that leads to the maximum energy growth up to a specified location. Unlike previous predictions that used a basic state obtained from a self-similar solution to the boundary layer equations, mean flow solutions based on the full Navier-Stokes (NS) equations are used in select cases to help account for the viscous-inviscid interaction near the leading edge of the plate and also for the weak shock wave emanating from that region. These differences in the base flow lead to an increasing reduction with Mach number in the magnitude of optimal growth relative to the predictions based on self-similar mean-flow approximation. Finally, the maximum optimal energy gain for the favorable pressure gradient boundary layer near a planar stagnation point is found to be substantially weaker than that in a zero pressure gradient Blasius boundary layer.
On Riemann boundary value problems for null solutions of the two dimensional Helmholtz equation
Bory Reyes, Juan; Abreu Blaya, Ricardo; Rodríguez Dagnino, Ramón Martin; Kats, Boris Aleksandrovich
2018-01-01
The Riemann boundary value problem (RBVP to shorten notation) in the complex plane, for different classes of functions and curves, is still widely used in mathematical physics and engineering. For instance, in elasticity theory, hydro and aerodynamics, shell theory, quantum mechanics, theory of orthogonal polynomials, and so on. In this paper, we present an appropriate hyperholomorphic approach to the RBVP associated to the two dimensional Helmholtz equation in R^2 . Our analysis is based on a suitable operator calculus.
Discrete maximum principle for Poisson equation with mixed boundary conditions solved by hp-FEM
Czech Academy of Sciences Publication Activity Database
Vejchodský, Tomáš; Šolín, P.
2009-01-01
Roč. 1, č. 2 (2009), s. 201-214 ISSN 2070-0733 R&D Projects: GA AV ČR IAA100760702; GA ČR(CZ) GA102/07/0496; GA ČR GA102/05/0629 Institutional research plan: CEZ:AV0Z10190503 Keywords : discrete maximum principle * hp-FEM * Poisson equation * mixed boundary conditions Subject RIV: BA - General Mathematics
International Nuclear Information System (INIS)
Makhan'kov, V.G.; Slavov, S.I.
1989-01-01
Vector nonlinear Schroedinger equations (VS3) is investigated under quasi-constant boundary conditions. New two-soliton solutions are obtained with such non-trivial dynamics that they may be called the breather solutions. A version of the basic Novikov-Dubrovin-Krichever algebro-geometrical approach is applied to obtain breather like solutions existing for all types of internal symmetry is specified are formulated in terms of the soliton velocity expressed via the parameters of the problem. 4 refs
International Nuclear Information System (INIS)
Karimov, Ruslan Kh; Kozhevnikova, Larisa M
2010-01-01
The first mixed problem with homogeneous Dirichlet boundary condition and initial function with compact support is considered for quasilinear second order parabolic equations in a cylindrical domain D=(0,∞)xΩ. Upper bounds are obtained, which give the rate of decay of the solutions as t→∞ as a function of the geometry of the unbounded domain Ω subset of R n , n≥2. Bibliography: 18 titles.
Multi-point boundary value problems for linear functional-differential equations
Czech Academy of Sciences Publication Activity Database
Domoshnitsky, A.; Hakl, Robert; Půža, Bedřich
2017-01-01
Roč. 24, č. 2 (2017), s. 193-206 ISSN 1072-947X Institutional support: RVO:67985840 Keywords : boundary value problems * linear functional- differential equations * functional- differential inequalities Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics Impact factor: 0.290, year: 2016 https://www.degruyter.com/view/j/gmj.2017.24.issue-2/gmj-2016-0076/gmj-2016-0076.xml
Directory of Open Access Journals (Sweden)
Domoshnitsky Alexander
2009-01-01
Full Text Available We obtain the maximum principles for the first-order neutral functional differential equation where , and are linear continuous operators, and are positive operators, is the space of continuous functions, and is the space of essentially bounded functions defined on . New tests on positivity of the Cauchy function and its derivative are proposed. Results on existence and uniqueness of solutions for various boundary value problems are obtained on the basis of the maximum principles.
Moving-boundary problems for the time-fractional diffusion equation
Directory of Open Access Journals (Sweden)
Sabrina D. Roscani
2017-02-01
Full Text Available We consider a one-dimensional moving-boundary problem for the time-fractional diffusion equation. The time-fractional derivative of order $\\alpha\\in (0,1$ is taken in the sense of Caputo. We study the asymptotic behaivor, as t tends to infinity, of a general solution by using a fractional weak maximum principle. Also, we give some particular exact solutions in terms of Wright functions.
Multi-point boundary value problems for linear functional-differential equations
Czech Academy of Sciences Publication Activity Database
Domoshnitsky, A.; Hakl, Robert; Půža, Bedřich
2017-01-01
Roč. 24, č. 2 (2017), s. 193-206 ISSN 1072-947X Institutional support: RVO:67985840 Keywords : boundary value problems * linear functional-differential equations * functional-differential inequalities Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics Impact factor: 0.290, year: 2016 https://www.degruyter.com/view/j/gmj.2017.24.issue-2/gmj-2016-0076/gmj-2016-0076. xml
An immersed-boundary method for conjugate heat transfer analysis
Energy Technology Data Exchange (ETDEWEB)
Song, Jeong Chul; Lee, Joon Sik [Seoul National University, Seoul (Korea, Republic of); Ahn, Joon [Kookmin University, Seoul (Korea, Republic of)
2017-05-15
An immersed-boundary method is proposed for the analysis of conjugate problems of convective heat transfer in conducting solids. In- side the solid body, momentum forcing is applied to set the velocity to zero. A thermal conductivity ratio and a heat capacity ratio, between the solid body and the fluid, are introduced so that the energy equation is reduced to the heat diffusion equation. At the solid fluid interface, an effective conductivity is introduced to satisfy the heat flux continuity. The effective thermal conductivity is obtained by considering the heat balance at the interface or by using a harmonic mean formulation. The method is first validated against the analytic solution to the heat transfer problem in a fully developed laminar channel flow with conducting solid walls. Then it is applied to a laminar channel flow with a heated, block-shaped obstacle to show its validity for geometry with sharp edges. Finally the validation for a curvilinear solid body is accomplished with a laminar flow through arrayed cylinders.
Numerical solution of sixth-order boundary-value problems using Legendre wavelet collocation method
Sohaib, Muhammad; Haq, Sirajul; Mukhtar, Safyan; Khan, Imad
2018-03-01
An efficient method is proposed to approximate sixth order boundary value problems. The proposed method is based on Legendre wavelet in which Legendre polynomial is used. The mechanism of the method is to use collocation points that converts the differential equation into a system of algebraic equations. For validation two test problems are discussed. The results obtained from proposed method are quite accurate, also close to exact solution, and other different methods. The proposed method is computationally more effective and leads to more accurate results as compared to other methods from literature.
Boundary Layers for the Navier-Stokes Equations Linearized Around a Stationary Euler Flow
Gie, Gung-Min; Kelliher, James P.; Mazzucato, Anna L.
2018-03-01
We study the viscous boundary layer that forms at small viscosity near a rigid wall for the solution to the Navier-Stokes equations linearized around a smooth and stationary Euler flow (LNSE for short) in a smooth bounded domain Ω \\subset R^3 under no-slip boundary conditions. LNSE is supplemented with smooth initial data and smooth external forcing, assumed ill-prepared, that is, not compatible with the no-slip boundary condition. We construct an approximate solution to LNSE on the time interval [0, T], 0Math J 45(3):863-916, 1996), Xin and Yanagisawa (Commun Pure Appl Math 52(4):479-541, 1999), and Gie (Commun Math Sci 12(2):383-400, 2014).
Analytic solutions to a family of boundary-value problems for Ginsburg-Landau type equations
Vassilev, V. M.; Dantchev, D. M.; Djondjorov, P. A.
2017-10-01
We consider a two-parameter family of nonlinear ordinary differential equations describing the behavior of a critical thermodynamic system, e.g., a binary liquid mixture, of film geometry within the framework of the Ginzburg-Landau theory by means of the order-parameter. We focus on the case in which the confining surfaces are strongly adsorbing but prefer different components of the mixture, i.e., the order-parameter tends to infinity at one of the boundaries and to minus infinity at the other one. We assume that the boundaries of the system are positioned at a finite distance from each other and give analytic solutions to the corresponding boundary-value problems in terms of Weierstrass and Jacobi elliptic functions.
''Free-space'' boundary conditions for the time-dependent wave equation
International Nuclear Information System (INIS)
Lindman, E.L.
1975-01-01
Boundary conditions for the discrete wave equation which act like an infinite region of free space in contact with the computational region can be constructed using projection operators. Propagating and evanescent waves coming from within the computational region generate no reflected waves as they cross the boundary. At the same time arbitrary waves may be launched into the computational region. Well known projection operators for one-dimensional waves may be used for this purpose in one dimension. Extensions of these operators to higher dimensions along with numerically efficient approximations to them are described for higher-dimensional problems. The separation of waves into ingoing and outgoing waves inherent in these boundary conditions greatly facilitates diagnostics
Energy Technology Data Exchange (ETDEWEB)
Bazalii, B V; Degtyarev, S P [Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk (Ukraine)
2013-07-31
An elliptic boundary-value problem for second-order equations with nonnegative characteristic form is investigated in the situation when there is a weak degeneracy on the boundary of the domain. A priori estimates are obtained for solutions and the problem is proved to be solvable in some weighted Hölder spaces. Bibliography: 18 titles.
International Nuclear Information System (INIS)
Adams, J.; Pneuman, G.W.
1976-01-01
A new method for computing potential magnetic field configurations in the solar atmosphere is described. A discrete approximation to Laplace's equation is solved in the domain R(Sun) 1 , 0 1 being an arbitrary radial distance from the solar center). The method utilizes the measured line-of-sight magnetic fields directly as the boundary condition at the solar surface and constrains the field to become radial at the outer boundary, R 1 . First the differential equation and boundary conditions are reduced to a set of two-dimensional equations in r, theta by Fourier transforming out the periodic phi dependence. Next each transformed boundary condition is converted to a Dirichlet surface condition. Then each two-dimensional equation with standard Dirichlet-Dirichlet boundary conditions is solved for the Fourier coefficient it determines. Finally, the solution of the original three dimensional equation is obtained through inverse Fourier transformation. The primary numerical tools in this technique are the use of a finite fast Fourier transform technique and also a generalized cyclic reduction algorithm developed at NCAR. Any extraneous monopole component present in the data can be removed if so desired. (Auth.)
Fundamental solutions and dual boundary element methods for fracture in plane Cosserat elasticity.
Atroshchenko, Elena; Bordas, Stéphane P A
2015-07-08
In this paper, both singular and hypersingular fundamental solutions of plane Cosserat elasticity are derived and given in a ready-to-use form. The hypersingular fundamental solutions allow to formulate the analogue of Somigliana stress identity, which can be used to obtain the stress and couple-stress fields inside the domain from the boundary values of the displacements, microrotation and stress and couple-stress tractions. Using these newly derived fundamental solutions, the boundary integral equations of both types are formulated and solved by the boundary element method. Simultaneous use of both types of equations (approach known as the dual boundary element method (BEM)) allows problems where parts of the boundary are overlapping, such as crack problems, to be treated and to do this for general geometry and loading conditions. The high accuracy of the boundary element method for both types of equations is demonstrated for a number of benchmark problems, including a Griffith crack problem and a plate with an edge crack. The detailed comparison of the BEM results and the analytical solution for a Griffith crack and an edge crack is given, particularly in terms of stress and couple-stress intensity factors, as well as the crack opening displacements and microrotations on the crack faces and the angular distributions of stresses and couple-stresses around the crack tip.
On numerical solution of Burgers' equation by homotopy analysis method
International Nuclear Information System (INIS)
Inc, Mustafa
2008-01-01
In this Letter, we present the Homotopy Analysis Method (shortly HAM) for obtaining the numerical solution of the one-dimensional nonlinear Burgers' equation. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Convergence of the solution and effects for the method is discussed. The comparison of the HAM results with the Homotopy Perturbation Method (HPM) and the results of [E.N. Aksan, Appl. Math. Comput. 174 (2006) 884; S. Kutluay, A. Esen, Int. J. Comput. Math. 81 (2004) 1433; S. Abbasbandy, M.T. Darvishi, Appl. Math. Comput. 163 (2005) 1265] are made. The results reveal that HAM is very simple and effective. The HAM contains the auxiliary parameter h, which provides us with a simple way to adjust and control the convergence region of solution series. The numerical solutions are compared with the known analytical and some numerical solutions
Cid, Jaime A; von Davier, Alina A
2015-05-01
Test equating is a method of making the test scores from different test forms of the same assessment comparable. In the equating process, an important step involves continuizing the discrete score distributions. In traditional observed-score equating, this step is achieved using linear interpolation (or an unscaled uniform kernel). In the kernel equating (KE) process, this continuization process involves Gaussian kernel smoothing. It has been suggested that the choice of bandwidth in kernel smoothing controls the trade-off between variance and bias. In the literature on estimating density functions using kernels, it has also been suggested that the weight of the kernel depends on the sample size, and therefore, the resulting continuous distribution exhibits bias at the endpoints, where the samples are usually smaller. The purpose of this article is (a) to explore the potential effects of atypical scores (spikes) at the extreme ends (high and low) on the KE method in distributions with different degrees of asymmetry using the randomly equivalent groups equating design (Study I), and (b) to introduce the Epanechnikov and adaptive kernels as potential alternative approaches to reducing boundary bias in smoothing (Study II). The beta-binomial model is used to simulate observed scores reflecting a range of different skewed shapes.
Numerov iteration method for second order integral-differential equation
International Nuclear Information System (INIS)
Zeng Fanan; Zhang Jiaju; Zhao Xuan
1987-01-01
In this paper, Numerov iterative method for second order integral-differential equation and system of equations are constructed. Numerical examples show that this method is better than direct method (Gauss elimination method) in CPU time and memoy requireing. Therefore, this method is an efficient method for solving integral-differential equation in nuclear physics
The boundary data immersion method for compressible flows with application to aeroacoustics
Energy Technology Data Exchange (ETDEWEB)
Schlanderer, Stefan C., E-mail: stefan.schlanderer@unimelb.edu.au [Faculty for Engineering and the Environment, University of Southampton, SO17 1BJ Southampton (United Kingdom); Weymouth, Gabriel D., E-mail: G.D.Weymouth@soton.ac.uk [Faculty for Engineering and the Environment, University of Southampton, SO17 1BJ Southampton (United Kingdom); Sandberg, Richard D., E-mail: richard.sandberg@unimelb.edu.au [Department of Mechanical Engineering, University of Melbourne, Melbourne VIC 3010 (Australia)
2017-03-15
This paper introduces a virtual boundary method for compressible viscous fluid flow that is capable of accurately representing moving bodies in flow and aeroacoustic simulations. The method is the compressible extension of the boundary data immersion method (BDIM, Maertens & Weymouth (2015), ). The BDIM equations for the compressible Navier–Stokes equations are derived and the accuracy of the method for the hydrodynamic representation of solid bodies is demonstrated with challenging test cases, including a fully turbulent boundary layer flow and a supersonic instability wave. In addition we show that the compressible BDIM is able to accurately represent noise radiation from moving bodies and flow induced noise generation without any penalty in allowable time step.
A Photon Free Method to Solve Radiation Transport Equations
International Nuclear Information System (INIS)
Chang, B
2006-01-01
The multi-group discrete-ordinate equations of radiation transfer is solved for the first time by Newton's method. It is a photon free method because the photon variables are eliminated from the radiation equations to yield a N group XN direction smaller but equivalent system of equations. The smaller set of equations can be solved more efficiently than the original set of equations. Newton's method is more stable than the Semi-implicit Linear method currently used by conventional radiation codes
International Nuclear Information System (INIS)
Fathizadeh, M.; Aroujalian, A.
2012-01-01
The boundary layer convective heat transfer equations with low pressure gradient over a flat plate are solved using Homotopy Perturbation Method, which is one of the semi-exact methods. The nonlinear equations of momentum and energy solved simultaneously via Homotopy Perturbation Method are in good agreement with results obtained from numerical methods. Using this method, a general equation in terms of Pr number and pressure gradient (λ) is derived which can be used to investigate velocity and temperature profiles in the boundary layer.
Adaptive finite element methods for differential equations
Bangerth, Wolfgang
2003-01-01
These Lecture Notes discuss concepts of `self-adaptivity' in the numerical solution of differential equations, with emphasis on Galerkin finite element methods. The key issues are a posteriori error estimation and it automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method for goal-oriented error estimation, is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. `Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost. At the end of each chapter some exercises are posed in order ...
Telescopic projective methods for parabolic differential equations
Gear, C W
2003-01-01
Projective methods were introduced in an earlier paper [C.W. Gear, I.G. Kevrekidis, Projective Methods for Stiff Differential Equations: problems with gaps in their eigenvalue spectrum, NEC Research Institute Report 2001-029, available from http://www.neci.nj.nec.com/homepages/cwg/projective.pdf Abbreviated version to appear in SISC] as having potential for the efficient integration of problems with a large gap between two clusters in their eigenvalue spectrum, one cluster containing eigenvalues corresponding to components that have already been damped in the numerical solution and one corresponding to components that are still active. In this paper we introduce iterated projective methods that allow for explicit integration of stiff problems that have a large spread of eigenvalues with no gaps in their spectrum as arise in the semi-discretization of PDEs with parabolic components.
Telescopic projective methods for parabolic differential equations
International Nuclear Information System (INIS)
Gear, C.W.; Kevrekidis, Ioannis G.
2003-01-01
Projective methods were introduced in an earlier paper [C.W. Gear, I.G. Kevrekidis, Projective Methods for Stiff Differential Equations: problems with gaps in their eigenvalue spectrum, NEC Research Institute Report 2001-029, available from http://www.neci.nj.nec.com/homepages/cwg/projective.pdf Abbreviated version to appear in SISC] as having potential for the efficient integration of problems with a large gap between two clusters in their eigenvalue spectrum, one cluster containing eigenvalues corresponding to components that have already been damped in the numerical solution and one corresponding to components that are still active. In this paper we introduce iterated projective methods that allow for explicit integration of stiff problems that have a large spread of eigenvalues with no gaps in their spectrum as arise in the semi-discretization of PDEs with parabolic components
Seo, Jung Hee; Mittal, Rajat
2011-08-10
A method for reducing the spurious pressure oscillations observed when simulating moving boundary flow problems with sharp-interface immersed boundary methods (IBMs) is proposed. By first identifying the primary cause of these oscillations to be the violation of the geometric conservation law near the immersed boundary, we adopt a cut-cell based approach to strictly enforce geometric conservation. In order to limit the complexity associated with the cut-cell method, the cut-cell based discretization is limited only to the pressure Poisson and velocity correction equations in the fractional-step method and the small-cell problem tackled by introducing a virtual cell-merging technique. The method is shown to retain all the desirable properties of the original finite-difference based IBM while at the same time, reducing pressure oscillations for moving boundaries by roughly an order of magnitude.
Energy Technology Data Exchange (ETDEWEB)
Buerger, R.; Frid, H.; Karlsen, K.H.
2002-07-01
We consider a free boundary problem of a quasilinear strongly degenerate parabolic equation arising from a model of pressure filtration of flocculated suspensions. We provide definitions of generalized solutions of the free boundary problem in the framework of L2 divergence-measure fields. The formulation of boundary conditions is based on a Gauss-Green theorem for divergence-measure fields on bounded domains with Lipschitz deformable boundaries and avoids referring to traces of the solution. This allows to consider generalized solutions from a larger class than BV. Thus it is not necessary to derive the usual uniform estimates on spatial and time derivatives of the solutions of the corresponding regularized problem requires in the BV approach. We first prove existence and uniqueness of the solution of the regularized parabolic free boundary problem and then apply the vanishing viscosity method to prove existence of a generalized solution to the degenerate free boundary problem. (author)
Directory of Open Access Journals (Sweden)
Bashir Ahmad
2012-06-01
Full Text Available We study boundary value problems of nonlinear fractional differential equations and inclusions of order $q in (m-1, m]$, $m ge 2$ with multi-strip boundary conditions. Multi-strip boundary conditions may be regarded as the generalization of multi-point boundary conditions. Our problem is new in the sense that we consider a nonlocal strip condition of the form: $$ x(1=sum_{i=1}^{n-2}alpha_i int^{eta_i}_{zeta_i} x(sds, $$ which can be viewed as an extension of a multi-point nonlocal boundary condition: $$ x(1=sum_{i=1}^{n-2}alpha_i x(eta_i. $$ In fact, the strip condition corresponds to a continuous distribution of the values of the unknown function on arbitrary finite segments $(zeta_i,eta_i$ of the interval $[0,1]$ and the effect of these strips is accumulated at $x=1$. Such problems occur in the applied fields such as wave propagation and geophysics. Some new existence and uniqueness results are obtained by using a variety of fixed point theorems. Some illustrative examples are also discussed.
Second-order wave diffraction by a circular cylinder using scaled boundary finite element method
International Nuclear Information System (INIS)
Song, H; Tao, L
2010-01-01
The scaled boundary finite element method (SBFEM) has achieved remarkable success in structural mechanics and fluid mechanics, combing the advantage of both FEM and BEM. Most of the previous works focus on linear problems, in which superposition principle is applicable. However, many physical problems in the real world are nonlinear and are described by nonlinear equations, challenging the application of the existing SBFEM model. A popular idea to solve a nonlinear problem is decomposing the nonlinear equation to a number of linear equations, and then solves them individually. In this paper, second-order wave diffraction by a circular cylinder is solved by SBFEM. By splitting the forcing term into two parts, the physical problem is described as two second-order boundary-value problems with different asymptotic behaviour at infinity. Expressing the velocity potentials as a series of depth-eigenfunctions, both of the 3D boundary-value problems are decomposed to a number of 2D boundary-value sub-problems, which are solved semi-analytically by SBFEM. Only the cylinder boundary is discretised with 1D curved finite-elements on the circumference of the cylinder, while the radial differential equation is solved completely analytically. The method can be extended to solve more complex wave-structure interaction problems resulting in direct engineering applications.
Institute of Scientific and Technical Information of China (English)
LIU FuPing; WANG AnLing; WANG AnXuan; CAO YueZu; CHEN Qiang; YANG ChangChun
2009-01-01
According to the electric potential of oblique multi-needle electrodes (OMNE) in biological tissue, the discrete equations based on the indetermination linear current density were established by the boundary element integral equations (BEIE). The non-uniform distribution of the current flowing from multi-needle electrodes to conductive biological tissues was imaged by solving a set of linear equa-tions. Then, the electric field and potential generated by OMNE in biological tissues at any point may be determined through the boundary element method (BEM). The time of program running and stability of computing method are examined by an example. It demonstrates that the algorithm possesses a quick speed and the steady computed results. It means that this method has an important referenced significance for computing the field and the potential generated by OMNE in bio-tissue, which is a fast, effective and accurate computing method.
Acoustic 3D modeling by the method of integral equations
Malovichko, M.; Khokhlov, N.; Yavich, N.; Zhdanov, M.
2018-02-01
This paper presents a parallel algorithm for frequency-domain acoustic modeling by the method of integral equations (IE). The algorithm is applied to seismic simulation. The IE method reduces the size of the problem but leads to a dense system matrix. A tolerable memory consumption and numerical complexity were achieved by applying an iterative solver, accompanied by an effective matrix-vector multiplication operation, based on the fast Fourier transform (FFT). We demonstrate that, the IE system matrix is better conditioned than that of the finite-difference (FD) method, and discuss its relation to a specially preconditioned FD matrix. We considered several methods of matrix-vector multiplication for the free-space and layered host models. The developed algorithm and computer code were benchmarked against the FD time-domain solution. It was demonstrated that, the method could accurately calculate the seismic field for the models with sharp material boundaries and a point source and receiver located close to the free surface. We used OpenMP to speed up the matrix-vector multiplication, while MPI was used to speed up the solution of the system equations, and also for parallelizing across multiple sources. The practical examples and efficiency tests are presented as well.
Aguareles, M.
2014-06-01
In this paper we consider an oscillatory medium whose dynamics are modeled by the complex Ginzburg-Landau equation. In particular, we focus on n-armed spiral wave solutions of the complex Ginzburg-Landau equation in a disk of radius d with homogeneous Neumann boundary conditions. It is well-known that such solutions exist for small enough values of the twist parameter q and large enough values of d. We investigate the effect of boundaries on the rotational frequency of the spirals, which is an unknown of the problem uniquely determined by the parameters d and q. We show that there is a threshold in the parameter space where the effect of the boundary on the rotational frequency switches from being algebraic to exponentially weak. We use the method of matched asymptotic expansions to obtain explicit expressions for the asymptotic wavenumber as a function of the twist parameter and the domain size for small values of q. © 2014 Elsevier B.V. All rights reserved.
An irrational trial equation method and its applications
Indian Academy of Sciences (India)
equation method which is different from those direct methods. Liu's key idea is that exact solution to a differential equation can be given by solving an integration. For example, consider a differential equation of u. We always assume that its exact solution satisfies a solvable equation u = F(u). Therefore, our task is just to find.
Deriving average soliton equations with a perturbative method
International Nuclear Information System (INIS)
Ballantyne, G.J.; Gough, P.T.; Taylor, D.P.
1995-01-01
The method of multiple scales is applied to periodically amplified, lossy media described by either the nonlinear Schroedinger (NLS) equation or the Korteweg--de Vries (KdV) equation. An existing result for the NLS equation, derived in the context of nonlinear optical communications, is confirmed. The method is then applied to the KdV equation and the result is confirmed numerically
International Nuclear Information System (INIS)
Zhang Zaiyun; Miao Xiujin; Chen Yuezhong; Liu Zhenhai
2011-01-01
In this paper, we prove the existence, uniqueness, and uniform stability of strong and weak solutions of the nonlinear generalized Klein-Gordon equation (1.1) 1 (see Sec. I) in bounded domains with nonlinear damped boundary conditions given by (1.1) 3 (see Sec. I) with some restrictions on function f(u), h(∇u), g(u t ), and b(x), we prove the existence and uniqueness by means of nonlinear semigroup method and obtain the uniform stabilization by using the multiplier technique.
Entropy methods for diffusive partial differential equations
Jüngel, Ansgar
2016-01-01
This book presents a range of entropy methods for diffusive PDEs devised by many researchers in the course of the past few decades, which allow us to understand the qualitative behavior of solutions to diffusive equations (and Markov diffusion processes). Applications include the large-time asymptotics of solutions, the derivation of convex Sobolev inequalities, the existence and uniqueness of weak solutions, and the analysis of discrete and geometric structures of the PDEs. The purpose of the book is to provide readers an introduction to selected entropy methods that can be found in the research literature. In order to highlight the core concepts, the results are not stated in the widest generality and most of the arguments are only formal (in the sense that the functional setting is not specified or sufficient regularity is supposed). The text is also suitable for advanced master and PhD students and could serve as a textbook for special courses and seminars.
Method of lines solution of Richards` equation
Energy Technology Data Exchange (ETDEWEB)
Kelley, C.T.; Miller, C.T.; Tocci, M.D.
1996-12-31
We consider the method of lines solution of Richard`s equation, which models flow through porous media, as an example of a situation in which the method can give incorrect results because of premature termination of the nonlinear corrector iteration. This premature termination arises when the solution has a sharp moving front and the Jacobian is ill-conditioned. While this problem can be solved by tightening the tolerances provided to the ODE or DAE solver used for the temporal integration, it is more efficient to modify the termination criteria of the nonlinear solver and/or recompute the Jacobian more frequently. In this paper we continue previous work on this topic by analyzing the modifications in more detail and giving a strategy on how the modifications can be turned on and off in response to changes in the character of the solution.
The reduced basis method for the electric field integral equation
International Nuclear Information System (INIS)
Fares, M.; Hesthaven, J.S.; Maday, Y.; Stamm, B.
2011-01-01
We introduce the reduced basis method (RBM) as an efficient tool for parametrized scattering problems in computational electromagnetics for problems where field solutions are computed using a standard Boundary Element Method (BEM) for the parametrized electric field integral equation (EFIE). This combination enables an algorithmic cooperation which results in a two step procedure. The first step consists of a computationally intense assembling of the reduced basis, that needs to be effected only once. In the second step, we compute output functionals of the solution, such as the Radar Cross Section (RCS), independently of the dimension of the discretization space, for many different parameter values in a many-query context at very little cost. Parameters include the wavenumber, the angle of the incident plane wave and its polarization.
On the economical solution method for a system of linear algebraic equations
Directory of Open Access Journals (Sweden)
Jan Awrejcewicz
2004-01-01
Full Text Available The present work proposes a novel optimal and exact method of solving large systems of linear algebraic equations. In the approach under consideration, the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped in ℝ3 is considered, where boundary value problems of first, second, or third order, or their combinations, are taken into account. The governing differential equations are reduced to algebraic ones with the help of the finite element and boundary element methods for different meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a nonhomogeneous shallow physically and geometrically nonlinear shell subject to transversal uniformly distributed load. The partial differential equations are reduced to a system of nonlinear algebraic equations with the error of O(hx12+hx22. The linearization process is realized through either Newton method or differentiation with respect to a parameter. In consequence, the relations of the boundary condition variations along the shell side and the conditions for the solution matching are reported.
Pagan Munoz, R.; Hornikx, M.C.J.
The wave-based Fourier Pseudospectral time-domain (Fourier-PSTD) method was shown to be an effective way of modeling outdoor acoustic propagation problems as described by the linearized Euler equations (LEE), but is limited to real-valued frequency independent boundary conditions and predominantly
International Nuclear Information System (INIS)
Lima E Silva, A.L.F.; Silveira-Neto, A.; Damasceno, J.J.R.
2003-01-01
In this work, a virtual boundary method is applied to the numerical simulation of a uniform flow over a cylinder. The force source term, added to the two-dimensional Navier-Stokes equations, guarantees the imposition of the no-slip boundary condition over the body-fluid interface. These equations are discretized, using the finite differences method. The immersed boundary is represented with a finite number of Lagrangian points, distributed over the solid-fluid interface. A Cartesian grid is used to solve the fluid flow equations. The key idea is to propose a method to calculate the interfacial force without ad hoc constants that should usually be adjusted for the type of flow and the type of the numerical method, when this kind of model is used. In the present work, this force is calculated using the Navier-Stokes equations applied to the Lagrangian points and then distributed over the Eulerian grid. The main advantage of this approach is that it enables calculation of this force field, even if the interface is moving or deforming. It is unnecessary to locate the Eulerian grid points near this immersed boundary. The lift and drag coefficients and the Strouhal number, calculated for an immersed cylinder, are compared with previous experimental and numerical results, for different Reynolds numbers
An efficient implicit direct forcing immersed boundary method for incompressible flows
International Nuclear Information System (INIS)
Cai, S-G; Ouahsine, A; Smaoui, H; Favier, J; Hoarau, Y
2015-01-01
A novel efficient implicit direct forcing immersed boundary method for incompressible flows with complex boundaries is presented. In the previous work [1], the calculation is performed on the Cartesian grid regardless of the immersed object, with a fictitious force evaluated on the Lagrangian points to mimic the presence of the physical boundaries. However the explicit direct forcing method [1] fails to accurately impose the non-slip boundary condition on the immersed interface. In the present work, the calculation is based on the implicit treatment of the artificial force while in an effective way of system iteration. The accuracy is also improved by solving the Navier-Stokes equation with the rotational incremental pressure- correction projection method of Guermond and Shen [2]. Numerical simulations performed with the proposed method are in good agreement with those in the literature
A multilevel correction adaptive finite element method for Kohn-Sham equation
Hu, Guanghui; Xie, Hehu; Xu, Fei
2018-02-01
In this paper, an adaptive finite element method is proposed for solving Kohn-Sham equation with the multilevel correction technique. In the method, the Kohn-Sham equation is solved on a fixed and appropriately coarse mesh with the finite element method in which the finite element space is kept improving by solving the derived boundary value problems on a series of adaptively and successively refined meshes. A main feature of the method is that solving large scale Kohn-Sham system is avoided effectively, and solving the derived boundary value problems can be handled efficiently by classical methods such as the multigrid method. Hence, the significant acceleration can be obtained on solving Kohn-Sham equation with the proposed multilevel correction technique. The performance of the method is examined by a variety of numerical experiments.
PARALLEL SOLUTION METHODS OF PARTIAL DIFFERENTIAL EQUATIONS
Directory of Open Access Journals (Sweden)
Korhan KARABULUT
1998-03-01
Full Text Available Partial differential equations arise in almost all fields of science and engineering. Computer time spent in solving partial differential equations is much more than that of in any other problem class. For this reason, partial differential equations are suitable to be solved on parallel computers that offer great computation power. In this study, parallel solution to partial differential equations with Jacobi, Gauss-Siedel, SOR (Succesive OverRelaxation and SSOR (Symmetric SOR algorithms is studied.
Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation
Directory of Open Access Journals (Sweden)
Vitanov Nikolay K.
2018-03-01
Full Text Available We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrödinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrödinger kind.
Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation
Vitanov, Nikolay K.; Dimitrova, Zlatinka I.
2018-03-01
We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrödinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrödinger kind.
Directory of Open Access Journals (Sweden)
Adel A.K. Mohsen
2010-07-01
Full Text Available The problem of nonuniqueness (NU of the solution of exterior acoustic problems via boundary integral equations is discussed in this article. The efficient implementation of the CHIEF (Combined Helmholtz Integral Equations Formulation method to axisymmetric problems is studied. Interior axial fields are used to indicate the solution error and to select proper CHIEF points. The procedure makes full use of LU-decomposition as well as the forward solution derived in the solution. Implementations of the procedure for hard spheres are presented. Accurate results are obtained up to a normalised radius of ka = 20.983, using only one CHIEF point. The radiation from a uniformly vibrating sphere is also considered. Accurate results for ka up to 16.927 are obtained using two CHIEF points.
International Nuclear Information System (INIS)
Feng Qing-Hua
2014-01-01
In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann—Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method, we apply this method to solve the space-time fractional Whitham—Broer—Kaup (WBK) equations and the nonlinear fractional Sharma—Tasso—Olever (STO) equation, and as a result, some new exact solutions for them are obtained. (general)
METHOD OF GREEN FUNCTIONS IN MATHEMATICAL MODELLING FOR TWO-POINT BOUNDARY-VALUE PROBLEMS
Directory of Open Access Journals (Sweden)
E. V. Dikareva
2015-01-01
Full Text Available Summary. In many applied problems of control, optimization, system theory, theoretical and construction mechanics, for problems with strings and nods structures, oscillation theory, theory of elasticity and plasticity, mechanical problems connected with fracture dynamics and shock waves, the main instrument for study these problems is a theory of high order ordinary differential equations. This methodology is also applied for studying mathematical models in graph theory with different partitioning based on differential equations. Such equations are used for theoretical foundation of mathematical models but also for constructing numerical methods and computer algorithms. These models are studied with use of Green function method. In the paper first necessary theoretical information is included on Green function method for multi point boundary-value problems. The main equation is discussed, notions of multi-point boundary conditions, boundary functionals, degenerate and non-degenerate problems, fundamental matrix of solutions are introduced. In the main part the problem to study is formulated in terms of shocks and deformations in boundary conditions. After that the main results are formulated. In theorem 1 conditions for existence and uniqueness of solutions are proved. In theorem 2 conditions are proved for strict positivity and equal measureness for a pair of solutions. In theorem 3 existence and estimates are proved for the least eigenvalue, spectral properties and positivity of eigenfunctions. In theorem 4 the weighted positivity is proved for the Green function. Some possible applications are considered for a signal theory and transmutation operators.
International Nuclear Information System (INIS)
Zhao, Zhonglong; Zhang, Yufeng; Han, Zhong; Rui, Wenjuan
2014-01-01
In this paper, the simplest equation method is used to construct exact traveling solutions of the (3+1)-dimensional KP equation and generalized Fisher equation. We summarize the main steps of the simplest equation method. The Bernoulli and Riccati equation are used as simplest equations. This method is straightforward and concise, and it can be applied to other nonlinear partial differential equations
Extreme Scale FMM-Accelerated Boundary Integral Equation Solver for Wave Scattering
AbdulJabbar, Mustafa Abdulmajeed
2018-03-27
Algorithmic and architecture-oriented optimizations are essential for achieving performance worthy of anticipated energy-austere exascale systems. In this paper, we present an extreme scale FMM-accelerated boundary integral equation solver for wave scattering, which uses FMM as a matrix-vector multiplication inside the GMRES iterative method. Our FMM Helmholtz kernels treat nontrivial singular and near-field integration points. We implement highly optimized kernels for both shared and distributed memory, targeting emerging Intel extreme performance HPC architectures. We extract the potential thread- and data-level parallelism of the key Helmholtz kernels of FMM. Our application code is well optimized to exploit the AVX-512 SIMD units of Intel Skylake and Knights Landing architectures. We provide different performance models for tuning the task-based tree traversal implementation of FMM, and develop optimal architecture-specific and algorithm aware partitioning, load balancing, and communication reducing mechanisms to scale up to 6,144 compute nodes of a Cray XC40 with 196,608 hardware cores. With shared memory optimizations, we achieve roughly 77% of peak single precision floating point performance of a 56-core Skylake processor, and on average 60% of peak single precision floating point performance of a 72-core KNL. These numbers represent nearly 5.4x and 10x speedup on Skylake and KNL, respectively, compared to the baseline scalar code. With distributed memory optimizations, on the other hand, we report near-optimal efficiency in the weak scalability study with respect to both the logarithmic communication complexity as well as the theoretical scaling complexity of FMM. In addition, we exhibit up to 85% efficiency in strong scaling. We compute in excess of 2 billion DoF on the full-scale of the Cray XC40 supercomputer.
Directory of Open Access Journals (Sweden)
Chuanzhi Bai
2010-06-01
Full Text Available This paper deals with the existence of positive solutions for a boundary value problem involving a nonlinear functional differential equation of fractional order $\\alpha$ given by $ D^{\\alpha} u(t + f(t, u_t = 0$, $t \\in (0, 1$, $2 < \\alpha \\le 3$, $ u^{\\prime}(0 = 0$, $u^{\\prime}(1 = b u^{\\prime}(\\eta$, $u_0 = \\phi$. Our results are based on the nonlinear alternative of Leray-Schauder type and Krasnosel'skii fixed point theorem.
Directory of Open Access Journals (Sweden)
Aqlan Mohammed H.
2016-01-01
Full Text Available We develop the existence theory for sequential fractional differential equations involving Liouville-Caputo fractional derivative equipped with anti-periodic type (non-separated and nonlocal integral boundary conditions. Several existence criteria depending on the nonlinearity involved in the problems are presented by means of a variety of tools of the fixed point theory. The applicability of the results is shown with the aid of examples. Our results are not only new in the given configuration but also yield some new special cases for specific choices of parameters involved in the problems.
Asymptotically linear Schrodinger equation with zero on the boundary of the spectrum
Directory of Open Access Journals (Sweden)
Dongdong Qin
2015-08-01
Full Text Available This article concerns the Schr\\"odinger equation $$\\displaylines{ -\\Delta u+V(xu=f(x, u, \\quad \\text{for } x\\in\\mathbb{R}^N,\\cr u(x\\to 0, \\quad \\text{as } |x| \\to \\infty, }$$ where V and f are periodic in x, and 0 is a boundary point of the spectrum $\\sigma(-\\Delta+V$. Assuming that f(x,u is asymptotically linear as $|u|\\to\\infty$, existence of a ground state solution is established using some new techniques.
Student Solutions Manual to Boundary Value Problems and Partial Differential Equations
Powers, David L
2005-01-01
This student solutions manual accompanies the text, Boundary Value Problems and Partial Differential Equations, 5e. The SSM is available in print via PDF or electronically, and provides the student with the detailed solutions of the odd-numbered problems contained throughout the book.Provides students with exercises that skillfully illustrate the techniques used in the text to solve science and engineering problemsNearly 900 exercises ranging in difficulty from basic drills to advanced problem-solving exercisesMany exercises based on current engineering applications
Fourier-Based Fast Multipole Method for the Helmholtz Equation
Cecka, Cris
2013-01-01
The fast multipole method (FMM) has had great success in reducing the computational complexity of solving the boundary integral form of the Helmholtz equation. We present a formulation of the Helmholtz FMM that uses Fourier basis functions rather than spherical harmonics. By modifying the transfer function in the precomputation stage of the FMM, time-critical stages of the algorithm are accelerated by causing the interpolation operators to become straightforward applications of fast Fourier transforms, retaining the diagonality of the transfer function, and providing a simplified error analysis. Using Fourier analysis, constructive algorithms are derived to a priori determine an integration quadrature for a given error tolerance. Sharp error bounds are derived and verified numerically. Various optimizations are considered to reduce the number of quadrature points and reduce the cost of computing the transfer function. © 2013 Society for Industrial and Applied Mathematics.
Analytic method for solitary solutions of some partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Ugurlu, Yavuz [Firat University, Department of Mathematics, 23119 Elazig (Turkey); Kaya, Dogan [Firat University, Department of Mathematics, 23119 Elazig (Turkey)], E-mail: dkaya@firat.edu.tr
2007-10-22
In this Letter by considering an improved tanh function method, we found some exact solutions of the clannish random walker's parabolic equation, the modified Korteweg-de Vries (KdV) equation, and the Sharma-Tasso-Olver (STO) equation with its fission and fusion, the Jaulent-Miodek equation.
Analytic method for solitary solutions of some partial differential equations
International Nuclear Information System (INIS)
Ugurlu, Yavuz; Kaya, Dogan
2007-01-01
In this Letter by considering an improved tanh function method, we found some exact solutions of the clannish random walker's parabolic equation, the modified Korteweg-de Vries (KdV) equation, and the Sharma-Tasso-Olver (STO) equation with its fission and fusion, the Jaulent-Miodek equation
The Laplace equation boundary value problems on bounded and unbounded Lipschitz domains
Medková, Dagmar
2018-01-01
This book is devoted to boundary value problems of the Laplace equation on bounded and unbounded Lipschitz domains. It studies the Dirichlet problem, the Neumann problem, the Robin problem, the derivative oblique problem, the transmission problem, the skip problem and mixed problems. It also examines different solutions - classical, in Sobolev spaces, in Besov spaces, in homogeneous Sobolev spaces and in the sense of non-tangential limit. It also explains relations between different solutions. The book has been written in a way that makes it as readable as possible for a wide mathematical audience, and includes all the fundamental definitions and propositions from other fields of mathematics. This book is of interest to research students, as well as experts in partial differential equations and numerical analysis.
Directory of Open Access Journals (Sweden)
Alexander Domoshnitsky
2014-01-01
Full Text Available The impulsive delay differential equation is considered (Lx(t=x′(t+∑i=1mpi(tx(t-τi(t=f(t, t∈[a,b], x(tj=βjx(tj-0, j=1,…,k, a=t0
Energy Technology Data Exchange (ETDEWEB)
Laboure, Vincent M.; Wang, Yaqi; DeHart, Mark D.
2016-05-01
In this paper, we study the Least-Squares (LS) PN form of the transport equation compatible with voids [1] in the context of Continuous Finite Element Methods (CFEM).We first deriveweakly imposed boundary conditions which make the LS weak formulation equivalent to the Self-Adjoint Angular Flux (SAAF) variational formulation with a void treatment [2], in the particular case of constant cross-sections and a uniform mesh. We then implement this method in Rattlesnake with the Multiphysics Object Oriented Simulation Environment (MOOSE) framework [3] using a spherical harmonics (PN) expansion to discretize in angle. We test our implementation using the Method of Manufactured Solutions (MMS) and find the expected convergence behavior both in angle and space. Lastly, we investigate the impact of the global non-conservation of LS by comparing the method with SAAF on a heterogeneous test problem.
Energy Technology Data Exchange (ETDEWEB)
Vincent M. Laboure; Yaqi Wang; Mark D. DeHart
2016-05-01
In this paper, we study the Least-Squares (LS) PN form of the transport equation compatible with voids in the context of Continuous Finite Element Methods (CFEM).We first deriveweakly imposed boundary conditions which make the LS weak formulation equivalent to the Self-Adjoint Angular Flux (SAAF) variational formulation with a void treatment, in the particular case of constant cross-sections and a uniform mesh. We then implement this method in Rattlesnake with the Multiphysics Object Oriented Simulation Environment (MOOSE) framework using a spherical harmonics (PN) expansion to discretize in angle. We test our implementation using the Method of Manufactured Solutions (MMS) and find the expected convergence behavior both in angle and space. Lastly, we investigate the impact of the global non-conservation of LS by comparing the method with SAAF on a heterogeneous test problem.
A system boundary identification method for life cycle assessment
DEFF Research Database (Denmark)
Li, Tao; Zhang, Hongchao; Liu, Zhichao
2014-01-01
, technical, geographical and temporal dimensions are presented to limit the boundaries of LCA. An algorithm is developed to identify an appropriate boundary by searching the process tree and evaluating the environmental impact contribution of each process while it is added into the studied system...... as processes are added. The two threshold rules and identification methods presented can be used to identify system boundary of LCA. The case study demonstrated that the methodology presented in this paper is an effective tool for the boundary identification....
A Line-Tau Collocation Method for Partial Differential Equations ...
African Journals Online (AJOL)
This paper deals with the numerical solution of second order linear partial differential equations with the use of the method of lines coupled with the tau collocation method. The method of lines is used to convert the partial differential equation (PDE) to a sequence of ordinary differential equations (ODEs) which is then ...
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Bashir Ahmad
2013-02-01
Full Text Available In this article, we discuss the existence of solutions for a boundary-value problem of integro-differential equations of fractional order with nonlocal fractional boundary conditions by means of some standard tools of fixed point theory. Our problem describes a more general form of fractional stochastic dynamic model for financial asset. An illustrative example is also presented.
International Nuclear Information System (INIS)
Choi, B.H.; Poe, R.T.; Tang, K.T.
1978-01-01
The body-fixed (BF) formulation for atom--diatom scatterings is developed to the extent that one can use it to perform accurate close-coupling calculation, without introducing further approximation except truncating a finite basis set of the target molecular wave function, on the same ground as one use the space-fixed (SF) formulation. In this formulation, the coupled differential equations are solved an the boundary conditions matched entirely in the BF coordinate system. A unitary transformation is used to obtain both the coupled differential equation and the boundary condition in BF system system from SF system. All properties of the solution with respect to parity are derived entirely from the transformation, without using the parity eignfunctions of the BF frame. Boundary conditions that yield the scattering (S) matrix and the reactance (R) matrix are presented for each parity in both the far asymptotic region (where the interaction and the centrifugal potentials are both negligible) and the near asymptotic region (where the interaction potential is negligible but the centrifugal potential is not). While our differential equations are the same as those derived by others with different methods, our asymptotic boundary conditions disagree with some existing ones. With a given form of the BF coupled differential equations, the acceptable boundary conditions are discussed
An approximate moving boundary method for American option pricing
Chockalingam, A.; Muthuraman, K.
2015-01-01
We present a method to solve the free-boundary problem that arises in the pricing of classical American options. Such free-boundary problems arise when one attempts to solve optimal-stopping problems set in continuous time. American option pricing is one of the most popular optimal-stopping problems
Energy Technology Data Exchange (ETDEWEB)
Namestnik, B; Skerget, L; Beadar, D [tehniska fakulteta, Maribor (Yugoslavia)
1989-07-01
The paper presents numerical method for evaluating heat transfer on two-dimensional ribbed surfaces. Governing elliptic partial differential equation is transformed to boundary integral equation, and solved by the boundary element method. Efficiency of fins is calculated from boundary heat flux balance. Several test cases have shown usefulness of the presented method. (author)
Ernst Equation and Riemann Surfaces: Analytical and Numerical Methods
International Nuclear Information System (INIS)
Ernst, Frederick J
2007-01-01
Shortly after Einstein published his general theory of relativity, the spherically symmetric solution of the vacuum field equations was discovered by Karl Schwarzschild, while Hermann Weyl showed that from any axisymmetric solution ψ of the Laplace equation ∇ 2 ψ = 0 (satisfying appropriate boundary conditions) the metric tensor of a static axisymmetric vacuum spacetime can be constructed. In particular, the Schwarzschild solution corresponds to a rather trivial solution of Laplace's equation expressed in terms of prolate spheroidal coordinates. It took about 45 years before Roy Kerr discovered what he called the 'rotating Schwarzschild solution', and an additional five years before I established that from any complex axisymmetric solution E of the nonlinear equation E∇ 2 E = ∇E·∇E (satisfying appropriate boundary conditions) the metric tensor of a stationary axisymmetric vacuum spacetime can be constructed. In particular, the Kerr solution corresponds to an extremely simple solution of this equation expressed in terms of prolate spheroidal coordinates. Ever more complicated solutions of this equation (using prolate spheroidal coordinates) were discovered by Tomimatsu and Sato, but for none of the associated spacetimes has a reasonable material source been suggested. What the present book describes are some of the heroic efforts that have been undertaken to construct physically significant spacetimes by solving the vacuum Ernst equation. Unfortunately, thus far, no one has made much progress extending the Ernst equation approach to facilitate the investigation of spacetime within a stationary axisymmetric material source, where, for example, the stress-energy tensor is that of a perfect fluid. However, in 1994, Meinel and Neugebauer had the novel idea of focusing attention upon global solutions such as the spacetime geometry associated with a rotating infinitely thin disk of dust, where the material source can be represented by discontinuities in the
Directory of Open Access Journals (Sweden)
Murat Osmanoglu
2013-01-01
Full Text Available We have considered linear partial differential algebraic equations (LPDAEs of the form , which has at least one singular matrix of . We have first introduced a uniform differential time index and a differential space index. The initial conditions and boundary conditions of the given system cannot be prescribed for all components of the solution vector here. To overcome this, we introduced these indexes. Furthermore, differential transform method has been given to solve LPDAEs. We have applied this method to a test problem, and numerical solution of the problem has been compared with analytical solution.
International Nuclear Information System (INIS)
Itagaki, Masafumi; Sahashi, Naoki.
1996-01-01
The multiple reciprocity method (MRM) in conjunction with the boundary element method has been employed to solve one-group eigenvalue problems described by the three-dimensional (3-D) neutron diffusion equation. The domain integral related to the fission source is transformed into a series of boundary-only integrals, with the aid of the higher order fundamental solutions based on the spherical and the modified spherical Bessel functions. Since each degree of the higher order fundamental solutions in the 3-D cases has a singularity of order (1/r), the above series of boundary integrals requires additional terms which do not appear in the 2-D MRM formulation. The critical eigenvalue itself can be also described using only boundary integrals. Test calculations show that Wielandt's spectral shift technique guarantees rapid and stable convergence of 3-D MRM computations. (author)
Survey of the status of finite element methods for partial differential equations
Temam, Roger
1986-01-01
The finite element methods (FEM) have proved to be a powerful technique for the solution of boundary value problems associated with partial differential equations of either elliptic, parabolic, or hyperbolic type. They also have a good potential for utilization on parallel computers particularly in relation to the concept of domain decomposition. This report is intended as an introduction to the FEM for the nonspecialist. It contains a survey which is totally nonexhaustive, and it also contains as an illustration, a report on some new results concerning two specific applications, namely a free boundary fluid-structure interaction problem and the Euler equations for inviscid flows.
The Extended Fractional Subequation Method for Nonlinear Fractional Differential Equations
Zhao, Jianping; Tang, Bo; Kumar, Sunil; Hou, Yanren
2012-01-01
An extended fractional subequation method is proposed for solving fractional differential equations by introducing a new general ansätz and Bäcklund transformation of the fractional Riccati equation with known solutions. Being concise and straightforward, this method is applied to the space-time fractional coupled Burgers’ equations and coupled MKdV equations. As a result, many exact solutions are obtained. It is shown that the considered method provides a very effective, convenient, and powe...
Semigroup methods for evolution equations on networks
Mugnolo, Delio
2014-01-01
This concise text is based on a series of lectures held only a few years ago and originally intended as an introduction to known results on linear hyperbolic and parabolic equations. Yet the topic of differential equations on graphs, ramified spaces, and more general network-like objects has recently gained significant momentum and, well beyond the confines of mathematics, there is a lively interdisciplinary discourse on all aspects of so-called complex networks. Such network-like structures can be found in virtually all branches of science, engineering and the humanities, and future research thus calls for solid theoretical foundations. This book is specifically devoted to the study of evolution equations – i.e., of time-dependent differential equations such as the heat equation, the wave equation, or the Schrödinger equation (quantum graphs) – bearing in mind that the majority of the literature in the last ten years on the subject of differential equations of graphs has been devoted to ellip...
Nanoscale roughness effect on Maxwell-like boundary conditions for the Boltzmann equation
Energy Technology Data Exchange (ETDEWEB)
Brull, S., E-mail: Stephane.Brull@math.u-bordeaux.fr; Charrier, P., E-mail: Pierre.Charrier@math.u-bordeaux.fr; Mieussens, L., E-mail: Luc.Mieussens@math.u-bordeaux.fr [University of Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400 Talence (France)
2016-08-15
It is well known that the roughness of the wall has an effect on microscale gas flows. This effect can be shown for large Knudsen numbers by using a numerical solution of the Boltzmann equation. However, when the wall is rough at a nanometric scale, it is necessary to use a very small mesh size which is much too expansive. An alternative approach is to incorporate the roughness effect in the scattering kernel of the boundary condition, such as the Maxwell-like kernel introduced by the authors in a previous paper. Here, we explain how this boundary condition can be implemented in a discrete velocity approximation of the Boltzmann equation. Moreover, the influence of the roughness is shown by computing the structure scattering pattern of mono-energetic beams of the incident gas molecules. The effect of the angle of incidence of these molecules, of their mass, and of the morphology of the wall is investigated and discussed in a simplified two-dimensional configuration. The effect of the azimuthal angle of the incident beams is shown for a three-dimensional configuration. Finally, the case of non-elastic scattering is considered. All these results suggest that our approach is a promising way to incorporate enough physics of gas-surface interaction, at a reasonable computing cost, to improve kinetic simulations of micro- and nano-flows.
Introduction to numerical methods for time dependent differential equations
Kreiss, Heinz-Otto
2014-01-01
Introduces both the fundamentals of time dependent differential equations and their numerical solutions Introduction to Numerical Methods for Time Dependent Differential Equations delves into the underlying mathematical theory needed to solve time dependent differential equations numerically. Written as a self-contained introduction, the book is divided into two parts to emphasize both ordinary differential equations (ODEs) and partial differential equations (PDEs). Beginning with ODEs and their approximations, the authors provide a crucial presentation of fundamental notions, such as the t
Natural convection in a composite fluid-porous cavity by the boundary element method
International Nuclear Information System (INIS)
Jecl, R.; Skerget, L.
2005-01-01
The main purpose of this work is to present the use of the boundary element method (BEM) for analyzing the convective fluid flow and heat transfer in composite fluid-porous media domain when the fluid is compressible. In our case the flow is modeled by utilizing the Brinkman extended Darcy momentum equation (Brinkman model) which is commonly used when it is important to satisfy the no-slip boundary condition and when one wishes to compare flows in porous medium with those in pure fluids. The Brinkman equation reduce to the classical Navier Stokes equation for clear fluid when the permeability tends to infinity (porosity is equal to unity), i.e. when the solid matrix in the porous medium disappears and, when the permeability is finite the equation is valid for porous medium. Therefore it is possible to handle porous medium free fluid interface problems by changing the properties of the medium in the computational domain appropriately. Our goal is to widen the applicability of the computational model based on the boundary domain integral method (BDIM) which is an extension of the classical BEM. The governing equations are transformed by using the velocity-vorticity variables formulation and therefore the computation scheme is partitioned into kinematic and kinetic part. (authors)
Initial value methods for boundary value problems
Meyer, Gunter H
1973-01-01
In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank mat
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
In this paper, we consider a two-point boundary value problem for a system of second order ordinary differential equations. Under some conditions, we show the existence of positive solution to the system of second order ordinary differential equa-tions.
International Nuclear Information System (INIS)
Gorshkov, A V
2003-01-01
The problem of the stabilization of a semilinear equation in the exterior of a bounded domain is considered. In view of the impossibility of an exponential stabilization of the form e -σt of the solution of a parabolic equation in an unbounded domain no matter what the boundary control is, one poses the problem of power-like stabilization by means of a boundary control. For a fixed initial condition and parameter k>0 of the rate of stabilization the existence of a boundary control such that the solution approaches zero at the rate 1/t k is demonstrated
A generalized simplest equation method and its application to the Boussinesq-Burgers equation.
Sudao, Bilige; Wang, Xiaomin
2015-01-01
In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method.
International Nuclear Information System (INIS)
Pinto, L C; Silvestrini, J H; Schettini, E B C
2011-01-01
In present paper, Navier-Stokes and Continuity equations for incompressible flow around an oscillating cylinder were numerically solved. Sixth order compact difference schemes were used to solve the spatial derivatives, while the time advance was carried out through second order Adams Bashforth accurate scheme. In order to represent the obstacle in the flow, the Immersed Boundary Method was adopted. In this method a force term is added to the Navier-Stokes equations representing the body. The simulations present results regarding the hydrodynamic coefficients and vortex wakes in agreement to experimental and numerical previous works and the physical lock-in phenomenon was identified. Comparing different methods to impose the IBM, it can be concluded that no alterations regarding the vortex shedding mode were observed. The Immersed Boundary Method techniques used here can represent the surface of an oscillating cylinder in the flow.
Sanderse, B.; Verstappen, R.W.C.P.; Koren, B.
2014-01-01
A discretization method for the incompressible Navier–Stokes equations conserving the secondary quantities kinetic energy and vorticity was introduced, besides the primary quantities mass and momentum. This method was extended to fourth order accuracy. In this paper we propose a new consistent
Boundary integral method for torsion of composite shafts
International Nuclear Information System (INIS)
Chou, S.I.; Mohr, J.A.
1987-01-01
The Saint-Venant torsion problem for homogeneous shafts with simply or multiply-connected regions has received a great deal of attention in the past. However, because of the mathematical difficulties inherent in the problem, very few problems of torsion of shafts with composite cross sections have been solved analytically. Muskhelishvili (1963) studied the torsion problem for shafts with cross sections having several solid inclusions surrounded by an elastic material. The problem of a circular shaft reinforced by a non-concentric round inclusion, a rectangular shaft composed of two rectangular parts made of different materials were solved. In this paper, a boundary integral equation method, which can be used to solve problems more complex than those considered by Katsikadelis et. al., is developed. Square shaft with two dissimilar rectangular parts, square shaft with a square inclusion are solved and the results compared with those given in the reference cited above. Finally, a square shaft composed of two rectangular parts with circular inclusion is solved. (orig./GL)
International Nuclear Information System (INIS)
Nepomechie, Rafael I
2013-01-01
An inhomogeneous T-Q equation has recently been proposed by Cao, Yang, Shi and Wang for the open spin-1/2 XXX chain with general (nondiagonal) boundary terms. We argue that a simplified version of this equation describes all the eigenvalues of the transfer matrix of this model. We also propose a generating function for the inhomogeneous T-Q equations of arbitrary spin. (fast track communication)
A high order multi-resolution solver for the Poisson equation with application to vortex methods
DEFF Research Database (Denmark)
Hejlesen, Mads Mølholm; Spietz, Henrik Juul; Walther, Jens Honore
A high order method is presented for solving the Poisson equation subject to mixed free-space and periodic boundary conditions by using fast Fourier transforms (FFT). The high order convergence is achieved by deriving mollified Green’s functions from a high order regularization function which...
Pseudospectral collocation methods for fourth order differential equations
Malek, Alaeddin; Phillips, Timothy N.
1994-01-01
Collocation schemes are presented for solving linear fourth order differential equations in one and two dimensions. The variational formulation of the model fourth order problem is discretized by approximating the integrals by a Gaussian quadrature rule generalized to include the values of the derivative of the integrand at the boundary points. Collocation schemes are derived which are equivalent to this discrete variational problem. An efficient preconditioner based on a low-order finite difference approximation to the same differential operator is presented. The corresponding multidomain problem is also considered and interface conditions are derived. Pseudospectral approximations which are C1 continuous at the interfaces are used in each subdomain to approximate the solution. The approximations are also shown to be C3 continuous at the interfaces asymptotically. A complete analysis of the collocation scheme for the multidomain problem is provided. The extension of the method to the biharmonic equation in two dimensions is discussed and results are presented for a problem defined in a nonrectangular domain.
A superlinear convergence estimate for an iterative method for the biharmonic equation
Energy Technology Data Exchange (ETDEWEB)
Horn, M.A. [Wichita State Univ., Wichita, KS (United States)
1996-12-31
In [CDH] a method for the solution of boundary value problems for the biharmonic equation using conformal mapping was investigated. The method is an implementation of the classical method of Muskhelishvili. In [CDH] it was shown, using the Hankel structure, that the linear system in [Musk] is the discretization of the identify plus a compact operator, and therefore the conjugate gradient method will converge superlinearly. The purpose of this paper is to give an estimate of the superlinear convergence in the case when the boundary curve is in a Hoelder class.
System and method for free-boundary surface extraction
Algarni, Marei
2017-10-26
A method of extracting surfaces in three-dimensional data includes receiving as inputs three-dimensional data and a seed point p located on a surface to be extracted. The method further includes propagating a front outwardly from the seed point p and extracting a plurality of ridge curves based on the propagated front. A surface boundary is detected based on a comparison of distances between adjacent ridge curves and the desired surface is extracted based on the detected surface boundary.
Applying homotopy analysis method for solving differential-difference equation
International Nuclear Information System (INIS)
Wang Zhen; Zou Li; Zhang Hongqing
2007-01-01
In this Letter, we apply the homotopy analysis method to solving the differential-difference equations. A simple but typical example is applied to illustrate the validity and the great potential of the generalized homotopy analysis method in solving differential-difference equation. Comparisons are made between the results of the proposed method and exact solutions. The results show that the homotopy analysis method is an attractive method in solving the differential-difference equations
Directory of Open Access Journals (Sweden)
Salih Yalcinbas
2016-01-01
Full Text Available In this study, a numerical approach is proposed to obtain approximate solutions of nonlinear system of second order boundary value problem. This technique is essentially based on the truncated Fermat series and its matrix representations with collocation points. Using the matrix method, we reduce the problem system of nonlinear algebraic equations. Numerical examples are also given to demonstrate the validity and applicability of the presented technique. The method is easy to implement and produces accurate results.
Adaptive integral equation methods in transport theory
International Nuclear Information System (INIS)
Kelley, C.T.
1992-01-01
In this paper, an adaptive multilevel algorithm for integral equations is described that has been developed with the Chandrasekhar H equation and its generalizations in mind. The algorithm maintains good performance when the Frechet derivative of the nonlinear map is singular at the solution, as happens in radiative transfer with conservative scattering and in critical neutron transport. Numerical examples that demonstrate the algorithm's effectiveness are presented
Symmetrized neutron transport equation and the fast Fourier transform method
International Nuclear Information System (INIS)
Sinh, N.Q.; Kisynski, J.; Mika, J.
1978-01-01
The differential equation obtained from the neutron transport equation by the application of the source iteration method in two-dimensional rectangular geometry is transformed into a symmetrized form with respect to one of the angular variables. The discretization of the symmetrized equation leads to finite difference equations based on the five-point scheme and solved by use of the fast Fourier transform method. Possible advantages of the approach are shown on test calculations
An analytical method for the inverse Cauchy problem of Lame equation in a rectangle
Grigor’ev, Yu
2018-04-01
In this paper, we present an analytical computational method for the inverse Cauchy problem of Lame equation in the elasticity theory. A rectangular domain is frequently used in engineering structures and we only consider the analytical solution in a two-dimensional rectangle, wherein a missing boundary condition is recovered from the full measurement of stresses and displacements on an accessible boundary. The essence of the method consists in solving three independent Cauchy problems for the Laplace and Poisson equations. For each of them, the Fourier series is used to formulate a first-kind Fredholm integral equation for the unknown function of data. Then, we use a Lavrentiev regularization method, and the termwise separable property of kernel function allows us to obtain a closed-form regularized solution. As a result, for the displacement components, we obtain solutions in the form of a sum of series with three regularization parameters. The uniform convergence and error estimation of the regularized solutions are proved.
Liska, Sebastian; Colonius, Tim
2017-02-01
A new parallel, computationally efficient immersed boundary method for solving three-dimensional, viscous, incompressible flows on unbounded domains is presented. Immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original (Peskin's) immersed boundary method. Unlike Peskin's method, boundary forces are regarded as Lagrange multipliers that are used to satisfy the no-slip condition. The incompressible Navier-Stokes equations are discretized on an unbounded staggered Cartesian grid and are solved in a finite number of operations using lattice Green's function techniques. These techniques are used to automatically enforce the natural free-space boundary conditions and to implement a novel block-wise adaptive grid that significantly reduces the run-time cost of solutions by limiting operations to grid cells in the immediate vicinity and near-wake region of the immersed surface. These techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half-explicit Runge-Kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation, incompressibility constraint, and no-slip constraint. Linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation-free nested projection technique. The algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems, e.g. discrete Poisson problems, that are compatible with recent parallel fast multipole methods for difference equations. Numerical experiments on low-aspect-ratio flat plates and spheres at Reynolds numbers up to 3700 are used to verify the accuracy and physical fidelity of the formulation.
International Nuclear Information System (INIS)
Tokuda, Shinji; Watanabe, Tomoko.
1996-08-01
The matching problem in resistive MagnetoHydroDynamic stability analysis by the asymptotic matching method has been reformulated as an initial-boundary value problem for the inner-layer equations describing the plasma dynamics in the thin layer around a rational surface. The third boundary conditions at boundaries of a finite interval are imposed on the inner layer equations in the formulation instead of asymptotic conditions at infinities. The finite difference method for this problem has been applied to model equations whose solutions are known in a closed form. It has been shown that the initial value problem and the associated eigenvalue problem for the model equations can be solved by the finite difference method with numerical stability. The formulation presented here enables the asymptotic matching method to be a practical method for the resistive MHD stability analysis. (author)
The Boundary Element Method Applied to the Two Dimensional Stefan Moving Boundary Problem
1991-03-15
Unc), - ( UGt )t - (UG,,),,] - (UG), If we integrate this equation with respect to r from 0 to t - c and with respect to and ij on the region 11(r...and others. "Moving Boundary Problems in Phase Change Mod- els," SIGNUM Newsletter, 20: 8-12 (1985). 21. Stefan, J. "Ober einige Probleme der Theorie ...ier Wirmelcitung," S.-B. \\Vein. Akad. Mat. Natur., 98: 173-484 (1889). 22.-. "flber (lie Theorie der Eisbildung insbesondere fiber die lisbildung im
Choi, Sae Il
2009-01-01
This study used simulation (a) to compare the kernel equating method to traditional equipercentile equating methods under the equivalent-groups (EG) design and the nonequivalent-groups with anchor test (NEAT) design and (b) to apply the parametric bootstrap method for estimating standard errors of equating. A two-parameter logistic item response…
The parabolic equation method for outdoor sound propagation
DEFF Research Database (Denmark)
Arranz, Marta Galindo
The parabolic equation method is a versatile tool for outdoor sound propagation. The present study has focused on the Cranck-Nicolson type Parabolic Equation method (CNPE). Three different applications of the CNPE method have been investigated. The first two applications study variations of the g......The parabolic equation method is a versatile tool for outdoor sound propagation. The present study has focused on the Cranck-Nicolson type Parabolic Equation method (CNPE). Three different applications of the CNPE method have been investigated. The first two applications study variations...
Simulation of Thermal Flow Problems via a Hybrid Immersed Boundary-Lattice Boltzmann Method
Directory of Open Access Journals (Sweden)
J. Wu
2012-01-01
Full Text Available A hybrid immersed boundary-lattice Boltzmann method (IB-LBM is presented in this work to simulate the thermal flow problems. In current approach, the flow field is resolved by using our recently developed boundary condition-enforced IB-LBM (Wu and Shu, (2009. The nonslip boundary condition on the solid boundary is enforced in simulation. At the same time, to capture the temperature development, the conventional energy equation is resolved. To model the effect of immersed boundary on temperature field, the heat source term is introduced. Different from previous studies, the heat source term is set as unknown rather than predetermined. Inspired by the idea in (Wu and Shu, (2009, the unknown is calculated in such a way that the temperature at the boundary interpolated from the corrected temperature field accurately satisfies the thermal boundary condition. In addition, based on the resolved temperature correction, an efficient way to compute the local and average Nusselt numbers is also proposed in this work. As compared with traditional implementation, no approximation for temperature gradients is required. To validate the present method, the numerical simulations of forced convection are carried out. The obtained results show good agreement with data in the literature.
Integral Equation Methods for Electromagnetic and Elastic Waves
Chew, Weng; Hu, Bin
2008-01-01
Integral Equation Methods for Electromagnetic and Elastic Waves is an outgrowth of several years of work. There have been no recent books on integral equation methods. There are books written on integral equations, but either they have been around for a while, or they were written by mathematicians. Much of the knowledge in integral equation methods still resides in journal papers. With this book, important relevant knowledge for integral equations are consolidated in one place and researchers need only read the pertinent chapters in this book to gain important knowledge needed for integral eq
Zhong, Jiaqi; Zeng, Cheng; Yuan, Yupeng; Zhang, Yuzhe; Zhang, Ye
2018-04-01
The aim of this paper is to present an explicit numerical algorithm based on improved spectral Galerkin method for solving the unsteady diffusion-convection-reaction equation. The principal characteristics of this approach give the explicit eigenvalues and eigenvectors based on the time-space separation method and boundary condition analysis. With the help of Fourier series and Galerkin truncation, we can obtain the finite-dimensional ordinary differential equations which facilitate the system analysis and controller design. By comparing with the finite element method, the numerical solutions are demonstrated via two examples. It is shown that the proposed method is effective.
Energy Technology Data Exchange (ETDEWEB)
Shao, Yan-Lin, E-mail: yanlin.shao@dnvgl.com; Faltinsen, Odd M.
2014-10-01
We propose a new efficient and accurate numerical method based on harmonic polynomials to solve boundary value problems governed by 3D Laplace equation. The computational domain is discretized by overlapping cells. Within each cell, the velocity potential is represented by the linear superposition of a complete set of harmonic polynomials, which are the elementary solutions of Laplace equation. By its definition, the method is named as Harmonic Polynomial Cell (HPC) method. The characteristics of the accuracy and efficiency of the HPC method are demonstrated by studying analytical cases. Comparisons will be made with some other existing boundary element based methods, e.g. Quadratic Boundary Element Method (QBEM) and the Fast Multipole Accelerated QBEM (FMA-QBEM) and a fourth order Finite Difference Method (FDM). To demonstrate the applications of the method, it is applied to some studies relevant for marine hydrodynamics. Sloshing in 3D rectangular tanks, a fully-nonlinear numerical wave tank, fully-nonlinear wave focusing on a semi-circular shoal, and the nonlinear wave diffraction of a bottom-mounted cylinder in regular waves are studied. The comparisons with the experimental results and other numerical results are all in satisfactory agreement, indicating that the present HPC method is a promising method in solving potential-flow problems. The underlying procedure of the HPC method could also be useful in other fields than marine hydrodynamics involved with solving Laplace equation.
Directory of Open Access Journals (Sweden)
I. C. Ramos
2015-10-01
Full Text Available We present the adaptation to non-free boundary conditions of a pseudospectral method based on the (complex Fourier transform. The method is applied to the numerical integration of the Oberbeck-Boussinesq equations in a Rayleigh-Bénard cell with no-slip boundary conditions for velocity and Dirichlet boundary conditions for temperature. We show the first results of a 2D numerical simulation of dry air convection at high Rayleigh number (. These results are the basis for the later study, by the same method, of wet convection in a solar still. Received: 20 Novembre 2014, Accepted: 15 September 2015; Edited by: C. A. Condat, G. J. Sibona; DOI:http://dx.doi.org/10.4279/PIP.070015 Cite as: I C Ramos, C B Briozzo, Papers in Physics 7, 070015 (2015
Stability analysis of a boundary layer over a hump using parabolized stability equations
Energy Technology Data Exchange (ETDEWEB)
Gao, B; Park, D H; Park, S O, E-mail: sopark@kaist.ac.kr [Division of Aerospace Engineering, Korea Advanced Institute of Science and Technology, Gusong-dong, Yusong-gu, Daejeon 305-701 (Korea, Republic of)
2011-10-15
Parabolized stability equations (PSEs) were used to investigate the stability of boundary layer flows over a small hump. The applicability of PSEs to flows with a small separation bubble was examined by comparing the result with DNS data. It was found that PSEs can efficiently track the disturbance waves with an acceptable accuracy in spite of a small separation bubble. A typical evolution scenario of Tollmien-Schlichting (TS) wave is presented. The adverse pressure gradient and the flow separation due to the hump have a strong effect on the amplification of the disturbances. The effect of hump width and height is also examined. When the width of the hump is reduced, the amplification factor is increased. The height of the hump is found to obviously influence the stability only when it is greater than the critical layer thickness.
Feshchenko, R. M.
Recently a new exact transparent boundary condition (TBC) for the 3D parabolic wave equation (PWE) in rectangular computational domain was derived. However in the obtained form it does not appear to be unconditionally stable when used with, for instance, the Crank-Nicolson finite-difference scheme. In this paper two new formulations of the TBC for the 3D PWE in rectangular computational domain are reported, which are likely to be unconditionally stable. They are based on an unconditionally stable fully discrete TBC for the Crank-Nicolson scheme for the 2D PWE. These new forms of the TBC can be used for numerical solution of the 3D PWE when a higher precision is required.
Stability analysis of a boundary layer over a hump using parabolized stability equations
International Nuclear Information System (INIS)
Gao, B; Park, D H; Park, S O
2011-01-01
Parabolized stability equations (PSEs) were used to investigate the stability of boundary layer flows over a small hump. The applicability of PSEs to flows with a small separation bubble was examined by comparing the result with DNS data. It was found that PSEs can efficiently track the disturbance waves with an acceptable accuracy in spite of a small separation bubble. A typical evolution scenario of Tollmien-Schlichting (TS) wave is presented. The adverse pressure gradient and the flow separation due to the hump have a strong effect on the amplification of the disturbances. The effect of hump width and height is also examined. When the width of the hump is reduced, the amplification factor is increased. The height of the hump is found to obviously influence the stability only when it is greater than the critical layer thickness.
Directory of Open Access Journals (Sweden)
Xue-Lian Jin
2017-01-01
Full Text Available The exponential stability of the monotubular heat exchanger equation with boundary observation possessing a time delay and inner control was investigated. Firstly, the close-loop system was translated into an abstract Cauchy problem in the suitable state space. A uniformly bounded C0-semigroup generated by the close-loop system, which implies that the unique solution of the system exists, was shown. Secondly, the spectrum configuration of the closed-loop system was analyzed and the eventual differentiability and the eventual compactness of the semigroup were shown by the resolvent estimates on some resolvent sets. This implies that the spectrum-determined growth assumption holds. Finally, a sufficient condition, which is related to the physical parameters in the system and is independent of the time delay, of the exponential stability of the closed-loop system was given.
Superfluid kinetic equation approach to the dynamics of the 3He A-B phase boundary
International Nuclear Information System (INIS)
Palmeri, J.
1990-01-01
The dynamics of the A-B phase boundary is studied using a nonequilibrium theory inspired by the microscopic approach to flux flow in type-II superconductors, namely a generalized two-fluid model consisting of coupled dynamical equations for the superfluid order parameter and the quasiparticle fluid. The interface mobility is obtained to lowest order in the front velocity in three different dynamical regimes: the gapless, hydrodynamic, and ballistic. Experiments have so far only been performed in the ballistic regime, and in this regime we find that, if only Andreev scattering processes are accounted for in the interface mobility, then the theoretical predictions for the terminal velocity of the planar interface are too big by a factor ∼2. From this we conclude that there may be other important contributions to the interface mobility in the ballistic regime, and we discuss a few possibilities
Solutions of hyperbolic equations with the CIP-BS method
International Nuclear Information System (INIS)
Utsumi, Takayuki; Koga, James; Yamagiwa, Mitsuru; Yabe, Takashi; Aoki, Takayuki
2004-01-01
In this paper, we show that a new numerical method, the Constrained Interpolation Profile - Basis Set (CIP-BS) method, can solve general hyperbolic equations efficiently. This method uses a simple polynomial basis set that is easily extendable to any desired higher-order accuracy. The interpolating profile is chosen so that the subgrid scale solution approaches the local real solution owing to the constraints from the spatial derivatives of the master equations. Then, introducing scalar products, the linear and nonlinear partial differential equations are uniquely reduced to the ordinary differential equations for values and spatial derivatives at the grid points. The method gives stable, less diffusive, and accurate results. It is successfully applied to the continuity equation, the Burgers equation, the Korteweg-de Vries equation, and one-dimensional shock tube problems. (author)
The modified simple equation method for solving some fractional ...
Indian Academy of Sciences (India)
... and processes in various areas of natural science. Thus, many effective and powerful methods have been established and improved. In this study, we establish exact solutions of the time fractional biological population model equation and nonlinearfractional Klein–Gordon equation by using the modified simple equation ...
A General Linear Method for Equating with Small Samples
Albano, Anthony D.
2015-01-01
Research on equating with small samples has shown that methods with stronger assumptions and fewer statistical estimates can lead to decreased error in the estimated equating function. This article introduces a new approach to linear observed-score equating, one which provides flexible control over how form difficulty is assumed versus estimated…
Differential and difference equations a comparison of methods of solution
Maximon, Leonard C
2016-01-01
This book, intended for researchers and graduate students in physics, applied mathematics and engineering, presents a detailed comparison of the important methods of solution for linear differential and difference equations - variation of constants, reduction of order, Laplace transforms and generating functions - bringing out the similarities as well as the significant differences in the respective analyses. Equations of arbitrary order are studied, followed by a detailed analysis for equations of first and second order. Equations with polynomial coefficients are considered and explicit solutions for equations with linear coefficients are given, showing significant differences in the functional form of solutions of differential equations from those of difference equations. An alternative method of solution involving transformation of both the dependent and independent variables is given for both differential and difference equations. A comprehensive, detailed treatment of Green’s functions and the associat...
Determining the equation of state of highly plasticised metals from boundary velocimetry
Hinch, E. J.
2010-01-01
This is a follow-up paper to that of Ockendon et al. (J.Eng.Math., this issue). A more detailed derivation is provided, along with a numerical method which determines directly the full equation of state relating pressure to density. The issue
Directory of Open Access Journals (Sweden)
A. Anguraj
2014-02-01
Full Text Available We study in this paper,the existence of solutions for fractional integro differential equations with impulsive and integral conditions by using fixed point method. We establish the Sufficient conditions and unique solution for given problem. An Example is also explained to the main results.
Free surface simulation of a two-layer fluid by boundary element method
Directory of Open Access Journals (Sweden)
Weoncheol Koo
2010-09-01
Full Text Available A two-layer fluid with free surface is simulated in the time domain by a two-dimensional potential-based Numerical Wave Tank (NWT. The developed NWT is based on the boundary element method and a leap-frog time integration scheme. A whole domain scheme including interaction terms between two layers is applied to solve the boundary integral equation. The time histories of surface elevations on both fluid layers in the respective wave modes are verified with analytic results. The amplitude ratios of upper to lower elevation for various density ratios and water depths are also compared.
Azis, Moh. Ivan; Kasbawati; Haddade, Amiruddin; Astuti Thamrin, Sri
2018-03-01
A boundary element method (BEM) is obtained for solving a boundary value problem of homogeneous anisotropic media governed by diffusion-convection equation. The application of the BEM is shown for two particular pollutant transport problems of Tello river and Unhas lake in Makassar Indonesia. For the two particular problems a variety of the coefficients of diffusion and the velocity components are taken. The results show that the solutions vary as the parameters change. And this suggests that one has to be careful in measuring or determining the values of the parameters.
A Comparison of Methods of Vertical Equating.
Loyd, Brenda H.; Hoover, H. D.
Rasch model vertical equating procedures were applied to three mathematics computation tests for grades six, seven, and eight. Each level of the test was composed of 45 items in three sets of 15 items, arranged in such a way that tests for adjacent grades had two sets (30 items) in common, and the sixth and eighth grades had 15 items in common. In…
A method for solving neutron transport equation
International Nuclear Information System (INIS)
Dimitrijevic, Z.
1993-01-01
The procedure for solving the transport equation by directly integrating for case one-dimensional uniform multigroup medium is shown. The solution is expressed in terms of linear combination of function H n (x,μ), and the coefficient is determined from given conditions. The solution is applied for homogeneous slab of critical thickness. (author)
International Nuclear Information System (INIS)
Tang, Bo; He, Yinnian; Wei, Leilei; Zhang, Xindong
2012-01-01
In this Letter, a generalized fractional sub-equation method is proposed for solving fractional differential equations with variable coefficients. Being concise and straightforward, this method is applied to the space–time fractional Gardner equation with variable coefficients. As a result, many exact solutions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. It is shown that the considered method provides a very effective, convenient and powerful mathematical tool for solving many other fractional differential equations in mathematical physics. -- Highlights: ► Study of fractional differential equations with variable coefficients plays a role in applied physical sciences. ► It is shown that the proposed algorithm is effective for solving fractional differential equations with variable coefficients. ► The obtained solutions may give insight into many considerable physical processes.
On the energetics of a damped beam-like equation for different boundary conditions
International Nuclear Information System (INIS)
Sandilo, S.H.; Sheikh, A.H.; Soomro, A.R.
2017-01-01
In this paper, the energy estimates for a damped linear homogeneous beam-like equation will be considered. The energy estimates will be studied for different BCs (Boundary Conditions) for the axially moving continuum. The problem has physical and engineering application. The applications are mostly occurring in models of conveyor belts and band-saw blades. The research study is focused on the Dirichlet, the Neumann and the Robin type of BCs. From physical point of view, the considered mathematical model expounds the transversal vibrations of a moving belt system or moving band-saw blade. It is assumed that a viscous damping parameter and the horizontal velocity are positive and constant. It will be shown in this paper that change in geometry or the physics of the boundaries can affect the stability properties of the system in general and stability depends on the axial direction of the motion. In all cases of the BCs, it will be shown that there is energy decay due to viscous damping parameter and it will also be shown that in some cases there is no conclusion whether the beam energy decreases or increases. The detailed physical interpretation of all terms and expressions is provided and studied in detail. (author)
On the Energetics of a Damped Beam-Like Equation for Different Boundary Conditions
Directory of Open Access Journals (Sweden)
SAJAD HUSSAIN SANDILO
2017-04-01
Full Text Available In this paper, the energy estimates for a damped linear homogeneous beam-like equation will be considered. The energy estimates will be studied for different BCs (Boundary Conditions for the axially moving continuum. The problem has physical and engineering application. The applications are mostly occurring in models of conveyor belts and band-saw blades. The research study is focused on the Dirichlet, the Neumann and the Robin type of BCs. From physical point of view, the considered mathematical model expounds the transversal vibrations of a moving belt system or moving band-saw blade. It is assumed that a viscous damping parameter and the horizontal velocity are positive and constant. It will be shown in this paper that change in geometry or the physics of the boundaries can affect the stability properties of the system in general and stability depends on the axial direction of the motion. In all cases of the BCs, it will be shown that there is energy decay due to viscous damping parameter and it will also be shown that in some cases there is no conclusion whether the beam energy decreases or increases. The detailed physical interpretation of all terms and expressions is provided and studied in detail.
Green's function method for perturbed Korteweg-de Vries equation
International Nuclear Information System (INIS)
Cai Hao; Huang Nianning
2003-01-01
The x-derivatives of squared Jost solution are the eigenfunctions with the zero eigenvalue of the linearized equation derived from the perturbed Korteweg-de Vries equation. A method similar to Green's function formalism is introduced to show the completeness of the squared Jost solutions in multi-soliton cases. It is not related to Lax equations directly, and thus it is beneficial to deal with the nonlinear equations with complicated Lax pair
Liu, Bingchen; Dong, Mengzhen; Li, Fengjie
2018-04-01
This paper deals with a reaction-diffusion problem with coupled nonlinear inner sources and nonlocal boundary flux. Firstly, we propose the critical exponents on nonsimultaneous blow-up under some conditions on the initial data. Secondly, we combine the scaling technique and the Green's identity method to determine four kinds of simultaneous blow-up rates. Thirdly, the lower and the upper bounds of blow-up time are derived by using Sobolev-type differential inequalities.
Exact solution of some linear matrix equations using algebraic methods
Djaferis, T. E.; Mitter, S. K.
1977-01-01
A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.
International Nuclear Information System (INIS)
Wang Qi; Chen Yong; Zhang Hongqing
2005-01-01
In this paper, we present a new Riccati equation rational expansion method to uniformly construct a series of exact solutions for nonlinear evolution equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general solutions. The solutions obtained in this paper include rational triangular periodic wave solutions, rational solitary wave solutions and rational wave solutions. The efficiency of the method can be demonstrated on (2 + 1)-dimensional Burgers equation
International Nuclear Information System (INIS)
Chen Yong; Wang Qi; Li Biao
2005-01-01
Based on a new general ansatz and a general subepuation, a new general algebraic method named elliptic equation rational expansion method is devised for constructing multiple travelling wave solutions in terms of rational special function for nonlinear evolution equations (NEEs). We apply the proposed method to solve Whitham-Broer-Kaup equation and explicitly construct a series of exact solutions which include rational form solitary wave solution, rational form triangular periodic wave solutions and rational wave solutions as special cases. In addition, the links among our proposed method with the method by Fan [Chaos, Solitons and Fractals 2004;20:609], are also clarified generally
International Nuclear Information System (INIS)
Saitoh, Ayumu; Kamitani, Atsushi; Takayama, Teruou; Nakamura, Hiroaki
2016-01-01
The extended boundary-node method (X-BNM) with the hierarchical-matrix (H-matrix) method has been developed and its performance has been investigated numerically. The results of computations show that the solver speed of the X-BNM with the H-matrix method is much faster than that of the standard X-BNM for the case where the number of boundary nodes exceeds a certain limit. Furthermore, the accuracy of the X-BNM with the H-matrix method is almost equal to that of the standard X-BNM. From these results, it is found that the H-matrix method is useful as the acceleration technique of the X-BNM. (author)
Modeling boundary-layer transition in DNS and LES using Parabolized Stability Equations
Lozano-Duran, Adrian; Hack, M. J. Philipp; Moin, Parviz
2016-11-01
The modeling of the laminar region and the prediction of the point of transition remain key challenges in the numerical simulation of boundary layers. The issue is of particular relevance for wall-modeled large eddy simulations which require 10 to 100 times higher grid resolution in the thin laminar region than in the turbulent regime. Our study examines the potential of the nonlinear parabolized stability equations (PSE) to provide an accurate, yet computationally efficient treatment of the growth of disturbances in the pre-transitional flow regime. The PSE captures the nonlinear interactions that eventually induce breakdown to turbulence, and can as such identify the onset of transition without relying on empirical correlations. Since the local PSE solution at the point of transition is the solution of the Navier-Stokes equations, it provides a natural inflow condition for large eddy and direct simulations by avoiding unphysical transients. We show that in a classical H-type transition scenario, a combined PSE/DNS approach can reproduce the skin-friction distribution obtained in reference direct numerical simulations. The computational cost in the laminar region is reduced by several orders of magnitude. Funded by the Air Force Office of Scientific Research.
Nodal spectrum method for solving neutron diffusion equation
International Nuclear Information System (INIS)
Sanchez, D.; Garcia, C. R.; Barros, R. C. de; Milian, D.E.
1999-01-01
Presented here is a new numerical nodal method for solving static multidimensional neutron diffusion equation in rectangular geometry. Our method is based on a spectral analysis of the nodal diffusion equations. These equations are obtained by integrating the diffusion equation in X, Y directions and then considering flat approximations for the current. These flat approximations are the only approximations that are considered in this method, as a result the numerical solutions are completely free from truncation errors. We show numerical results to illustrate the methods accuracy for coarse mesh calculations
OpenBEM - An open source Boundary Element Method software in Acoustics
DEFF Research Database (Denmark)
Cutanda Henriquez, Vicente; Juhl, Peter Møller
2010-01-01
OpenBEM is a collection of open source programs for solving the Helmholtz Equation using the Boundary Element Method. The collection is written in Matlab by the authors and contains codes for dealing with exterior and interior problems in two or three dimensions as well as implementation of axi...... with examples of its use. Previous research results where OpenBEM was employed will be mentioned....
Short climatology of the atmospheric boundary layer using acoustic methods
International Nuclear Information System (INIS)
Schubert, J.F.
1975-06-01
A climatology of the boundary layer of the atmosphere at the Savannah River Laboratory is being compiled using acoustic methods. The atmospheric phenomenon as depicted on the facsimile recorder is classified and then placed into one of sixteen categories. After classification, the height of the boundary layer is measured. From this information, frequency tables of boundary layer height and category are created and then analyzed for the percentage of time that each category was detected by the acoustic sounder. The sounder also accurately depicts the diurnal cycle of the boundary layer and, depending on the sensitivity of the system, shows microstructure that is normally unavailable using other methods of profiling. The acoustic sounder provides a means for continuous, real time measurements of the time rate of change of the depth of the boundary layer. This continuous record of the boundary layer with its convective cells, gravity waves, inversions, and frontal system passages permits the synoptic and complex climatology of the local area to be compiled. (U.S.)
The boundary element method applied to 3D magneto-electro-elastic dynamic problems
Igumnov, L. A.; Markov, I. P.; Kuznetsov, Iu A.
2017-11-01
Due to the coupling properties, the magneto-electro-elastic materials possess a wide number of applications. They exhibit general anisotropic behaviour. Three-dimensional transient analyses of magneto-electro-elastic solids can hardly be found in the literature. 3D direct boundary element formulation based on the weakly-singular boundary integral equations in Laplace domain is presented in this work for solving dynamic linear magneto-electro-elastic problems. Integral expressions of the three-dimensional fundamental solutions are employed. Spatial discretization is based on a collocation method with mixed boundary elements. Convolution quadrature method is used as a numerical inverse Laplace transform scheme to obtain time domain solutions. Numerical examples are provided to illustrate the capability of the proposed approach to treat highly dynamic problems.
Directory of Open Access Journals (Sweden)
Alsaedi Ahmed
2009-01-01
Full Text Available A generalized quasilinearization technique is developed to obtain a sequence of approximate solutions converging monotonically and quadratically to a unique solution of a boundary value problem involving Duffing type nonlinear integro-differential equation with integral boundary conditions. The convergence of order for the sequence of iterates is also established. It is found that the work presented in this paper not only produces new results but also yields several old results in certain limits.
Numerical solution of multi group-Two dimensional- Adjoint equation with finite element method
International Nuclear Information System (INIS)
Poursalehi, N.; Khalafi, H.; Shahriari, M.; Minoochehr
2008-01-01
Adjoint equation is used for perturbation theory in nuclear reactor design. For numerical solution of adjoint equation, usually two methods are applied. These are Finite Element and Finite Difference procedures. Usually Finite Element Procedure is chosen for solving of adjoint equation, because it is more use able in variety of geometries. In this article, Galerkin Finite Element method is discussed. This method is applied for numerical solving multi group, multi region and two dimensional (X, Y) adjoint equation. Typical reactor geometry is partitioned with triangular meshes and boundary condition for adjoint flux is considered zero. Finally, for a case of defined parameters, Finite Element Code was applied and results were compared with Citation Code
Analysis and Modeling of Boundary Layer Separation Method (BLSM).
Pethő, Dóra; Horváth, Géza; Liszi, János; Tóth, Imre; Paor, Dávid
2010-09-01
Nowadays rules of environmental protection strictly regulate pollution material emission into environment. To keep the environmental protection laws recycling is one of the useful methods of waste material treatment. We have developed a new method for the treatment of industrial waste water and named it boundary layer separation method (BLSM). We apply the phenomena that ions can be enriched in the boundary layer of the electrically charged electrode surface compared to the bulk liquid phase. The main point of the method is that the boundary layer at correctly chosen movement velocity can be taken out of the waste water without being damaged, and the ion-enriched boundary layer can be recycled. Electrosorption is a surface phenomenon. It can be used with high efficiency in case of large electrochemically active surface of electrodes. During our research work two high surface area nickel electrodes have been prepared. The value of electrochemically active surface area of electrodes has been estimated. The existence of diffusion part of the double layer has been experimentally approved. The electrical double layer capacity has been determined. Ion transport by boundary layer separation has been introduced. Finally we have tried to estimate the relative significance of physical adsorption and electrosorption.
Directory of Open Access Journals (Sweden)
Mohammad Siddique
2010-08-01
Full Text Available Parabolic partial differential equations with nonlocal boundary conditions arise in modeling of a wide range of important application areas such as chemical diffusion, thermoelasticity, heat conduction process, control theory and medicine science. In this paper, we present the implementation of positivity- preserving Padé numerical schemes to the two-dimensional diffusion equation with nonlocal time dependent boundary condition. We successfully implemented these numerical schemes for both Homogeneous and Inhomogeneous cases. The numerical results show that these Padé approximation based numerical schemes are quite accurate and easily implemented.
International Nuclear Information System (INIS)
Mano, Toshiyuki
2010-01-01
We study analytic properties of solutions to the q-Painlevé VI equation (q-P VI ), which was derived by Jimbo and Sakai as the compatibility condition for a connection preserving deformation (CPD) of a linear q-difference equation. We investigate local behaviours of solutions to q-P VI around a boundary point making use of the structure of the CPD. We also give a formula connecting the local behaviours of a solution around two boundary points. The results in this paper should be useful in future for studying more detailed global properties of solutions to q-P VI or exploring new special solutions with remarkable analytic properties
Sirenko, Kostyantyn; Asirim, Ozum Emre; Bagci, Hakan
2014-01-01
Discontinuous Galerkin time-domain method (DGTD) has been used extensively in computational electromagnetics for analyzing transient electromagnetic wave interactions on structures described with linear constitutive relations. DGTD expands unknown fields independently on disconnected mesh elements and uses numerical flux to realize information exchange between fields on different elements (J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Method, 2008). The numerical flux of choice for 'linear' Maxwell equations is the upwind flux, which mimics accurately the physical behavior of electromagnetic waves on discontinuous boundaries. It is obtained from the analytical solution of the Riemann problem defined on the boundary of two neighboring mesh elements.
Sirenko, Kostyantyn
2014-07-01
Discontinuous Galerkin time-domain method (DGTD) has been used extensively in computational electromagnetics for analyzing transient electromagnetic wave interactions on structures described with linear constitutive relations. DGTD expands unknown fields independently on disconnected mesh elements and uses numerical flux to realize information exchange between fields on different elements (J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Method, 2008). The numerical flux of choice for \\'linear\\' Maxwell equations is the upwind flux, which mimics accurately the physical behavior of electromagnetic waves on discontinuous boundaries. It is obtained from the analytical solution of the Riemann problem defined on the boundary of two neighboring mesh elements.
Enriched reproducing kernel particle method for fractional advection-diffusion equation
Ying, Yuping; Lian, Yanping; Tang, Shaoqiang; Liu, Wing Kam
2018-06-01
The reproducing kernel particle method (RKPM) has been efficiently applied to problems with large deformations, high gradients and high modal density. In this paper, it is extended to solve a nonlocal problem modeled by a fractional advection-diffusion equation (FADE), which exhibits a boundary layer with low regularity. We formulate this method on a moving least-square approach. Via the enrichment of fractional-order power functions to the traditional integer-order basis for RKPM, leading terms of the solution to the FADE can be exactly reproduced, which guarantees a good approximation to the boundary layer. Numerical tests are performed to verify the proposed approach.
Directory of Open Access Journals (Sweden)
A. Sakabekov
2016-01-01
Full Text Available We prove existence and uniqueness of the solution of the problem with initial and Maxwell-Auzhan boundary conditions for nonstationary nonlinear one-dimensional Boltzmann’s six-moment system equations in space of functions continuous in time and summable in square by a spatial variable. In order to obtain a priori estimation of the initial and boundary value problem for nonstationary nonlinear one-dimensional Boltzmann’s six-moment system equations we get the integral equality and then use the spherical representation of vector. Then we obtain the initial value problem for Riccati equation. We have managed to obtain a particular solution of this equation in an explicit form.
Variational iteration method for solving coupled-KdV equations
International Nuclear Information System (INIS)
Assas, Laila M.B.
2008-01-01
In this paper, the He's variational iteration method is applied to solve the non-linear coupled-KdV equations. This method is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. This technique provides a sequence of functions which converge to the exact solution of the coupled-KdV equations. This procedure is a powerful tool for solving coupled-KdV equations
Exp-function method for solving fractional partial differential equations.
Zheng, Bin
2013-01-01
We extend the Exp-function method to fractional partial differential equations in the sense of modified Riemann-Liouville derivative based on nonlinear fractional complex transformation. For illustrating the validity of this method, we apply it to the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation. As a result, some new exact solutions for them are successfully established.
Hashimoto, Itsuko
2016-01-01
We investigate the large-time behavior of the radially symmetric solution for Burgers equation on the exterior of a small ball in multi-dimensional space, where the boundary data and the data at the far field are prescribed. In a previous paper [1], we showed that, for the case in which the boundary data is equal to $0$ or negative, the asymptotic stability is the same as that for the viscous conservation law. In the present paper, it is proved that if the boundary data i...
The Initial and Neumann Boundary Value Problem for a Class Parabolic Monge-Ampère Equation
Directory of Open Access Journals (Sweden)
Juan Wang
2013-01-01
Full Text Available We consider the existence, uniqueness, and asymptotic behavior of a classical solution to the initial and Neumann boundary value problem for a class nonlinear parabolic equation of Monge-Ampère type. We show that such solution exists for all times and is unique. It converges eventually to a solution that satisfies a Neumann type problem for nonlinear elliptic equation of Monge-Ampère type.
Generalized differential transform method to differential-difference equation
International Nuclear Information System (INIS)
Zou Li; Wang Zhen; Zong Zhi
2009-01-01
In this Letter, we generalize the differential transform method to solve differential-difference equation for the first time. Two simple but typical examples are applied to illustrate the validity and the great potential of the generalized differential transform method in solving differential-difference equation. A Pade technique is also introduced and combined with GDTM in aim of extending the convergence area of presented series solutions. Comparisons are made between the results of the proposed method and exact solutions. Then we apply the differential transform method to the discrete KdV equation and the discrete mKdV equation, and successfully obtain solitary wave solutions. The results reveal that the proposed method is very effective and simple. We should point out that generalized differential transform method is also easy to be applied to other nonlinear differential-difference equation.
Energy Technology Data Exchange (ETDEWEB)
Goncalez, Tifani T. [Universidade Federal do Rio Grande do Sul (PROMEC/UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica; Segatto, Cynthia F.; Vilhena, Marco Tullio, E-mail: csegatto@pq.cnpq.b, E-mail: vilhena@pq.cnpq.b [Universidade Federal do Rio Grande do Sul (DMPA/UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Matematica Aplicada
2011-07-01
In this work, we report an analytical solution for the set of S{sub N} equations for the angular flux, in a rectangle, using the double Laplace transform technique. Its main idea comprehends the steps: application of the Laplace transform in one space variable, solution of the resulting equation by the LTS{sub N} method and reconstruction of the double Laplace transformed angular flux using the inversion theorem of the Laplace transform. We must emphasize that we perform the Laplace inversion by the LTS{sub N} method in the x direction, meanwhile we evaluate the inversion in the y direction performing the calculation of the corresponding line integral solution by the Stefest method. We have also to figure out that the application of Laplace transform to this type of boundary value problem introduces additional unknown functions associated to the partial derivatives of the angular flux at boundary. Based on the good results attained by the nodal LTS{sub N} method, we assume that the angular flux at boundary is also approximated by an exponential function. By analytical we mean that no approximation is done along the solution derivation except for the exponential hypothesis for the exiting angular flux at boundary. For sake of completeness, we report numerical comparisons of the obtained results against the ones of the literature. (author)
Optimization method for identifying the source term in an inverse wave equation
Directory of Open Access Journals (Sweden)
Arumugam Deiveegan
2017-08-01
Full Text Available In this work, we investigate the inverse problem of identifying a space-wise dependent source term of wave equation from the measurement on the boundary. On the basis of the optimal control framework, the inverse problem is transformed into an optimization problem. The existence and necessary condition of the minimizer for the cost functional are obtained. The projected gradient method and two-parameter model function method are applied to the minimization problem and numerical results are illustrated.
The Integral Equation Method and the Neumann Problem for the Poisson Equation on NTA Domains
Czech Academy of Sciences Publication Activity Database
Medková, Dagmar
2009-01-01
Roč. 63, č. 21 (2009), s. 227-247 ISSN 0378-620X Institutional research plan: CEZ:AV0Z10190503 Keywords : Poisson equation * Neumann problem * integral equation method Subject RIV: BA - General Mathematics Impact factor: 0.477, year: 2009
Receptor binding kinetics equations: Derivation using the Laplace transform method.
Hoare, Sam R J
Measuring unlabeled ligand receptor binding kinetics is valuable in optimizing and understanding drug action. Unfortunately, deriving equations for estimating kinetic parameters is challenging because it involves calculus; integration can be a frustrating barrier to the pharmacologist seeking to measure simple rate parameters. Here, a well-known tool for simplifying the derivation, the Laplace transform, is applied to models of receptor-ligand interaction. The method transforms differential equations to a form in which simple algebra can be applied to solve for the variable of interest, for example the concentration of ligand-bound receptor. The goal is to provide instruction using familiar examples, to enable investigators familiar with handling equilibrium binding equations to derive kinetic equations for receptor-ligand interaction. First, the Laplace transform is used to derive the equations for association and dissociation of labeled ligand binding. Next, its use for unlabeled ligand kinetic equations is exemplified by a full derivation of the kinetics of competitive binding equation. Finally, new unlabeled ligand equations are derived using the Laplace transform. These equations incorporate a pre-incubation step with unlabeled or labeled ligand. Four equations for measuring unlabeled ligand kinetics were compared and the two new equations verified by comparison with numerical solution. Importantly, the equations have not been verified with experimental data because no such experiments are evident in the literature. Equations were formatted for use in the curve-fitting program GraphPad Prism 6.0 and fitted to simulated data. This description of the Laplace transform method will enable pharmacologists to derive kinetic equations for their model or experimental paradigm under study. Application of the transform will expand the set of equations available for the pharmacologist to measure unlabeled ligand binding kinetics, and for other time
Weighted particle method for solving the Boltzmann equation
International Nuclear Information System (INIS)
Tohyama, M.; Suraud, E.
1990-01-01
We propose a new, deterministic, method of solution of the nuclear Boltzmann equation. In this Weighted Particle Method two-body collisions are treated by a Master equation for an occupation probability of each numerical particle. We apply the method to the quadrupole motion of 12 C. A comparison with usual stochastic methods is made. Advantages and disadvantages of the Weighted Particle Method are discussed
Determining the equation of state of highly plasticised metals from boundary velocimetry
Hinch, E. J.
2010-07-08
This is a follow-up paper to that of Ockendon et al. (J.Eng.Math., this issue). A more detailed derivation is provided, along with a numerical method which determines directly the full equation of state relating pressure to density. The issue of whether or not the problem is an inverse problem is discussed. © 2010 Springer Science+Business Media B.V.
Level Set Projection Method for Incompressible Navier-Stokes on Arbitrary Boundaries
Williams-Rioux, Bertrand
2012-01-12
Second order level set projection method for incompressible Navier-Stokes equations is proposed to solve flow around arbitrary geometries. We used rectilinear grid with collocated cell centered velocity and pressure. An explicit Godunov procedure is used to address the nonlinear advection terms, and an implicit Crank-Nicholson method to update viscous effects. An approximate pressure projection is implemented at the end of the time stepping using multigrid as a conventional fast iterative method. The level set method developed by Osher and Sethian [17] is implemented to address real momentum and pressure boundary conditions by the advection of a distance function, as proposed by Aslam [3]. Numerical results for the Strouhal number and drag coefficients validated the model with good accuracy for flow over a cylinder in the parallel shedding regime (47 < Re < 180). Simulations for an array of cylinders and an oscillating cylinder were performed, with the latter demonstrating our methods ability to handle dynamic boundary conditions.
Variable-mesh method of solving differential equations
Van Wyk, R.
1969-01-01
Multistep predictor-corrector method for numerical solution of ordinary differential equations retains high local accuracy and convergence properties. In addition, the method was developed in a form conducive to the generation of effective criteria for the selection of subsequent step sizes in step-by-step solution of differential equations.
Discontinuous Galerkin finite element methods for hyperbolic differential equations
van der Vegt, Jacobus J.W.; van der Ven, H.; Boelens, O.J.; Boelens, O.J.; Toro, E.F.
2002-01-01
In this paper a suryey is given of the important steps in the development of discontinuous Galerkin finite element methods for hyperbolic partial differential equations. Special attention is paid to the application of the discontinuous Galerkin method to the solution of the Euler equations of gas
Insights: A New Method to Balance Chemical Equations.
Garcia, Arcesio
1987-01-01
Describes a method designed to balance oxidation-reduction chemical equations. Outlines a method which is based on changes in the oxidation number that can be applied to both molecular reactions and ionic reactions. Provides examples and delineates the steps to follow for each type of equation balancing. (TW)
International Nuclear Information System (INIS)
Pontedeiro, E.M.B.D.; Maiorino, J.R.
1982-01-01
The linear equation transport, monoenergetic, with anysotropic scattering, in multiregions, by F sub(N) method, is resolved. The mathematical analysis used for this method consists in to use parcially the expansion method in singular autofunctions, or Case's method, aiming to derive a set of integral equations coupled to the angular distribution in the boundaries and interfaces, and then to approximate these distributions by polynomics of N order, aiming to derive, with the use of these boundary and continuity conditions in the interfaces, a set of algebric equations for the coef. of polynomical approximation. With the goal to obtain numerical results, a computer code (FNAM-1) with options for the number of regions, boundary conditions, F sub(N) approx order, were developed. Numerical results were then obtained for various sample problems and compared with the results published in the literature with the objective to demonstrate the precision and applicability of the F sub(N) method. (E.G.) [pt
The Split Coefficient Matrix method for hyperbolic systems of gasdynamic equations
Chakravarthy, S. R.; Anderson, D. A.; Salas, M. D.
1980-01-01
The Split Coefficient Matrix (SCM) finite difference method for solving hyperbolic systems of equations is presented. This new method is based on the mathematical theory of characteristics. The development of the method from characteristic theory is presented. Boundary point calculation procedures consistent with the SCM method used at interior points are explained. The split coefficient matrices that define the method for steady supersonic and unsteady inviscid flows are given for several examples. The SCM method is used to compute several flow fields to demonstrate its accuracy and versatility. The similarities and differences between the SCM method and the lambda-scheme are discussed.
Numerical simulation of the regularized long wave equation by He's homotopy perturbation method
Energy Technology Data Exchange (ETDEWEB)
Inc, Mustafa [Department of Mathematics, Firat University, 23119 Elazig (Turkey)], E-mail: minc@firat.edu.tr; Ugurlu, Yavuz [Department of Mathematics, Firat University, 23119 Elazig (Turkey)
2007-09-17
In this Letter, we present the homotopy perturbation method (shortly HPM) for obtaining the numerical solution of the RLW equation. We obtain the exact and numerical solutions of the Regularized Long Wave (RLW) equation for certain initial condition. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Comparison of the results with those of other methods have led us to significant consequences. The numerical solutions are compared with the known analytical solutions.
Numerical simulation of the regularized long wave equation by He's homotopy perturbation method
International Nuclear Information System (INIS)
Inc, Mustafa; Ugurlu, Yavuz
2007-01-01
In this Letter, we present the homotopy perturbation method (shortly HPM) for obtaining the numerical solution of the RLW equation. We obtain the exact and numerical solutions of the Regularized Long Wave (RLW) equation for certain initial condition. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Comparison of the results with those of other methods have led us to significant consequences. The numerical solutions are compared with the known analytical solutions
Cellular Neural Network Method for Critical Slab with Albedo Boundary Condition
International Nuclear Information System (INIS)
Pirouzmanda, A.; Hadada, K.; Suh, K. Y.
2010-01-01
The neutron transport problems have been studied theoretically and numerically for years. A number of researchers have studied the criticality problems of one-speed neutrons in homogeneous slabs and spheres using various methods. The Chebyshev polynomial approximation method (T N method) has lately been developed and improved for the neutron transport equation in slab geometry. The one-speed time-dependent neutron transport equation using the Cellular Neural Network (CNN) for the vacuum boundary condition has previously been solved. In this paper, we demonstrate the capacity of CNN in calculating the critical slab thickness for different boundary conditions and its variation with moments N. The architecture of the CNN has already been dealt with thoroughly. Essentially, the CNN is used to model a first-order system of the partial differential equations (PDEs). The original equations in the T N approximation are also a set of PDEs. The CNN approach lends itself to analog VLSI implementation. In this study, the CNN model is implemented using the HSpice software package
Patel, Jitendra Kumar; Natarajan, Ganesh
2018-05-01
We present an interpolation-free diffuse interface immersed boundary method for multiphase flows with moving bodies. A single fluid formalism using the volume-of-fluid approach is adopted to handle multiple immiscible fluids which are distinguished using the volume fractions, while the rigid bodies are tracked using an analogous volume-of-solid approach that solves for the solid fractions. The solution to the fluid flow equations are carried out using a finite volume-immersed boundary method, with the latter based on a diffuse interface philosophy. In the present work, we assume that the solids are filled with a "virtual" fluid with density and viscosity equal to the largest among all fluids in the domain. The solids are assumed to be rigid and their motion is solved using Newton's second law of motion. The immersed boundary methodology constructs a modified momentum equation that reduces to the Navier-Stokes equations in the fully fluid region and recovers the no-slip boundary condition inside the solids. An implicit incremental fractional-step methodology in conjunction with a novel hybrid staggered/non-staggered approach is employed, wherein a single equation for normal momentum at the cell faces is solved everywhere in the domain, independent of the number of spatial dimensions. The scalars are all solved for at the cell centres, with the transport equations for solid and fluid volume fractions solved using a high-resolution scheme. The pressure is determined everywhere in the domain (including inside the solids) using a variable coefficient Poisson equation. The solution to momentum, pressure, solid and fluid volume fraction equations everywhere in the domain circumvents the issue of pressure and velocity interpolation, which is a source of spurious oscillations in sharp interface immersed boundary methods. A well-balanced algorithm with consistent mass/momentum transport ensures robust simulations of high density ratio flows with strong body forces. The
Field Method for Integrating the First Order Differential Equation
Institute of Scientific and Technical Information of China (English)
JIA Li-qun; ZHENG Shi-wang; ZHANG Yao-yu
2007-01-01
An important modern method in analytical mechanics for finding the integral, which is called the field-method, is used to research the solution of a differential equation of the first order. First, by introducing an intermediate variable, a more complicated differential equation of the first order can be expressed by two simple differential equations of the first order, then the field-method in analytical mechanics is introduced for solving the two differential equations of the first order. The conclusion shows that the field-method in analytical mechanics can be fully used to find the solutions of a differential equation of the first order, thus a new method for finding the solutions of the first order is provided.
International Nuclear Information System (INIS)
Kirkpatrick, M.P.; Armfield, S.W.; Kent, J.H.
2003-01-01
A method is presented for representing curved boundaries for the solution of the Navier-Stokes equations on a non-uniform, staggered, three-dimensional Cartesian grid. The approach involves truncating the Cartesian cells at the boundary surface to create new cells which conform to the shape of the surface. We discuss in some detail the problems unique to the development of a cut cell method on a staggered grid. Methods for calculating the fluxes through the boundary cell faces, for representing pressure forces and for calculating the wall shear stress are derived and it is verified that the new scheme retains second-order accuracy in space. In addition, a novel 'cell-linking' method is developed which overcomes problems associated with the creation of small cells while avoiding the complexities involved with other cell-merging approaches. Techniques are presented for generating the geometric information required for the scheme based on the representation of the boundaries as quadric surfaces. The new method is tested for flow through a channel placed oblique to the grid and flow past a cylinder at Re=40 and is shown to give significant improvement over a staircase boundary formulation. Finally, it is used to calculate unsteady flow past a hemispheric protuberance on a plate at a Reynolds number of 800. Good agreement is obtained with experimental results for this flow
Discontinuous Galerkin Subgrid Finite Element Method for Heterogeneous Brinkman’s Equations
Iliev, Oleg P.
2010-01-01
We present a two-scale finite element method for solving Brinkman\\'s equations with piece-wise constant coefficients. This system of equations model fluid flows in highly porous, heterogeneous media with complex topology of the heterogeneities. We make use of the recently proposed discontinuous Galerkin FEM for Stokes equations by Wang and Ye in [12] and the concept of subgrid approximation developed for Darcy\\'s equations by Arbogast in [4]. In order to reduce the error along the coarse-grid interfaces we have added a alternating Schwarz iteration using patches around the coarse-grid boundaries. We have implemented the subgrid method using Deal.II FEM library, [7], and we present the computational results for a number of model problems. © 2010 Springer-Verlag Berlin Heidelberg.
Time reversal method with stabilizing boundary conditions for Photoacoustic tomography
International Nuclear Information System (INIS)
Chervova, Olga; Oksanen, Lauri
2016-01-01
We study an inverse initial source problem that models photoacoustic tomography measurements with array detectors, and introduce a method that can be viewed as a modification of the so called back and forth nudging method. We show that the method converges at an exponential rate under a natural visibility condition, with data given only on a part of the boundary of the domain of wave propagation. In this paper we consider the case of noiseless measurements. (paper)
Introduction to partial differential equations and Hilbert space methods
Gustafson, Karl E
1997-01-01
Easy-to-use text examines principal method of solving partial differential equations, 1st-order systems, computation methods, and much more. Over 600 exercises, with answers for many. Ideal for a 1-semester or full-year course.
students' preference of method of solving simultaneous equations
African Journals Online (AJOL)
Ugboduma,Samuel.O.
substitution method irrespective of their gender for solving simultaneous equations. A recommendation ... advantage given to one method over others. Students' interest .... from two (2) single girls' schools, two (2) single boys schools and ten.
Xie, Hai-Yang; Liu, Qian; Li, Jia-Hao; Fan, Liu-Yin; Cao, Cheng-Xi
2013-02-21
A novel moving redox reaction boundary (MRRB) model was developed for studying electrophoretic behaviors of analytes involving redox reaction on the principle of moving reaction boundary (MRB). Traditional potassium permanganate method was used to create the boundary model in agarose gel electrophoresis because of the rapid reaction rate associated with MnO(4)(-) ions and Fe(2+) ions. MRB velocity equation was proposed to describe the general functional relationship between velocity of moving redox reaction boundary (V(MRRB)) and concentration of reactant, and can be extrapolated to similar MRB techniques. Parameters affecting the redox reaction boundary were investigated in detail. Under the selected conditions, good linear relationship between boundary movement distance and time were obtained. The potential application of MRRB in electromigration redox reaction titration was performed in two different concentration levels. The precision of the V(MRRB) was studied and the relative standard deviations were below 8.1%, illustrating the good repeatability achieved in this experiment. The proposed MRRB model enriches the MRB theory and also provides a feasible realization of manual control of redox reaction process in electrophoretic analysis.
Description of internal flow problems by a boundary integral method with dipole panels
International Nuclear Information System (INIS)
Krieg, R.; Hailfinger, G.
1979-01-01
In reactor safety studies the failure of single components is postulated or sudden accident loadings are assumed and the consequences are investigated. Often as a first consequence highly transient three dimensional flow problems occur. In contrast to classical flow problems, in most of the above cases the fluid velocities are relatively small whereas the accelerations assume high values. As a consequence both, viscosity effects and dynamic pressures which are proportional to the square of the fluid velocities are usually negligible. For cases, where the excitation times are considerably longer than the times necessary for a wave to traverse characteristic regions of the fluid field, also the fluid compressibility is negligible. Under these conditions boundary integral methods are an appropriate tool to deal with the problem. Flow singularities are distributed over the fluid boundaries in such a way that pressure and velocity fields are obtained which satisfy the boundary conditions. In order to facilitate the numerical treatment the fluid boundaries are approximated by a finite number of panels with uniform singularity distributions on each of them. Consequently the pressure and velocity field of the given problem may be obtained by superposition of the corresponding fields due to these panels with their singularity intensities as unknown factors. Then satisfying the boundary conditions in so many boundary points as panels have been introduced, yields a system of linear equations which in general allows for a unique determination of the unknown intensities. (orig./RW)
Li, Zhiyuan; Huang, Xinchi; Yamamoto, Masahiro
2018-01-01
In this paper, we discuss an initial-boundary value problem (IBVP) for the multi-term time-fractional diffusion equation with x-dependent coefficients. By means of the Mittag-Leffler functions and the eigenfunction expansion, we reduce the IBVP to an equivalent integral equation to show the unique existence and the analyticity of the solution for the equation. Especially, in the case where all the coefficients of the time-fractional derivatives are non-negative, by the Laplace and inversion L...
Approximate Method for Solving the Linear Fuzzy Delay Differential Equations
Directory of Open Access Journals (Sweden)
S. Narayanamoorthy
2015-01-01
Full Text Available We propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using Adomian decomposition method. The detailed algorithm of the approach is provided. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show this proper features of this proposed method, numerical example is illustrated.
Directory of Open Access Journals (Sweden)
Juergen Saal
2007-02-01
Full Text Available It is proved under mild regularity assumptions on the data that the Navier-Stokes equations in bounded and unbounded noncylindrical regions admit a unique local-in-time strong solution. The result is based on maximal regularity estimates for the in spatial regions with a moving boundary obtained in [16] and the contraction mapping principle.
Denche, M.; Marhoune, A. L.
2001-01-01
We study a mixed problem with integral boundary conditions for a third-order partial differential equation of mixed type. We prove the existence and uniqueness of the solution. The proof is based on two-sided a priori estimates and on the density of the range of the operator generated by the considered problem.
Czech Academy of Sciences Publication Activity Database
Rontó, András; Samoilenko, A. M.
2007-01-01
Roč. 41, - (2007), s. 115-136 ISSN 1512-0015 R&D Projects: GA ČR(CZ) GA201/06/0254 Institutional research plan: CEZ:AV0Z10190503 Keywords : two-point problem * functional differential equation * singular boundary problem Subject RIV: BA - General Mathematics
Five-equation and robust three-equation methods for solution verification of large eddy simulation
Dutta, Rabijit; Xing, Tao
2018-02-01
This study evaluates the recently developed general framework for solution verification methods for large eddy simulation (LES) using implicitly filtered LES of periodic channel flows at friction Reynolds number of 395 on eight systematically refined grids. The seven-equation method shows that the coupling error based on Hypothesis I is much smaller as compared with the numerical and modeling errors and therefore can be neglected. The authors recommend five-equation method based on Hypothesis II, which shows a monotonic convergence behavior of the predicted numerical benchmark ( S C ), and provides realistic error estimates without the need of fixing the orders of accuracy for either numerical or modeling errors. Based on the results from seven-equation and five-equation methods, less expensive three and four-equation methods for practical LES applications were derived. It was found that the new three-equation method is robust as it can be applied to any convergence types and reasonably predict the error trends. It was also observed that the numerical and modeling errors usually have opposite signs, which suggests error cancellation play an essential role in LES. When Reynolds averaged Navier-Stokes (RANS) based error estimation method is applied, it shows significant error in the prediction of S C on coarse meshes. However, it predicts reasonable S C when the grids resolve at least 80% of the total turbulent kinetic energy.
Three-dimensional wake field analysis by boundary element method
International Nuclear Information System (INIS)
Miyata, K.
1987-01-01
A computer code HERTPIA was developed for the calculation of electromagnetic wake fields excited by charged particles travelling through arbitrarily shaped accelerating cavities. This code solves transient wave problems for a Hertz vector. The numerical analysis is based on the boundary element method. This program is validated by comparing its results with analytical solutions in a pill-box cavity
Fast sweeping method for the factored eikonal equation
Fomel, Sergey; Luo, Songting; Zhao, Hongkai
2009-09-01
We develop a fast sweeping method for the factored eikonal equation. By decomposing the solution of a general eikonal equation as the product of two factors: the first factor is the solution to a simple eikonal equation (such as distance) or a previously computed solution to an approximate eikonal equation. The second factor is a necessary modification/correction. Appropriate discretization and a fast sweeping strategy are designed for the equation of the correction part. The key idea is to enforce the causality of the original eikonal equation during the Gauss-Seidel iterations. Using extensive numerical examples we demonstrate that (1) the convergence behavior of the fast sweeping method for the factored eikonal equation is the same as for the original eikonal equation, i.e., the number of iterations for the Gauss-Seidel iterations is independent of the mesh size, (2) the numerical solution from the factored eikonal equation is more accurate than the numerical solution directly computed from the original eikonal equation, especially for point sources.
International Nuclear Information System (INIS)
Zhang Huiqun
2009-01-01
By using a new coupled Riccati equations, a direct algebraic method, which was applied to obtain exact travelling wave solutions of some complex nonlinear equations, is improved. And the exact travelling wave solutions of the complex KdV equation, Boussinesq equation and Klein-Gordon equation are investigated using the improved method. The method presented in this paper can also be applied to construct exact travelling wave solutions for other nonlinear complex equations.
Numerical methods to solve the two-dimensional heat conduction equation
International Nuclear Information System (INIS)
Santos, R.S. dos.
1981-09-01
A class of numerical methods, called 'Hopscotch Algorithms', was used to solve the heat conduction equation in cylindrical geometry. Using a time dependent heat source, the temperature versus time behaviour of cylindric rod was analysed. Numerical simulation was used to study the stability and the convergence of each different method. Another test had the temperature specified on the outer surface as boundary condition. The various Hopscotch methods analysed exhibit differing degrees of accuracy, few of them being so accurate as the ADE method, but requiring more computational operations than the later, were observed. Finally, compared with the so called ODD-EVEN method, two other Hopscotch methods, are more time consuming. (Author) [pt
Angelidis, Dionysios; Chawdhary, Saurabh; Sotiropoulos, Fotis
2016-11-01
A novel numerical method is developed for solving the 3D, unsteady, incompressible Navier-Stokes equations on locally refined fully unstructured Cartesian grids in domains with arbitrarily complex immersed boundaries. Owing to the utilization of the fractional step method on an unstructured Cartesian hybrid staggered/non-staggered grid layout, flux mismatch and pressure discontinuity issues are avoided and the divergence free constraint is inherently satisfied to machine zero. Auxiliary/hanging nodes are used to facilitate the discretization of the governing equations. The second-order accuracy of the solver is ensured by using multi-dimension Lagrange interpolation operators and appropriate differencing schemes at the interface of regions with different levels of refinement. The sharp interface immersed boundary method is augmented with local near-boundary refinement to handle arbitrarily complex boundaries. The discrete momentum equation is solved with the matrix free Newton-Krylov method and the Krylov-subspace method is employed to solve the Poisson equation. The second-order accuracy of the proposed method on unstructured Cartesian grids is demonstrated by solving the Poisson equation with a known analytical solution. A number of three-dimensional laminar flow simulations of increasing complexity illustrate the ability of the method to handle flows across a range of Reynolds numbers and flow regimes. Laminar steady and unsteady flows past a sphere and the oblique vortex shedding from a circular cylinder mounted between two end walls demonstrate the accuracy, the efficiency and the smooth transition of scales and coherent structures across refinement levels. Large-eddy simulation (LES) past a miniature wind turbine rotor, parameterized using the actuator line approach, indicates the ability of the fully unstructured solver to simulate complex turbulent flows. Finally, a geometry resolving LES of turbulent flow past a complete hydrokinetic turbine illustrates
Conference on Boundary and Interior Layers : Computational and Asymptotic Methods
2015-01-01
This volume offers contributions reflecting a selection of the lectures presented at the international conference BAIL 2014, which was held from 15th to 19th September 2014 at the Charles University in Prague, Czech Republic. These are devoted to the theoretical and/or numerical analysis of problems involving boundary and interior layers and methods for solving these problems numerically. The authors are both mathematicians (pure and applied) and engineers, and bring together a large number of interesting ideas. The wide variety of topics treated in the contributions provides an excellent overview of current research into the theory and numerical solution of problems involving boundary and interior layers. .
International Nuclear Information System (INIS)
Prinari, Barbara; Ablowitz, Mark J.; Biondini, Gino
2006-01-01
The inverse scattering transform for the vector defocusing nonlinear Schroedinger (NLS) equation with nonvanishing boundary values at infinity is constructed. The direct scattering problem is formulated on a two-sheeted covering of the complex plane. Two out of the six Jost eigenfunctions, however, do not admit an analytic extension on either sheet of the Riemann surface. Therefore, a suitable modification of both the direct and the inverse problem formulations is necessary. On the direct side, this is accomplished by constructing two additional analytic eigenfunctions which are expressed in terms of the adjoint eigenfunctions. The discrete spectrum, bound states and symmetries of the direct problem are then discussed. In the most general situation, a discrete eigenvalue corresponds to a quartet of zeros (poles) of certain scattering data. The inverse scattering problem is formulated in terms of a generalized Riemann-Hilbert (RH) problem in the upper/lower half planes of a suitable uniformization variable. Special soliton solutions are constructed from the poles in the RH problem, and include dark-dark soliton solutions, which have dark solitonic behavior in both components, as well as dark-bright soliton solutions, which have one dark and one bright component. The linear limit is obtained from the RH problem and is shown to correspond to the Fourier transform solution obtained from the linearized vector NLS system
Modified Chebyshev Collocation Method for Solving Differential Equations
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M Ziaul Arif
2015-05-01
Full Text Available This paper presents derivation of alternative numerical scheme for solving differential equations, which is modified Chebyshev (Vieta-Lucas Polynomial collocation differentiation matrices. The Scheme of modified Chebyshev (Vieta-Lucas Polynomial collocation method is applied to both Ordinary Differential Equations (ODEs and Partial Differential Equations (PDEs cases. Finally, the performance of the proposed method is compared with finite difference method and the exact solution of the example. It is shown that modified Chebyshev collocation method more effective and accurate than FDM for some example given.
Some free boundary problems in potential flow regime usinga based level set method
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Garzon, M.; Bobillo-Ares, N.; Sethian, J.A.
2008-12-09
Recent advances in the field of fluid mechanics with moving fronts are linked to the use of Level Set Methods, a versatile mathematical technique to follow free boundaries which undergo topological changes. A challenging class of problems in this context are those related to the solution of a partial differential equation posed on a moving domain, in which the boundary condition for the PDE solver has to be obtained from a partial differential equation defined on the front. This is the case of potential flow models with moving boundaries. Moreover the fluid front will possibly be carrying some material substance which will diffuse in the front and be advected by the front velocity, as for example the use of surfactants to lower surface tension. We present a Level Set based methodology to embed this partial differential equations defined on the front in a complete Eulerian framework, fully avoiding the tracking of fluid particles and its known limitations. To show the advantages of this approach in the field of Fluid Mechanics we present in this work one particular application: the numerical approximation of a potential flow model to simulate the evolution and breaking of a solitary wave propagating over a slopping bottom and compare the level set based algorithm with previous front tracking models.
Solving hyperbolic equations with finite volume methods
Vázquez-Cendón, M Elena
2015-01-01
Finite volume methods are used in numerous applications and by a broad multidisciplinary scientific community. The book communicates this important tool to students, researchers in training and academics involved in the training of students in different science and technology fields. The selection of content is based on the author’s experience giving PhD and master courses in different universities. In the book the introduction of new concepts and numerical methods go together with simple exercises, examples and applications that contribute to reinforce them. In addition, some of them involve the execution of MATLAB codes. The author promotes an understanding of common terminology with a balance between mathematical rigor and physical intuition that characterizes the origin of the methods. This book aims to be a first contact with finite volume methods. Once readers have studied it, they will be able to follow more specific bibliographical references and use commercial programs or open source software withi...
Quadratic algebras in the noncommutative integration method of wave equation
International Nuclear Information System (INIS)
Varaksin, O.L.
1995-01-01
The paper deals with the investigation of applications of the method of noncommutative integration of linear differential equations by partial derivatives. Nontrivial example was taken for integration of three-dimensions wave equation with the use of non-Abelian quadratic algebras
Application of the trial equation method for solving some nonlinear ...
Indian Academy of Sciences (India)
Therefore, our aim is just to find the function F. Liu has obtained a number of exact solutions to many nonlinear differential equations when F(u) is a polynomial or a rational function. ... In this study, we apply the trial equation method to seek exact solutions of the ... twice and setting the integration constant to zero, we have.
An auxiliary differential equation FDTD method for anisotropic magnetized plasmas
International Nuclear Information System (INIS)
Liu Shaobin; Mo Jinjun; Yuan Naichang
2004-01-01
An auxiliary differential equation finite-difference time-domain (ADE-FDTD) methodology for anisotropic magnetized plasmas is derived. The method is based on a difference approximation of the auxiliary differential equation. A comparison with the JEC method is included. The CPU time saving by several times and accuracy of the method are confirmed by computing the reflection and transmission through a magnetized plasma layer with the direction of propagation parallel to the direction of the biasing field
Systems of evolution equations and the singular perturbation method
International Nuclear Information System (INIS)
Mika, J.
Several fundamental theorems are presented important for the solution of linear evolution equations in the Banach space. The algorithm is deduced extending the solution of the system of singularly perturbed evolution equations into an asymptotic series with respect to a small positive parameter. The asymptotic convergence is shown of an approximate solution to the accurate solution. Singularly perturbed evolution equations of the resonance type were analysed. The special role is considered of the asymptotic equivalence of P1 equations obtained as the first order approximation if the spherical harmonics method is applied to the linear Boltzmann equation, and the diffusion equations of the linear transport theory where the small parameter approaches zero. (J.B.)
The Galerkin Finite Element Method for A Multi-term Time-Fractional Diffusion equation
Jin, Bangti; Lazarov, Raytcho; Liu, Yikan; Zhou, Zhi
2014-01-01
We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite...
Dynamic-stiffness matrix of embedded and pile foundations by indirect boundary-element method
International Nuclear Information System (INIS)
Wolf, J.P.; Darbre, G.R.
1984-01-01
The boundary-integral equation method is well suited for the calculation of the dynamic-stiffness matrix of foundations embedded in a layered visco-elastic halfspace (or a transmitting boundary of arbitrary shape), which represents an unbounded domain. It also allows pile groups to be analyzed, taking pile-soil-pile interaction into account. The discretization of this boundary-element method is restricted to the structure-soil interface. All trial functions satisfy exactly the field equations and the radiation condition at infinity. In the indirect boundary-element method distributed source loads of initially unknown intensities act on a source line located in the excavated part of the soil and are determined such that the prescribed boundary conditions on the structure-soil interface are satisfied in an average sense. In the two-dimensional case the variables are expanded in a Fourier integral in the wave number domain, while in three dimensions, Fourier series in the circumferential direction and bessel functions of the wave number domain, while in three dimensions, Fourier series in the circumferential direction and Bessel functions of the wave number in the radial direction are selected. Accurate results arise with a small number of parameters of the loads acting on a source line which should coincide with the structure-soil interface. In a parametric study the dynamic-stiffness matrices of rectangular foundations of various aspect ratios embedded in a halfplane and in a layer built-in at its base are calculated. For the halfplane, the spring coefficients for the translational directions hardly depend on the embedment, while the corresponding damping coefficients increase for larger embedments, this tendency being more pronounced in the horizontal direction. (orig.)
Numerical methods for stochastic partial differential equations with white noise
Zhang, Zhongqiang
2017-01-01
This book covers numerical methods for stochastic partial differential equations with white noise using the framework of Wong-Zakai approximation. The book begins with some motivational and background material in the introductory chapters and is divided into three parts. Part I covers numerical stochastic ordinary differential equations. Here the authors start with numerical methods for SDEs with delay using the Wong-Zakai approximation and finite difference in time. Part II covers temporal white noise. Here the authors consider SPDEs as PDEs driven by white noise, where discretization of white noise (Brownian motion) leads to PDEs with smooth noise, which can then be treated by numerical methods for PDEs. In this part, recursive algorithms based on Wiener chaos expansion and stochastic collocation methods are presented for linear stochastic advection-diffusion-reaction equations. In addition, stochastic Euler equations are exploited as an application of stochastic collocation methods, where a numerical compa...
Memory allocation and computations for Laplace’s equation of 3-D arbitrary boundary problems
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Tsay Tswn-Syau
2017-01-01
Full Text Available Computation iteration schemes and memory allocation technique for finite difference method were presented in this paper. The transformed form of a groundwater flow problem in the generalized curvilinear coordinates was taken to be the illustrating example and a 3-dimensional second order accurate 19-point scheme was presented. Traditional element-by-element methods (e.g. SOR are preferred since it is simple and memory efficient but time consuming in computation. For efficient memory allocation, an index method was presented to store the sparse non-symmetric matrix of the problem. For computations, conjugate-gradient-like methods were reported to be computationally efficient. Among them, using incomplete Choleski decomposition as preconditioner was reported to be good method for iteration convergence. In general, the developed index method in this paper has the following advantages: (1 adaptable to various governing and boundary conditions, (2 flexible for higher order approximation, (3 independence of problem dimension, (4 efficient for complex problems when global matrix is not symmetric, (5 convenience for general sparse matrices, (6 computationally efficient in the most time consuming procedure of matrix multiplication, and (7 applicable to any developed matrix solver.
Study on boundary search method for DFM mesh generation
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Li Ri
2012-08-01
Full Text Available The boundary mesh of the casting model was determined by direct calculation on the triangular facets extracted from the STL file of the 3D model. Then the inner and outer grids of the model were identified by the algorithm in which we named Inner Seed Grid Method. Finally, a program to automatically generate a 3D FDM mesh was compiled. In the paper, a method named Triangle Contraction Search Method (TCSM was put forward to ensure not losing the boundary grids; while an algorithm to search inner seed grids to identify inner/outer grids of the casting model was also brought forward. Our algorithm was simple, clear and easy to construct program. Three examples for the casting mesh generation testified the validity of the program.
A conservative finite difference method for the numerical solution of plasma fluid equations
International Nuclear Information System (INIS)
Colella, P.; Dorr, M.R.; Wake, D.D.
1999-01-01
This paper describes a numerical method for the solution of a system of plasma fluid equations. The fluid model is similar to those employed in the simulation of high-density, low-pressure plasmas used in semiconductor processing. The governing equations consist of a drift-diffusion model of the electrons, together with an internal energy equation, coupled via Poisson's equation to a system of Euler equations for each ion species augmented with electrostatic force, collisional, and source/sink terms. The time integration of the full system is performed using an operator splitting that conserves space charge and avoids dielectric relaxation timestep restrictions. The integration of the individual ion species and electrons within the time-split advancement is achieved using a second-order Godunov discretization of the hyperbolic terms, modified to account for the significant role of the electric field in the propagation of acoustic waves, combined with a backward Euler discretization of the parabolic terms. Discrete boundary conditions are employed to accommodate the plasma sheath boundary layer on underresolved grids. The algorithm is described for the case of a single Cartesian grid as the first step toward an implementation on a locally refined grid hierarchy in which the method presented here may be applied on each refinement level
ICM: an Integrated Compartment Method for numerically solving partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Yeh, G.T.
1981-05-01
An integrated compartment method (ICM) is proposed to construct a set of algebraic equations from a system of partial differential equations. The ICM combines the utility of integral formulation of finite element approach, the simplicity of interpolation of finite difference approximation, and the flexibility of compartment analyses. The integral formulation eases the treatment of boundary conditions, in particular, the Neumann-type boundary conditions. The simplicity of interpolation provides great economy in computation. The flexibility of discretization with irregular compartments of various shapes and sizes offers advantages in resolving complex boundaries enclosing compound regions of interest. The basic procedures of ICM are first to discretize the region of interest into compartments, then to apply three integral theorems of vectors to transform the volume integral to the surface integral, and finally to use interpolation to relate the interfacial values in terms of compartment values to close the system. The Navier-Stokes equations are used as an example of how to derive the corresponding ICM alogrithm for a given set of partial differential equations. Because of the structure of the algorithm, the basic computer program remains the same for cases in one-, two-, or three-dimensional problems.
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Benoist, P; Kavenoky, A [Commissariat a l' Energie Atomique, Saclay (France). Centre d' Etudes Nucleaires
1968-01-15
In a new method of approximation of the Boltzmann equation, one starts from a particular form of the equation which involves only the angular flux at the boundary of the considered medium and where the space variable does not appear explicitly. Expanding in orthogonal polynomials the angular flux of neutrons leaking from the medium and making no assumption about the angular flux within the medium, very good approximations to several classical plane geometry problems, i.e. the albedo of slabs and the transmission by slabs, the extrapolation length of the Milne problem, the spectrum of neutrons reflected by a semi-infinite slowing down medium. The method can be extended to other geometries. (authors) [French] On etablit une nouvelle methode d'approximation pour l'equation de Boltzmann en partant d'une forme particuliere de cette equation qui n'implique que le flux angulaire a la frontiere du milieu et ou les variables d'espace n'apparaissent pas explicitement. Par un developpement en polynomes orthogonaux du flux angulaire sortant du milieu et sans faire d'hypothese sur le flux angulaire a l'interieur du milieu, on obtient de tres bonnes approximations pour plusieurs problemes classiques en geometrie plane: l'albedo et le facteur de transmission des plaques, la longueur d'extrapolation du probleme de Milne, le spectre des neutrons reflechis par un milieu semi-infini ralentisseur. La methode se generalise a d'autres geometries. (auteurs)
An element-free Galerkin (EFG) method for generalized Fisher equations (GFE)
International Nuclear Information System (INIS)
Shi Ting-Yu; Ge Hong-Xia; Cheng Rong-Jun
2013-01-01
A generalized Fisher equation (GFE) relates the time derivative of the average of the intrinsic rate of growth to its variance. The exact mathematical result of the GFE has been widely used in population dynamics and genetics, where it originated. Many researchers have studied the numerical solutions of the GFE, up to now. In this paper, we introduce an element-free Galerkin (EFG) method based on the moving least-square approximation to approximate positive solutions of the GFE from population dynamics. Compared with other numerical methods, the EFG method for the GFE needs only scattered nodes instead of meshing the domain of the problem. The Galerkin weak form is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. In comparison with the traditional method, numerical solutions show that the new method has higher accuracy and better convergence. Several numerical examples are presented to demonstrate the effectiveness of the method
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A. D. Chernyshov
2017-01-01
Full Text Available The brief presentation of the method of fast expansions is given to solve nonlinear differential equations. Application rules of the operator of fast expansions are specified for solving differential equations. According to the method of fast expansions, an unknown function can be represented as the sum of the boundary function and Fourier series sines and cosines for one variable. The special construction of the boundary functions leads to reasonably fast convergence of the Fourier series, so that for engineering calculations, it is sufficient to consider only the first three members. The method is applicable both to linear and nonlinear integro-differential systems. By means of applying the method of fast expansions to nonlinear Navier-Stokes equations the problem is reduced to a closed system of ordinary differential equations, which solution doesn't represent special difficulties. We can reapply the method of fast expansions to the resulting system of differential equations and reduce the original problem to a system of algebraic equations. If the problem is n-dimensional, then after n-fold application of the method of fast expansions the problem will be reduced to a closed algebraic system. Finally, we obtain an analytic-form solution of complicated boundary value problem in partial derivatives. The flow of an incompressible viscous fluid of Navier–Stokes is considered in a curvilinear pipe. The problem is reduced to solving a closed system of ordinary differential equations with boundary conditions by the method of fast expansions. The article considers peculiarities of finding the coefficients of boundary functions and Fourier coefficients for the zero-order and first-order operators of fast expansions. Obtaining the analytic-form solution is of great interest, because it allows to analyze and to investigate the influence of various factors on the properties of the viscous fluid in specific cases.
Numerical Methods for Partial Differential Equations.
1984-01-09
iteration or the conjugate gradient method. The smoothing sweeps are used to annihilate the highly oscillatory (compared to the grid spacing) components of...53 52 "- 33 41 *32 * . 31 * 21 - 11 O- carrius plane rotacions o I ~~arr: ’.trix vrS2-0 Cf A Figure 4. QM fiitorization of a BLTE (1,2) mnitrix
Galerkin method for solving diffusion equations
International Nuclear Information System (INIS)
Tsapelkin, E.S.
1975-01-01
A programme for the solution of the three-dimensional two-group multizone neutron diffusion problem in (x, y, z)-geometry is described. The programme XYZ-5 gives the currents of both groups, the effective neutron multiplication coefficient and several integral properties of the reactor. The solution was found with the Galerkin method using speciallly constructed and chosen coordinate functions. The programme is written in ALGOL-60 and consists of 5 parts. Its text is given
Incompressible spectral-element method: Derivation of equations
Deanna, Russell G.
1993-01-01
A fractional-step splitting scheme breaks the full Navier-Stokes equations into explicit and implicit portions amenable to the calculus of variations. Beginning with the functional forms of the Poisson and Helmholtz equations, we substitute finite expansion series for the dependent variables and derive the matrix equations for the unknown expansion coefficients. This method employs a new splitting scheme which differs from conventional three-step (nonlinear, pressure, viscous) schemes. The nonlinear step appears in the conventional, explicit manner, the difference occurs in the pressure step. Instead of solving for the pressure gradient using the nonlinear velocity, we add the viscous portion of the Navier-Stokes equation from the previous time step to the velocity before solving for the pressure gradient. By combining this 'predicted' pressure gradient with the nonlinear velocity in an explicit term, and the Crank-Nicholson method for the viscous terms, we develop a Helmholtz equation for the final velocity.
Knezevic, David; Patera, Anthony T.; Huynh, Dinh Bao Phuong
2010-01-01
We present a certified reduced basis (RB) method for the heat equation and wave equation. The critical ingredients are certified RB approximation of the Laplace transform; the inverse Laplace transform to develop the time-domain RB output approximation and rigorous error bound; a (Butterworth) filter in time to effect the necessary “modal” truncation; RB eigenfunction decomposition and contour integration for Offline–Online decomposition. We present numerical results to demonstrate the accura...
International Nuclear Information System (INIS)
Dirin, M.M.; Karimian, S.M.H.; Maerefat, M.
2003-01-01
An engineering method has been modified for the prediction of aerodynamic heating of the hypersonic bodies in the leeward region. This is achieved using our proposed new method for determining streamlines in the leeward region. The modified form of Maslen's second order relation, which calculates pressure in the shock layer explicitly, is employed. The inviscid outer flow within the shock layer is first solved. The calculated solution, then, is used to determine the flow properties at the boundary layer edge and the orientation of the surface streamlines. Boundary layer equations, written in the streamline coordinates, are integrated along the surface to obtain the rate of heat transferred to the body surface. The present method is an inverse method in which the body shape is obtained according to the shape of the shock. In general, inviscid-boundary layer engineering methods calculate accurately the orientation of streamlines in the windward side only, and therefore they are not usually applicable in the leeward region. In the present study, a new method is proposed to determine the orientation of the surface streamlines in the leeward region. Using the present method, three-dimensional hypersonic flow is solved fast and easy all around a cone. The obtained results show that the corrections presented in this study extend excellently the application of the method to the leeward region. (author)
Various Newton-type iterative methods for solving nonlinear equations
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Manoj Kumar
2013-10-01
Full Text Available The aim of the present paper is to introduce and investigate new ninth and seventh order convergent Newton-type iterative methods for solving nonlinear equations. The ninth order convergent Newton-type iterative method is made derivative free to obtain seventh-order convergent Newton-type iterative method. These new with and without derivative methods have efficiency indices 1.5518 and 1.6266, respectively. The error equations are used to establish the order of convergence of these proposed iterative methods. Finally, various numerical comparisons are implemented by MATLAB to demonstrate the performance of the developed methods.
A pseudospectra-based approach to non-normal stability of embedded boundary methods
Rapaka, Narsimha; Samtaney, Ravi
2017-11-01
We present non-normal linear stability of embedded boundary (EB) methods employing pseudospectra and resolvent norms. Stability of the discrete linear wave equation is characterized in terms of the normalized distance of the EB to the nearest ghost node (α) in one and two dimensions. An important objective is that the CFL condition based on the Cartesian grid spacing remains unaffected by the EB. We consider various discretization methods including both central and upwind-biased schemes. Stability is guaranteed when α Funds under Award No. URF/1/1394-01.