A Boundary Integral Equation Approach for Boundary Problem of Laplace Equation
Institute of Scientific and Technical Information of China (English)
SUNJian-she; YELiu-qing
2003-01-01
Using the second Green formula, the boundary problem of Laplace equation satisfied by potential function of static electric field is transformed to the problem of the boundary integral equation,and then a boundary integral equation approach is established by partitioning boundary using linear boundary element.
Iterative solution of Hermite boundary integral equations
Energy Technology Data Exchange (ETDEWEB)
Gray, Leonard J [ORNL; Nintcheu Fata, Sylvain [ORNL; Ma, Ding [ORNL
2008-01-01
An efficient iterative method for the solution of the linear equations arising from a Hermite boundary integral approximation has been developed. Along with equations for the boundary unknowns, the Hermite system incorporates equations for the first-order surface derivatives (gradient) of the potential, and is therefore substantially larger than the matrix for a corresponding linear approximation. However, by exploiting the structure of the Hermite matrix, a two-level iterative algorithm has been shown to provide a very efficient solution algorithm. In this approach, the boundary function unknowns are treated separately from the gradient, taking advantage of the sparsity and near-positive definiteness of the gradient equations. In test problems, the new algorithm significantly reduced computation time compared to iterative solution applied to the full matrix. This approach should prove to be even more effective for the larger systems encountered in three-dimensional analysis, and increased efficiency should come from pre-conditioning of the non-sparse matrix component.
A NOVEL BOUNDARY INTEGRAL EQUATION METHOD FOR LINEAR ELASTICITY--NATURAL BOUNDARY INTEGRAL EQUATION
Institute of Scientific and Technical Information of China (English)
Niu Zhongrong; Wang Xiuxi; Zhou Huanlin; Zhang Chenli
2001-01-01
The boundary integral equation (BIE) of displacement derivatives is put at a disadvantage for the difficulty involved in the evaluation of the hypersingular integrals. In this paper, the operators δij and εij are used to act on the derivative BIE. The boundary displacements, tractions and displacement derivatives are transformed into a set of new boundary tensors as boundary variables. A new BIE formulation termed natural boundary integral equation (NBIE) is obtained. The NBIE is applied to solving two-dimensional elasticity problems. In the NBIE only the strongly singular integrals are contained. The Cauchy principal value integrals occurring in the NBIE are evaluated. A combination of the NBIE and displacement BIE can be used to directly calculate the boundary stresses. The numerical results of several examples demonstrate the accuracy of the NBIE.
APPLICATION OF BOUNDARY INTEGRAL EQUATION METHOD FOR THERMOELASTICITY PROBLEMS
Vorona Yu.V.; Kara I.D.
2015-01-01
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.
APPLICATION OF BOUNDARY INTEGRAL EQUATION METHOD FOR THERMOELASTICITY PROBLEMS
Directory of Open Access Journals (Sweden)
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.
On the Implementation of 3D Galerkin Boundary Integral Equations
Energy Technology Data Exchange (ETDEWEB)
Nintcheu Fata, Sylvain [ORNL; Gray, Leonard J [ORNL
2010-01-01
In this article, a reverse contribution technique is proposed to accelerate the construction of the dense influence matrices associated with a Galerkin approximation of singular and hypersingular boundary integral equations of mixed-type in potential theory. In addition, a general-purpose sparse preconditioner for boundary element methods has also been developed to successfully deal with ill-conditioned linear systems arising from the discretization of mixed boundary-value problems on non-smooth surfaces. The proposed preconditioner, which originates from the precorrected-FFT method, is sparse, easy to generate and apply in a Krylov subspace iterative solution of discretized boundary integral equations. Moreover, an approximate inverse of the preconditioner is implicitly built by employing an incomplete LU factorization. Numerical experiments involving mixed boundary-value problems for the Laplace equation are included to illustrate the performance and validity of the proposed techniques.
Boundary Integral Equations and A Posteriori Error Estimates
Institute of Scientific and Technical Information of China (English)
YU Dehao; ZHAO Longhua
2005-01-01
Adaptive methods have been rapidly developed and applied in many fields of scientific and engineering computing. Reliable and efficient a posteriori error estimates play key roles for both adaptive finite element and boundary element methods. The aim of this paper is to develop a posteriori error estimates for boundary element methods. The standard a posteriori error estimates for boundary element methods are obtained from the classical boundary integral equations. This paper presents hyper-singular a posteriori error estimates based on the hyper-singular integral equations. Three kinds of residuals are used as the estimates for boundary element errors. The theoretical analysis and numerical examples show that the hyper-singular residuals are good a posteriori error indicators in many adaptive boundary element computations.
NOVEL REGULARIZED BOUNDARY INTEGRAL EQUATIONS FOR POTENTIAL PLANE PROBLEMS
Institute of Scientific and Technical Information of China (English)
ZHANG Yao-ming; L(U) He-xiang; WANG Li-min
2006-01-01
The universal practices have been centralizing on the research of regularization to the direct boundary integal equations (DBIEs). The character is elimination of singularities by using the simple solutions. However, up to now the research of regularization to the first kind integral equations for plane potential problems has never been found in previous literatures. The presentation is mainly devoted to the research on the regularization of the singular boundaryintegral equations with indirect unknowns. A novel view and idea is presented herein, in which the regularized boundary integral equations with indirect unknowns without including the Cauchy principal value (CPV) and Hadamard-finite-part (HFP) integrals are established for the plane potential problems.With some numerical results, it is shown that the better accuracy and higher efficiency,especially on the boundary, can be achieved by the present system.
Galerkin Boundary Integral Analysis for the 3D Helmholtz Equation
Energy Technology Data Exchange (ETDEWEB)
Swager, Melissa [Emporia State University; Gray, Leonard J [ORNL; Nintcheu Fata, Sylvain [ORNL
2010-01-01
A linear element Galerkin boundary integral analysis for the three-dimensional Helmholtz equation is presented. The emphasis is on solving acoustic scattering by an open (crack) surface, and to this end both a dual equation formulation and a symmetric hypersingular formulation have been developed. All singular integrals are defined and evaluated via a boundary limit process, facilitating the evaluation of the (finite) hypersingular Galerkin integral. This limit process is also the basis for the algorithm for post-processing of the surface gradient. The analytic integrations required by the limit process are carried out by employing a Taylor series expansion for the exponential factor in the Helmholtz fundamental solutions. For the open surface, the implementations are validated by comparing the numerical results obtained by using the two different methods.
Advanced applications of boundary-integral equation methods
International Nuclear Information System (INIS)
The BIE (boundary integral equation) method is based on the numerical solution of a set of integral constraint equations which couple boundary tractions (stresses) to boundary displacements. Thus the dimensionality of the problem is reduced by one; only boundary geometry and data are discretized. Stresses at any set of selected interior points are computed following the boundary solution without any further numerical approximations. Thus, the BIE method has inherently greater resolution capability for stress gradients than does the finite element method. Conversely, the BIE method is not efficient for problems involving significant inhomogeneity such as in multi-thin-layered materials, or in elastoplasticity. Some progress in applyiing the BIE method to the latter problem has been made but much more work remains. Further, the BIE method is only optional for problems with significant stress risers, and only when boundary stresses are most important. Interior stress calculations are expensive, per point, and can drive the solution costs up rapidly. The current report summarizes some of the advanced elastic applications of fracture mechanics and three-dimensional stress analysis, while referencing some of the much broader developmental effort. Future emphasis is needed to exploit the BIE method in conjunction with other techniques such as the finite element method through the creation of hybrid stress analysis methods
Advanced applications of boundary-integral equation methods
International Nuclear Information System (INIS)
Numerical analysis has become the basic tool for both design and research problems in solid mechanics. The boundary-integral equation (BIE) method is based on classical mathematical techniques but is finding new life as a basic stress analysis tool for engineering applications. The BIE method is based on the numerical solution of a set of integral constraint equations which couple boundary tractions (stresses) to boundary displacements. Thus the dimensionality of the problem is reduced by one; only boundary geometry and data are discretized. Stresses at any set of selected interior points are computed following the boundary solution without any further numerical approximations. Thus, the BIE method has inherently greater resolution capability for stress gradients than does the finite element method. Conversely, the BIE method is not efficient for problems involving significant inhomogeneity such as in multi-thin-layered materials, or in elastoplasticity. Some progress in applying the BIE method to the latter problem has been made but much more work remains. Further, the BIE method is only optional for problems with significant stress risers, and only when boundary stresses are more important. Interior stress calculations are expensive, per point, and can drive the solution costs up rapidly. The current report summarizes some of the advanced elastic applications of fracture mechanics and three-dimensional stress analysis, while referring some of the much broader developmental effort. (Auth.)
Numerical analysis of Weyl's method for integrating boundary layer equations
Najfeld, I.
1982-01-01
A fast method for accurate numerical integration of Blasius equation is proposed. It is based on the limit interchange in Weyl's fixed point method formulated as an iterated limit process. Each inner limit represents convergence to a discrete solution. It is shown that the error in a discrete solution admits asymptotic expansion in even powers of step size. An extrapolation process is set up to operate on a sequence of discrete solutions to reach the outer limit. Finally, this method is extended to related boundary layer equations.
Tokamak plasma shape identification based on the boundary integral equations
International Nuclear Information System (INIS)
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)
A spectral boundary integral equation method for the 2D Helmholtz equation
International Nuclear Information System (INIS)
In this paper, we present a new numerical formulation of solving the boundary integral equations reformulated from the Helmholtz equation. The boundaries of the problems are assumed to be smooth closed contours. The solution on the boundary is treated as a periodic function, which is in turn approximated by a truncated Fourier series. A Fourier collocation method is followed in which the boundary integral equation is transformed into a system of algebraic equations. It is shown that in order to achieve spectral accuracy for the numerical formulation, the non-smoothness of the integral kernels, associated with the Helmholtz equation, must be carefully removed. The emphasis of the paper is on investigating the essential elements of removing the non-smoothness of the integral kernels in the spectral implementation. The present method is robust for a general smooth boundary contour. Aspects of efficient implementation of the method using FFT are also discussed. Numerical examples of wave scattering are given in which the exponential accuracy of the present numerical method is demonstrated. 15 refs., 3 figs., 4 tabs
Advanced applications of boundary-integral equation methods
International Nuclear Information System (INIS)
Numerical analysis has become the basic tool for both design and research problems in solid mechanics. The need for accuracy and detail, plus the availablity of the high speed computer has led to the development of many new modeling methods ranging from general purpose structural analysis finite element programs to special purpose research programs. The boundary-integral equation (BIE) method is based on classical mathematical techniques but is finding new life as a basic stress analysis tool for engineering applications. The paper summarizes some advanced elastic applications of fracture mechanics and three-dimensional stress analysis, while referencing some of the much broader developmental effort. Future emphasis is needed to exploit the BIE method in conjunction with other techniques such as the finite element method through the creation of hybrid stress analysis methods. (Auth.)
International Nuclear Information System (INIS)
The reflection equation algebra of Sklyanin is extended to the supersymmetric case. A graded reflection equation algebra is proposed and the corresponding graded (supersymmetric) boundary quantum inverse scattering method (QISM) is formulated. As an application, integrable open-boundary conditions for the doped spin-1 chain of the supersymmetric t-J model are studied in the framework of the boundary QISM. Diagonal boundary K-matrices are found and four classes of integrable boundary terms are determined. (author)
RESTRICTED NONLINEAR APPROXIMATION AND SINGULAR SOLUTIONS OF BOUNDARY INTEGRAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
Reinhard Hochmuth
2002-01-01
This paper studies several problems, which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1 ] are chosen as a starting point for characterizations of functions in Besov spaces B , (0,1) with 0＜σ＜∞ and (1+σ)-1＜τ＜∞. Such function spaces are known to be related to nonlinear approximation. Then so called restricted nonlinear approximation procedures with respect to Sobolev space norms are considered. Besides characterization results Jackson type estimates for various tree-type and tresholding algorithms are investigated. Finally known approximation results for geometry induced singularity functions of boundary integeral equations are combined with the characterization results for restricted nonlinear approximation to show Besov space regularity results.
Moiseiwitsch, B L
2005-01-01
Two distinct but related approaches hold the solutions to many mathematical problems--the forms of expression known as differential and integral equations. The method employed by the integral equation approach specifically includes the boundary conditions, which confers a valuable advantage. In addition, the integral equation approach leads naturally to the solution of the problem--under suitable conditions--in the form of an infinite series.Geared toward upper-level undergraduate students, this text focuses chiefly upon linear integral equations. It begins with a straightforward account, acco
A well-conditioned boundary integral equation for transmission problems of electromagnetism
Levadoux, David; Millot, Florence; Pernet, Sébastien
2015-01-01
We propose a new well-conditioned boundary integral equation to solve transmission problems of electromagnetism. This equation is well posed and appears as a compact perturbation of the identity leading to fast iterative solutions without the help of any preconditioner. Some numerical experiments confirm this result.
Boundary integral equation Neumann-to-Dirichlet map method for gratings in conical diffraction.
Wu, Yumao; Lu, Ya Yan
2011-06-01
Boundary integral equation methods for diffraction gratings are particularly suitable for gratings with complicated material interfaces but are difficult to implement due to the quasi-periodic Green's function and the singular integrals at the corners. In this paper, the boundary integral equation Neumann-to-Dirichlet map method for in-plane diffraction problems of gratings [Y. Wu and Y. Y. Lu, J. Opt. Soc. Am. A26, 2444 (2009)] is extended to conical diffraction problems. The method uses boundary integral equations to calculate the so-called Neumann-to-Dirichlet maps for homogeneous subdomains of the grating, so that the quasi-periodic Green's functions can be avoided. Since wave field components are coupled on material interfaces with the involvement of tangential derivatives, a least squares polynomial approximation technique is developed to evaluate tangential derivatives along these interfaces for conical diffraction problems. Numerical examples indicate that the method performs equally well for dielectric or metallic gratings. PMID:21643404
Energy Technology Data Exchange (ETDEWEB)
Shih, H.R. [Jackson State Univ., MS (United States); Duffield, R.C.; Lin, J. [Univ. of Missouri, Columbia, MO (United States)
1996-10-01
An integral equation formulation and a numerical procedure for a boundary-finite element technique are developed for the static analysis of a stiffened plate with eccentric stiffeners. This formulation employs the fundamental solution associated with unstiffened plate bending and plane stress problems. With this approach, the resulting integral equation not only contained integrals along the perimeter of the stiffened but additional integrals along the stiffeners and the interface between the plate and its stiffeners. Thus the domain of the plate has to be divided into zones between the stiffeners. Each zone is modeled by boundary elements and stiffeners by finite elements. In this paper, the boundary element solution procedures for plate bending and in-plane problems are presented. The zone technique which permits coupling of unstiffened plate boundary element with stiffener finite elements is presented as well. Numerical example is given to demonstrate the effectiveness of this approach.
Analytical Solution of Boundary Integral Equations for 2-D Steady Linear Wave Problems
Institute of Scientific and Technical Information of China (English)
J.M. Chuang
2005-01-01
Based on the Fourier transform, the analytical solution of boundary integral equations formulated for the complex velocity of a 2-D steady linear surface flow is derived. It has been found that before the radiation condition is imposed,free waves appear both far upstream and downstream. In order to cancel the free waves in far upstream regions, the eigensolution of a specific eigenvalue, which satisfies the homogeneous boundary integral equation, is found and superposed to the analytical solution. An example, a submerged vortex, is used to demonstrate the derived analytical solution. Furthermore,an analytical approach to imposing the radiation condition in the numerical solution of boundary integral equations for 2-D steady linear wave problems is proposed.
On approximation of nonlinear boundary integral equations for the combined method
Energy Technology Data Exchange (ETDEWEB)
Gregus, M.; Khoromsky, B.N.; Mazurkevich, G.E.; Zhidkov, E.P.
1989-09-22
The nonlinear boundary integral equations that arise in research of nonlinear magnetostatic problems are investigated in combined formulation on an unbounded domain. Approximations of the derived operator equations are studied based on the Galerkin method. The investigated boundary operators are strongly monotone, Lipschitz-continuous, potential and have a symmetrical Gateaux derivative. The error estimates of the Galerkin's approximation in Sobolev spaces of fractional powers are obtained using the above-mentioned properties of the operators, too. The problem has been studied on surfaces in two and three-dimensional spaces. We answer also some questions on convergence connected with the discretized systems of equations. 21 refs.
Perfectly-matched-layer boundary integral equation method for wave scattering in a layered medium
Lu, Wangtao; Qian, Jianliang
2016-01-01
For scattering problems of time-harmonic waves, the boundary integral equation (BIE) methods are highly competitive, since they are formulated on lower-dimension boundaries or interfaces, and can automatically satisfy outgoing radiation conditions. For scattering problems in a layered medium, standard BIE methods based on the Green's function of the background medium must evaluate the expensive Sommefeld integrals. Alternative BIE methods based on the free-space Green's function give rise to integral equations on unbounded interfaces which are not easy to truncate, since the wave fields on these interfaces decay very slowly. We develop a BIE method based on the perfectly matched layer (PML) technique. The PMLs are widely used to suppress outgoing waves in numerical methods that directly discretize the physical space. Our PML-based BIE method uses the Green's function of the PML-transformed free space to define the boundary integral operators. The method is efficient, since the Green's function of the PML-tran...
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...
Numerical solution of multiple hole problem by using boundary integral equation
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
This paper studies a numerical solution of multiple hole problem by using a boundary integral equation.The studied problem can be considered as a supposition of many single hole problems.After considering the interaction among holes,an algebraic equation is formulated,which is then solved by using an iteration technique.The hoop stress around holes can be finally determined. One numerical example is provided to check its accuracy.
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.
Boundary integral equation formulations for free-surface flow problems in two and three dimensions
Romate, J.E.; Zandbergen, P.J.
1989-01-01
To compute the transient solution of free-surface flow problems in two and three dimensions boundary integral equation formulations are considered. Consistent lower and higher order approximations based on small curvature expansions are compared and applied to a time-dependent, linear free-surface wave problem.
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
Directory of Open Access Journals (Sweden)
Sulym Heorhiy
2016-03-01
Full Text Available This paper studies a thermoelastic anisotropic bimaterial with thermally imperfect interface and internal inhomogeneities. Based on the complex variable calculus and the extended Stroh formalism a new approach is proposed for obtaining the Somigliana type integral formulae and corresponding boundary integral equations for a thermoelastic bimaterial consisting of two half-spaces with different thermal and mechanical properties. The half-spaces are bonded together with mechanically perfect and thermally imperfect interface, which model interfacial adhesive layers present in bimaterial solids. Obtained integral equations are introduced into the modified boundary element method that allows solving arbitrary 2D thermoelacticity problems for anisotropic bimaterial solids with imperfect thin thermo-resistant inter-facial layer, which half-spaces contain cracks and thin inclusions. Presented numerical examples show the effect of thermal resistance of the bimaterial interface on the stress intensity factors at thin inhomogeneities.
The Application of a Boundary Integral Equation Method to the Prediction of Ducted Fan Engine Noise
Dunn, M. H.; Tweed, J.; Farassat, F.
1999-01-01
The prediction of ducted fan engine noise using a boundary integral equation method (BIEM) is considered. Governing equations for the BIEM are based on linearized acoustics and describe the scattering of incident sound by a thin, finite-length cylindrical duct in the presence of a uniform axial inflow. A classical boundary value problem (BVP) is derived that includes an axisymmetric, locally reacting liner on the duct interior. Using potential theory, the BVP is recast as a system of hypersingular boundary integral equations with subsidiary conditions. We describe the integral equation derivation and solution procedure in detail. The development of the computationally efficient ducted fan noise prediction program TBIEM3D, which implements the BIEM, and its utility in conducting parametric noise reduction studies are discussed. Unlike prediction methods based on spinning mode eigenfunction expansions, the BIEM does not require the decomposition of the interior acoustic field into its radial and axial components which, for the liner case, avoids the solution of a difficult complex eigenvalue problem. Numerical spectral studies are presented to illustrate the nexus between the eigenfunction expansion representation and BIEM results. We demonstrate BIEM liner capability by examining radiation patterns for several cases of practical interest.
Kitanine, N; Niccoli, G
2014-01-01
We solve the longstanding problem to define a functional characterization of the spectrum of the transfer matrix associated to the most general spin-1/2 representations of the 6-vertex reflection algebra for general inhomogeneous chains. The corresponding homogeneous limit reproduces the spectrum of the Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most general integrable boundaries. The spectrum is characterized by a second order finite difference functional equation of Baxter type with an inhomogeneous term which vanishes only for some special but yet interesting non-diagonal boundary conditions. This functional equation is shown to be equivalent to the known separation of variable (SOV) representation hence proving that it defines a complete characterization of the transfer matrix spectrum. The polynomial character of the Q-function allows us then to show that a finite system of equations of generalized Bethe type can be similarly used to describe the complete transfer matrix spectru...
The Prediction of Ducted Fan Engine Noise Via a Boundary Integral Equation Method
Farassat, F.; Dunn, M. H.; Tweed, J.
1996-01-01
A computationally efficient Boundary Integral Equation Method (BIEM) for the prediction of ducted fan engine noise is presented. The key features of the BIEM are its versatility and the ability to compute rapidly any portion of the sound field without the need to compute the entire field. Governing equations for the BIEM are based on the assumptions that all acoustic processes are linear, generate spinning modes, and occur in a uniform flow field. An exterior boundary value problem (BVP) is defined that describes the scattering of incident sound by an engine duct with arbitrary profile. Boundary conditions on the duct walls are derived that allow for passive noise control treatment. The BVP is recast as a system of hypersingular boundary integral equations for the unknown duct surface quantities. BIEM solution methodology is demonstrated for the scattering of incident sound by a thin cylindrical duct with hard walls. Numerical studies are conducted for various engine parameters and continuous portions of the total pressure field are computed. Radiation and duct propagation results obtained are in agreement with the classical results of spinning mode theory for infinite ducts.
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
The Reduction of Ducted Fan Engine Noise Via a Boundary Integral Equation Method
Tweed, John
2000-01-01
Engineering studies for reducing ducted fan engine noise were conducted using the noise prediction code TBIEM3D. To conduct parametric noise reduction calculations, it was necessary to advance certain theoretical and computational aspects of the boundary integral equation method (BIEM) described in and implemented in TBIEM3D. Also, enhancements and upgrades to TBIEM3D were made for facilitating the code's use in this research and by the aeroacoustics engineering community.
Fatma Kanca
2013-01-01
This paper investigates the inverse problem of finding a time-dependent diffusion coefficient in a parabolic equation with the periodic boundary and integral overdetermination conditions. Under some assumption on the data, the existence, uniqueness, and continuous dependence on the data of the solution are shown by using the generalized Fourier method. The accuracy and computational efficiency of the proposed method are verified with the help of the numerical examples.
Connectivity as an alternative to boundary integral equations: Construction of bases
Herrera, Ismael; Sabina, Federico J.
1978-01-01
In previous papers Herrera developed a theory of connectivity that is applicable to the problem of connecting solutions defined in different regions, which occurs when solving partial differential equations and many problems of mechanics. In this paper we explain how complete connectivity conditions can be used to replace boundary integral equations in many situations. We show that completeness is satisfied not only in steady-state problems such as potential, reduced wave equation and static and quasi-static elasticity, but also in time-dependent problems such as heat and wave equations and dynamical elasticity. A method to obtain bases of connectivity conditions, which are independent of the regions considered, is also presented. PMID:16592522
Li, Yuan; Dang, HuaYang; Xu, GuangTao; Fan, CuiYing; Zhao, MingHao
2016-08-01
The extended displacement discontinuity boundary integral equation (EDDBIE) and boundary element method is developed for the analysis of planar cracks of arbitrary shape in the isotropic plane of three-dimensional (3D) transversely isotropic thermo-magneto-electro-elastic (TMEE) media. The extended displacement discontinuities (EDDs) include conventional displacement discontinuity, electric potential discontinuity, magnetic potential discontinuity, as well as temperature discontinuity across crack faces; correspondingly, the extended stresses represent conventional stress, electric displacement, magnetic induction and heat flux. Employing a Hankel transformation, the fundamental solutions for unit point EDDs in 3D transversely isotropic TMEE media are derived. The EDDBIEs for a planar crack of arbitrary shape in the isotropic plane of a 3D transversely isotropic TMEE medium are then established. Using the boundary integral equation method, the singularities of near-crack border fields are obtained and the extended stress field intensity factors are expressed in terms of the EDDs on crack faces. According to the analogy between the EDDBIEs for an isotropic thermoelastic material and TMEE medium, an analogical solution method for crack problems of a TMEE medium is proposed for coupled multi-field loadings. Employing constant triangular elements, the EDDBIEs are discretized and numerically solved. As an application, the problems of an elliptical crack subjected to combined mechanical-electric-magnetic-thermal loadings are investigated.
Tetervin, Neal; Lin, Chia Chiao
1951-01-01
A general integral form of the boundary-layer equation, valid for either laminar or turbulent incompressible boundary-layer flow, is derived. By using the experimental finding that all velocity profiles of the turbulent boundary layer form essentially a single-parameter family, the general equation is changed to an equation for the space rate of change of the velocity-profile shape parameter. The lack of precise knowledge concerning the surface shear and the distribution of the shearing stress across turbulent boundary layers prevented the attainment of a reliable method for calculating the behavior of turbulent boundary layers.
Solving the hypersingular boundary integral equation for the Burton and Miller formulation.
Langrenne, Christophe; Garcia, Alexandre; Bonnet, Marc
2015-11-01
This paper presents an easy numerical implementation of the Burton and Miller (BM) formulation, where the hypersingular Helmholtz integral is regularized by identities from the associated Laplace equation and thus needing only the evaluation of weakly singular integrals. The Helmholtz equation and its normal derivative are combined directly with combinations at edge or corner collocation nodes not used when the surface is not smooth. The hypersingular operators arising in this process are regularized and then evaluated by an indirect procedure based on discretized versions of the Calderón identities linking the integral operators for associated Laplace problems. The method is valid for acoustic radiation and scattering problems involving arbitrarily shaped three-dimensional bodies. Unlike other approaches using direct evaluation of hypersingular integrals, collocation points still coincide with mesh nodes, as is usual when using conforming elements. Using higher-order shape functions (with the boundary element method model size kept fixed) reduces the overall numerical integration effort while increasing the solution accuracy. To reduce the condition number of the resulting BM formulation at low frequencies, a regularized version α = ik/(k(2 )+ λ) of the classical BM coupling factor α = i/k is proposed. Comparisons with the combined Helmholtz integral equation Formulation method of Schenck are made for four example configurations, two of them featuring non-smooth surfaces. PMID:26627805
Hu, Fang Q.; Pizzo, Michelle E.; Nark, Douglas M.
2016-01-01
Based on the time domain boundary integral equation formulation of the linear convective wave equation, a computational tool dubbed Time Domain Fast Acoustic Scattering Toolkit (TD-FAST) has recently been under development. The time domain approach has a distinct advantage that the solutions at all frequencies are obtained in a single computation. In this paper, the formulation of the integral equation, as well as its stabilization by the Burton-Miller type reformulation, is extended to cases of a constant mean flow in an arbitrary direction. In addition, a "Source Surface" is also introduced in the formulation that can be employed to encapsulate regions of noise sources and to facilitate coupling with CFD simulations. This is particularly useful for applications where the noise sources are not easily described by analytical source terms. Numerical examples are presented to assess the accuracy of the formulation, including a computation of noise shielding by a thin barrier motivated by recent Historical Baseline F31A31 open rotor noise shielding experiments. Furthermore, spatial resolution requirements of the time domain boundary element method are also assessed using point per wavelength metrics. It is found that, using only constant basis functions and high-order quadrature for surface integration, relative errors of less than 2% may be obtained when the surface spatial resolution is 5 points-per-wavelength (PPW) or 25 points-per-wavelength squared (PPW2).
A Fast Spectral Galerkin Method for Hypersingular Boundary Integral Equations in Potential Theory
Energy Technology Data Exchange (ETDEWEB)
Nintcheu Fata, Sylvain [ORNL; Gray, Leonard J [ORNL
2009-01-01
This research is focused on the development of a fast spectral method to accelerate the solution of three-dimensional hypersingular boundary integral equations of potential theory. Based on a Galerkin approximation, the Fast Fourier Transform and local interpolation operators, the proposed method is a generalization of the Precorrected-FFT technique to deal with double-layer potential kernels, hypersingular kernels and higher-order basis functions. Numerical examples utilizing piecewise linear shape functions are included to illustrate the performance of the method.
TIME–HARMONIC BEHAVIOUR OF CRACKED PIEZOELECTRIC SOLID BY BOUNDARY INTEGRAL EQUATION METHOD
Directory of Open Access Journals (Sweden)
Rangelov Tsviatko
2014-03-01
Full Text Available Anti-plane cracked functionally graded finite piezoelectric solid under time-harmonic elecromechanical load is studied by a non-hypersingular traction boundary integral equation method (BIEM. Exponentially varying material properties are considered. Numerical solutions are obtained by using Mathematica. The dependance of the intensity factors (IF - mechanical stress intensity factor (SIF and electrical field intensity factor (FIF on the inhomogeneous material parameters, on the type and frequency of the dynamic load and on the crack position are analyzed by numerical illustrative examples
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.
Boundary integral equation method for added mass in arrays of cylinders
International Nuclear Information System (INIS)
The dynamic behavior of a group of cylinders in fluid, such as heat exchanger tubes and nuclear fuel assemblies, is strongly influenced by the surrounding fluid. Although, the added mass of such clusters of cylinders has been studied by many researchers with various analytical methods and numerical methods, no attempt has been made so far to analyze these problems by the Boundary Integral Equation Method (BIEM). This paper presents a BIEM model simulating the added mass arising when clusters of cylinders vibrate in an inviscid and incompressible fluid. In this model, perturbed fluid pressure is described by a two- dimensional Laplace equation. The primary advantage of this approach compared with other numerical methods, e.g., the finite element method (FEM), is that the integration and discretization of the model are only needed on boundary rather than in whole domain. Therefore, the proposed approach is much more economical than the finite element method. Various numerical examples are subsequently presented in this paper to illustrate the methodology and to demonstrate its accuracy
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
This paper investigates the existence and multiplicity of nonnegative solutions to a singular nonlinear boundary value problem of second order differential equations with integral boundary conditions in a Banach space. The arguments are based on the construction of a nonempty bounded open convex set and fixed point index theory. Our nonlinearity possesses singularity and first derivative which makes it different with that in [10].
Institute of Scientific and Technical Information of China (English)
龙述尧; 熊渊博
2004-01-01
The meshless local boundary integral equation method is a currently developed numerical method, which combines the advantageous features of Galerkin finite element method(GFEM), boundary element method(BEM) and element free Galerkin method(EFGM), and is a truly meshless method possessing wide prospects in engineering applications.The companion solution and all the other formulas required in the meshless local boundary integral equation for a thin plate were presented, in order to make this method apply to solve the thin plate problem.
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...
Boundary integral equation method calculations of surface regression effects in flame spreading
Altenkirch, R. A.; Rezayat, M.; Eichhorn, R.; Rizzo, F. J.
1982-01-01
A solid-phase conduction problem that is a modified version of one that has been treated previously in the literature and is applicable to flame spreading over a pyrolyzing fuel is solved using a boundary integral equation (BIE) method. Results are compared to surface temperature measurements that can be found in the literature. In addition, the heat conducted through the solid forward of the flame, the heat transfer responsible for sustaining the flame, is also computed in terms of the Peclet number based on a heated layer depth using the BIE method and approximate methods based on asymptotic expansions. Agreement between computed and experimental results is quite good as is agreement between the BIE and the approximate results.
Directory of Open Access Journals (Sweden)
Jingfu Jin
2012-04-01
Full Text Available This article shows the existence of a positive solution for the singular fractional differential equation with integral boundary condition $$displaylines{ {}^C!D^p u(t=lambda h(tf(t, u(t, quad tin(0, 1, cr u(0-au(1=int^1_0g_0(su(s,ds, cr u'(0-b,{}^C!D^qu(1=int^1_0g_1(su(s,ds, cr u''(0=u'''(0=dots =u^{(n-1}(0=0, }$$ where $lambda $ is a parameter and the nonlinear term is allowed to be singular at $t=0, 1$ and $u=0$. We obtain an explicit interval for $lambda$ such that for any $lambda$ in this interval, existence of at least one positive solution is guaranteed. Our approach is by a fixed point theory in cones combined with linear operator theory.
Institute of Scientific and Technical Information of China (English)
ZHAO Ming-hao; LI Dong-xia; SHEN Ya-peng
2005-01-01
The integral-differential equations for three-dimensional planar interfacial cracks of arbitrary shape in transversely isotropic bimaterials were derived by virtue of the Somigliana identity and the fundamental solutions, in which the displacement discontinuities across the crack faces are the unknowns to be determined. The interface is parallel to both the planes of isotropy. The singular behaviors of displacement and stress near the crack border were analyzed and the stress singularity indexes were obtained by integral equation method. The stress intensity factors were expressed in terms of the displacement discontinuities. In the non-oscillatory case, the hyper-singular boundary integral-differential equations were reduced to hyper-singular boundary integral equations similar to those of homogeneously isotropie materials.
A hybrid boundary-integral/thin-sheet equation for subduction modelling
Xu, Bingrui; Ribe, Neil M.
2016-09-01
Subducting oceanic lithosphere is an example of a thin sheet-like object whose characteristic lateral dimension greatly exceeds its thickness. Here we exploit this property to derive a new hybrid boundary-integral/thin sheet (BITS) representation of subduction that combines in a single equation all the forces acting on the sheet: gravity, internal resistance to bending and stretching, and the tractions exerted by the ambient mantle. For simplicity, we limit ourselves to 2-D. We solve the BITS equations using a discrete Lagrangian approach in which the sheet is represented by a set of vertices connected by edges. Instantaneous solutions for the sinking speed of a slab attached to a trailing flat sheet obey a scaling law of the form V/VStokes = fct(St), where VStokes is a characteristic Stokes sinking speed and St is the sheet's flexural stiffness. Time-dependent solutions for the evolution of the sheet's shape and thickness show that these are controlled by the viscosity ratio between the sheet and its surroundings. An important advantage of the BITS approach is the possibility of generalizing the sheet's rheology, either to a viscosity that varies along the sheet or to a non-Newtonian shear-thinning rheology.
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.
Boundary transfer matrices and boundary quantum KZ equations
Energy Technology Data Exchange (ETDEWEB)
Vlaar, Bart, E-mail: Bart.Vlaar@nottingham.ac.uk [School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD (United Kingdom)
2015-07-15
A simple relation between inhomogeneous transfer matrices and boundary quantum Knizhnik-Zamolodchikov (KZ) equations is exhibited for quantum integrable systems with reflecting boundary conditions, analogous to an observation by Gaudin for periodic systems. Thus, the boundary quantum KZ equations receive a new motivation. We also derive the commutativity of Sklyanin’s boundary transfer matrices by merely imposing appropriate reflection equations, in particular without using the conditions of crossing symmetry and unitarity of the R-matrix.
Directory of Open Access Journals (Sweden)
Chatthai Thaiprayoon
2014-01-01
Full Text Available By developing a new comparison result and using the monotone iterative technique, we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for first-order impulsive functional integrodifferential equations with integral-jump conditions. An example is also given to illustrate our results.
Yuan, Zhengwen; Xiao, Hong; Xie, Hongbiao
2014-02-01
Precise strip-shape control theory is significant to improve rolled strip quality, and roll flattening theory is a primary part of the strip-shape theory. To improve the accuracy of roll flattening calculation based on semi-infinite body model, a new and more accurate roll flattening model is proposed in this paper, which is derived based on boundary integral equation method. The displacement fields of the finite length semi-infinite body on left and right sides are simulated by using finite element method (FEM) and displacement decay functions on left and right sides are established. Based on the new roll flattening model, a new 4Hi mill deformation model is established and verified by FEM. The new model is compared with Foppl formula and semi-infinite body model in different strip width, roll shifting value and bending force. The results show that the pressure and flattening between rolls calculated by the new model are more precise than other two models, especially near the two roll barrel edges.
Directory of Open Access Journals (Sweden)
Nittaya Pongarm
2013-01-01
Full Text Available This paper studies sufficient conditions for the existence of solutions to the problem of sequential derivatives of nonlinear q-difference equations with three-point q-integral boundary conditions. Our results are concerned with several quantum numbers of derivatives and integrals. By using Banach's contraction mapping, Krasnoselskii's fixed-point theorem, and Leray-Schauder degree theory, some new existence results are obtained. Two examples illustrate our results.
Directory of Open Access Journals (Sweden)
D. Diederen
2015-06-01
Full Text Available We present a new equation describing the hydrodynamics in infinitely long tidal channels (i.e., no reflection under the influence of oceanic forcing. The proposed equation is a simple relationship between partial derivatives of water level and velocity. It is formally derived for a progressive wave in a frictionless, prismatic, tidal channel with a horizontal bed. Assessment of a large number of numerical simulations, where an open boundary condition is posed at a certain distance landward, suggests that it can also be considered accurate in the more natural case of converging estuaries with nonlinear friction and a bed slope. The equation follows from the open boundary condition and is therefore a part of the problem formulation for an infinite tidal channel. This finding provides a practical tool for evaluating tidal wave dynamics, by reconstructing the temporal variation of the velocity based on local observations of the water level, providing a fully local open boundary condition and allowing for local friction calibration.
A Collocation Method for Volterra Integral Equations with Diagonal and Boundary Singularities
Kolk, Marek; Pedas, Arvet; Vainikko, Gennadi
2009-08-01
We propose a smoothing technique associated with piecewise polynomial collocation methods for solving linear weakly singular Volterra integral equations of the second kind with kernels which, in addition to a diagonal singularity, may have a singularity at the initial point of the interval of integration.
Stenroos, Matti
2016-01-01
Boundary element methods (BEM) are used for forward computation of bioelectromagnetic fields in multi-compartment volume conductor models. Most BEM approaches assume that each compartment is in contact with at most one external compartment. In this work, I present a general surface integral equation and BEM discretization that remove this limitation and allow BEM modeling of general piecewise-homogeneous medium. The new integral equation allows positioning of field points at junctioned boundary of more than two compartments, enabling the use of linear collocation BEM in such a complex geometry. A modular BEM implementation is presented for linear collocation and Galerkin approaches, starting from standard formulation. The approach and resulting solver are verified in three ways, including comparison to finite element method (FEM). In a two-compartment split-sphere model with two spaced monopoles, the results obtained with high-resolution FEM and the BEMs were almost identical (relative difference < 0.003).
Abarbanel, Saul; Gottlieb, David; Carpenter, Mark H.
1994-01-01
It has been previously shown that the temporal integration of hyperbolic partial differential equations (PDE's) may, because of boundary conditions, lead to deterioration of accuracy of the solution. A procedure for removal of this error in the linear case has been established previously. In the present paper we consider hyperbolic (PDE's) (linear and non-linear) whose boundary treatment is done via the SAT-procedure. A methodology is present for recovery of the full order of accuracy, and has been applied to the case of a 4th order explicit finite difference scheme.
Institute of Scientific and Technical Information of China (English)
LongShuyao; HuDe'an
2003-01-01
The meshless method is a new numerical technique presented in recent years .It uses the moving least square (MLS) approximation as a shape function . The smoothness of the MLS approximation is determined by that of the basic function and of the weight function, and is mainly determined by that of the weight function. Therefore, the weight function greatly affects the accuracy of results obtained. Different kinds of weight functions, such as the spline function, the Gauss function and so on, are proposed recently by many researchers. In the present work, the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method. The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed. Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and a in Gauss and exponential weight functions are in the range of reasonable values, respectively, and the higher the smoothness of the weight function, the better the features of the solutions.
Geophysical interpretation using integral equations
Eskola, L
1992-01-01
Along with the general development of numerical methods in pure and applied to apply integral equations to geophysical modelling has sciences, the ability improved considerably within the last thirty years or so. This is due to the successful derivation of integral equations that are applicable to the modelling of complex structures, and efficient numerical algorithms for their solution. A significant stimulus for this development has been the advent of fast digital computers. The purpose of this book is to give an idea of the principles by which boundary-value problems describing geophysical models can be converted into integral equations. The end results are the integral formulas and integral equations that form the theoretical framework for practical applications. The details of mathematical analysis have been kept to a minimum. Numerical algorithms are discussed only in connection with some illustrative examples involving well-documented numerical modelling results. The reader is assu med to have a back...
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.
Numerical Solution for the Helmholtz Equation with Mixed Boundary Condition
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
We consider the numerical solution for the Helmholtz equation in R2 with mixed boundary conditions. The solvability of this mixed boundary value problem is established by the boundary integral equation method. Based on the Green formula, we express the solution in terms of the boundary data. The key to the numerical realization of this method is the computation of weakly singular integrals. Numerical performances show the validity and feasibility of our method. The numerical schemes proposed in this paper have been applied in the realization of probe method for inverse scattering problems.
Integral equations and computation problems
International Nuclear Information System (INIS)
Volterra's Integral Equations and Fredholm's Integral Equations of the second kind are discussed. Computational problems are found in the derivations and the computations. The theorem of the solution of the Fredholm's Integral Equation is discussed in detail. (author)
BOUNDARY CONTROL OF MKDV-BURGERS EQUATION
Institute of Scientific and Technical Information of China (English)
TIAN Li-xin; ZHAO Zhi-feng; WANG Jing-feng
2006-01-01
The boundary control of MKdV-Burgers equation was considered by feedback control on the domain [0,1]. The existence of the solution of MKdV-Burgers equation with the feedback control law was proved. On the base, priori estimates for the solution was given. At last, the existence of the weak solution of MKdV-Burgers equation was proved and the global-exponential and asymptotic stability of the solution of MKdV-Burgers equation was given.
Energy Technology Data Exchange (ETDEWEB)
Orlowska, S [Ecole Centrale de Lyon, Centre de Genie Electrique de Lyon, CNRS UMR 5005, 69134 Ecully (France); Beroual, A [Ecole Centrale de Lyon, Centre de Genie Electrique de Lyon, CNRS UMR 5005, 69134 Ecully (France); Fleszynski, J [Institute of Fundamental Electrotechnics and Electrotechnology, University of Technology of Wroclaw, Wroclaw (Poland)
2002-10-21
The heterogeneous mixture properties depend on its constituents' characteristics. We examine the effective permittivity of a two-phase composite material made of epoxy resin host matrix and barium titanate (BaTiO{sub 3}) filler for different volume fractions in the matrix. The task we undertake consists in finding a model of BaTiO{sub 3} particles through the computer simulations executed in PHI3D-electric field calculating package, based on the resolution of the Laplace equation using boundary integral equation method. With this aim in view we compare the measured results of the effective permittivity of the BaTiO{sub 3}-epoxy resin composite samples with the simulation results for different BaTiO{sub 3} particle geometric models and for the same experimental conditions, with regard to the given volume fraction of the powder in the matrix. The experimental results are obtained through the measurements with an impedance meter in the range of frequencies from 50 Hz to 1 MHz.
On Certain Dual Integral Equations
Directory of Open Access Journals (Sweden)
R. S. Pathak
1974-01-01
Full Text Available Dual integral equations involving H-Functions have been solved by using the theory of Mellin transforms. The proof is analogous to that of Busbridge on solutions of dual integral equations involving Bessel functions.
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
Boundary stabilization of wave equations with variable coefficients
Institute of Scientific and Technical Information of China (English)
FENG; Shaoji
2001-01-01
［1］Chen, G., Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures. & Appl., 1979, 58: 249.［2］Komornik, V., Exact controllability and stabilization, Research in Applied Mathematics (Series Editors: Ciarlet, P. G., Lions, J.), New York: Masson/John Wiley copublication, 1994.［3］Komornik, V., Zuazua, E., A direct method for the boundary stabilization of the wave equation, J. Math. Pures. & Appl., 1990, 69: 33.［4］Lagnese, J., Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations, 1983, 50: 163.［5］Lasiecka, I., Triggiani, R., Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. & Optim., 1992, 25: 189.［6］Wyler, A., Stability of wave equations with dissipative boundary conditions in a bounded domain, Differential and Integral Equations,1994, 7: 345.［7］Yao, P. F., On the observability inequality for exact controllability of wave equations with variable coefficients, SIAM J. Control & Optimization, 1999, 37, 5: 1568.［8］Wu, H., Shen, C. L., Yu, Y. L., Introduction to Riemannian Geometry (in Chinese), Beijing: Peking University Press, 1989.
Enclosing Solutions of Integral Equations
DEFF Research Database (Denmark)
Madsen, Kaj; NA NA NA Caprani, Ole; Stauning, Ole
1996-01-01
We present a method for enclosing the solution of an integral equation. It is assumed that a solution exists and that the corresponding integral operator T is a contraction near y. When solving the integral equation by iteration we obtain a result which is normally different from y because...
The Dirichlet problem for the two-dimensional Helmholtz equation for an open boundary
Hayashi, Y.
1973-01-01
Development of a complete theory of the two-dimensional Dirichlet problem for an open boundary. It is shown that the solution of the Dirichlet problem for an open boundary requires the solution of a Fredholm integral equation of the first kind. Although a Fredholm integral equation of the first kind usually has no solution if the kernel is continuous, owing to the logarithmic singularity of the kernel, the equation in this case is converted to a singular integral equation with a Cauchy kernel. It is proven that the homogeneous adjoint equation of the singular integral equation has no nonzero solution. By virtue of this result, and with the aid of an existence theorem known in the theory of singular integral equations, the existence of solutions of the singular integral equation, and then of the unique solution of the Fredholm integral equation of the first kind is proved.
Institute of Scientific and Technical Information of China (English)
马杭; 黄兴
2003-01-01
Based on the fact that the singular boundary integrals in the sense of Cauchy principal value can be represented approxi-mately by the mean values of two companion nearly singular boundary integrals, a vary general approach was developed in the paper.In the approach, the approximate formulation before discretization was constructed to cope with the difficulties encountered in the cor-ner treatment in the formulations of hypersingular boundary integral equations. This makes it possible to solve the hypersingularboundary integral equation numerically in a non-regularized form and in a local manner by using conforming C0 quadratic boundary ele-ments and standard Gaussian quadratures similar to those employed in the conventional displacement-BIE formulations. The approxi-mate formulation is very convenient to use because the corner information is comprised naturally in the representations of those ap-proximate integrals. Numerical examples in plane elasticity show that with the present approach, the compatible or better results canbe achieved in comparison with those of the conventional BIE formulations.
Singular-boundary reductions of type-Q ABS equations
Atkinson, James; Joshi, Nalini
2011-01-01
We study the fully discrete elliptic integrable model Q4 and its immediate trigonometric and rational counterparts (Q3, Q2 and Q1). Singular boundary problems for these equations are systematised in the framework of global singularity analysis. We introduce a technique to obtain solutions of such problems, in particular constructing the exact solution on a regular singularity-bounded strip. The solution technique is based on the multidimensional consistency and uses new insights into these eq...
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.
Institute of Scientific and Technical Information of China (English)
王家玉; 陈金兰
2001-01-01
By using the concept of abstract cone,improved comparable theoremand fixed point theorem of nonlinear operator and by discussing the periodic boundary condition of nonlinear integral-differential equation in Banach spaces,this paper gave two monotonic iterative sequences,also verified the minimum and maximum theorem for nonlinear integral-differential equation with periodic boundary condition in Banach spaces.%本文考虑Banach空间中非线性积分－微分方程的周期边值问题，利用抽象锥、推广了的比较定理及非线性算子的不动点定理，构造出两个单调迭代序列，证明了Banach空间中非线性积分－微分方程具有周期边值的最小解、最大解存在定理。
Ardema, M. D.; Yang, L.
1985-01-01
A method of solving the boundary-layer equations that arise in singular-perturbation analysis of flightpath optimization problems is presented. The method is based on Picard iterations of the integrated form of the equations and does not require iteration to find unknown boundary conditions. As an example, the method is used to develop a solution algorithm for the zero-order boundary-layer equations of the aircraft minimum-time-to-climb problem.
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
Discrete holomorphicity and integrability in loop models with open boundaries
de Gier, Jan; Rasmussen, Jorgen
2012-01-01
We consider boundary conditions compatible with discrete holomorphicity for the dilute O(n) and C_2^(1) loop models. In each model, for a general set of boundary plaquettes, multiple types of loops can appear. A generalisation of Smirnov's parafermionic observable is therefore required in order to maintain the discrete holomorphicity property in the bulk. We show that there exist natural boundary conditions for this observable which are consistent with integrability, that is to say that, by imposing certain boundary conditions, we obtain a set of linear equations whose solutions also satisfy the corresponding reflection equation. In both loop models, several new sets of integrable weights are found using this approach.
Recovering an obstacle using integral equations
Rundell, William
2009-05-01
We consider the inverse problem of recovering the shape, location and surface properties of an object where the surrounding medium is both conductive and homogeneous and we measure Cauchy data on an accessible part of the exterior boundary. It is assumed that the physical situation is modelled by harmonic functions and the boundary condition on the obstacle is one of Dirichlet type. The purpose of this paper is to answer some of the questions raised in a recent paper that introduced a nonlinear integral equation approach for the solution of this type of problem.
Free Boundary Value Problems for Abstract Elliptic Equations and Applications
Institute of Scientific and Technical Information of China (English)
Veli SHAKHMUROV
2011-01-01
The free boundary value problems for elliptic differential-operator equations are studied.Several conditions for the uniform maximal regularity with respect to boundary parameters and the Fredholmness in abstract Lp-spaces are given.In application,the nonlocal free boundary problems for finite or infinite systems of elliptic and anisotropic type equations are studied.
LINEAR SINGULAR INTEGRAL EQUATION ON DOMAINS COMPOSED BY BALLS
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
For domains composed by balls in Cn, this paper studies the boundary behaviour of Cauchy type integrals with discrete holomorphic kernels and the corresponding linear singular integral equation on each piece of smooth lower dimensional edges on the boundary of the domain.
Calculation of Turbulent Boundary Layers Using the Dissipation Integral Method
Institute of Scientific and Technical Information of China (English)
MatthiasBuschmann
1999-01-01
This paper gives an introduction into the dissipation integral method.The general integral equations for the three-dimensional case are derved.It is found that for a practical calculation algorithm the integral monentum equation and the integral energy equation are msot useful.Using Two different sets of mean velocity profiles the hyperbolical character of a dissipation integral method is shown.Test cases for two-and three-dimensional boundary layers are analysed and discussed.The paper concludes with a discussion of the advantages and limits of dissipation integral methods.
Half-space Kinetic Equations with General Boundary Conditions
Li, Qin; Sun, Weiran
2015-01-01
We study half-space linear kinetic equations with general boundary conditions that consist of both given incoming data and various type of reflections, extending our previous work [LLS14] on half-space equations with incoming boundary conditions. As in [LLS14], the main technique is a damping adding-removing procedure. We establish the well-posedness of linear (or linearized) half-space equations with general boundary conditions and quasi-optimality of the numerical scheme. The numerical method is validated by examples including a two-species transport equation, a multi-frequency transport equation, and the linearized BGK equation in 2D velocity space.
The free boundary Euler equations with large surface tension
Disconzi, Marcelo M.; Ebin, David G.
2015-01-01
We study the free boundary Euler equations with surface tension in three spatial dimensions, showing that the equations are well-posed if the coefficient of surface tension is positive. Then we prove that under natural assumptions, the solutions of the free boundary motion converge to solutions of the Euler equations in a domain with fixed boundary when the coefficient of surface tension tends to infinity.
The free boundary Euler equations with large surface tension
Disconzi, Marcelo M.; Ebin, David G.
2016-07-01
We study the free boundary Euler equations with surface tension in three spatial dimensions, showing that the equations are well-posed if the coefficient of surface tension is positive. Then we prove that under natural assumptions, the solutions of the free boundary motion converge to solutions of the Euler equations in a domain with fixed boundary when the coefficient of surface tension tends to infinity.
INHOMOGENEOUS INITIAL-BOUNDARY VALUE PROBLEM FOR GINZBURG-LANDAU EQUATIONS
Institute of Scientific and Technical Information of China (English)
杨灵娥; 郭柏灵; 徐海祥
2004-01-01
Some integral identities of smooth solution of inhomogeneous initial boundary value problem of Ginzburg-Landau equations were deduced, by which a priori estimates of the square norm on boundary of normal derivative and the square norm of partial derivatives were obtained. Then the existence of global weak solution of inhomogeneous initial-boundary value problem of Ginzburg-Landau equations was proved by the method of approximation technique and a priori estimates and making limit.
SOLVABILITY FOR NONLINEAR ELLIPTIC EQUATION WITH BOUNDARY PERTURBATION
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
The solvability of nonlinear elliptic equation with boundary perturbation is considered. The perturbed solution of original problem is obtained and the uniformly valid expansion of solution is proved.
Integral equations on time scales
Georgiev, Svetlin G
2016-01-01
This book offers the reader an overview of recent developments of integral equations on time scales. It also contains elegant analytical and numerical methods. This book is primarily intended for senior undergraduate students and beginning graduate students of engineering and science courses. The students in mathematical and physical sciences will find many sections of direct relevance. The book contains nine chapters and each chapter is pedagogically organized. This book is specially designed for those who wish to understand integral equations on time scales without having extensive mathematical background.
The Pressure Boundary Conditions for the Incompressible Navier—Stokes Equations Computation
Institute of Scientific and Technical Information of China (English)
YaosongCHEN; TaoJIANG
1996-01-01
This article devotes to the correct formulation of the pressure boundary conditions for the incompressible Navier-Stokes equation and gives the method of integration.The example has been shown for its feasibility.
Khairullin, Ermek
2016-08-01
In this paper we consider a special boundary value problem for multidimensional parabolic integro-differential equation with boundary conditions that contains as a boundary condition containing derivatives of order higher than the order of the equation. The solution is sought in the form of a thermal potential of a double layer. Shows lemma of finding the limits of the derivatives of the unknown function in the neighborhood of the hyperplane. Using the boundary condition and lemma obtained integral-differential equation (IDE) of parabolic operators, whĐţre an unknown function under the integral contains higher-order space variables derivatives. IDE is reduced to a singular integral equation (SIE), when an unknown function in the spatial variables satisfies the Holder. The characteristic part is solved in the class of distribution function using method of transformation of Fourier-Laplace. Found an algebraic condition for the transition to the classical generalized solution. Integral equation of the resolvent for the characteristic part of SIE is obtained. Integro-differential equation is reduced to the Volterra-Fredholm type integral equation of the second kind by method of regularization. It is shown that the solution of SIE is a solution of IDE. Obtain a theorem on the solvability of the boundary value problem of multidimensional parabolic integro-differential equation, when a known function of the spatial variables belongs to the Holder class and satisfies the solvability conditions.
Boundary Integral Solutions to Three-Dimensional Unconfined Darcy's Flow
Lennon, Gerard P.; Liu, Philip L.-F.; Liggett, James A.
1980-08-01
The boundary integral equation method (BIEM) is used to solve three-dimensional potential flow problems in porous media. The problems considered here are time dependent and have a nonlinear boundary condition on the free surface. The entire boundary, including the moving free surface, discretized into linear finite elements for the purpose of evaluating the boundary integrals. The technique allows transient, three-dimensional problems to be solved with reasonable computational costs. Numerical examples include recharge through rectangular and circular areas and seepage flow from a surface pond. The examples are used to illustrate the method and show the nonlinear effects.
Treatment of domain integrals in boundary element methods
Energy Technology Data Exchange (ETDEWEB)
Nintcheu Fata, Sylvain [ORNL
2012-01-01
A systematic and rigorous technique to calculate domain integrals without a volume-fitted mesh has been developed and validated in the context of a boundary element approximation. In the proposed approach, a domain integral involving a continuous or weakly-singular integrand is first converted into a surface integral by means of straight-path integrals that intersect the underlying domain. Then, the resulting surface integral is carried out either via analytic integration over boundary elements or by use of standard quadrature rules. This domain-to-boundary integral transformation is derived from an extension of the fundamental theorem of calculus to higher dimension, and the divergence theorem. In establishing the method, it is shown that the higher-dimensional version of the first fundamental theorem of calculus corresponds to the well-known Poincare lemma. The proposed technique can be employed to evaluate integrals defined over simply- or multiply-connected domains with Lipschitz boundaries which are embedded in an Euclidean space of arbitrary but finite dimension. Combined with the singular treatment of surface integrals that is widely available in the literature, this approach can also be utilized to effectively deal with boundary-value problems involving non-homogeneous source terms by way of a collocation or a Galerkin boundary integral equation method using only the prescribed surface discretization. Sample problems associated with the three-dimensional Poisson equation and featuring the Newton potential are successfully solved by a constant element collocation method to validate this study.
Explicit Expressions for 3D Boundary Integrals in Potential Theory
Energy Technology Data Exchange (ETDEWEB)
Nintcheu Fata, Sylvain [ORNL
2009-01-01
On employing isoparametric, piecewise linear shape functions over a flat triangular domain, exact expressions are derived for all surface potentials involved in the numerical solution of three-dimensional singular and hyper-singular boundary integral equations of potential theory. These formulae, which are valid for an arbitrary source point in space, are represented as analytic expressions over the edges of the integration triangle. They can be used to solve integral equations defined on polygonal boundaries via the collocation method or may be utilized as analytic expressions for the inner integrals in the Galerkin technique. Also, the constant element approximation can be directly obtained with no extra effort. Sample problems solved by the collocation boundary element method for the Laplace equation are included to validate the proposed formulae.
The Pauli equation with complex boundary conditions
Kochan, D; Novak, R; Siegl, P
2012-01-01
We consider one-dimensional Pauli Hamiltonians in a bounded interval with possibly non-self-adjoint Robin-type boundary conditions. We study the influence of the spin-magnetic interaction on the interplay between the type of boundary conditions and the spectrum. A special attention is paid to PT-symmetric boundary conditions with the physical choice of the time-reversal operator T.
Numerical integration of asymptotic solutions of ordinary differential equations
Thurston, Gaylen A.
1989-01-01
Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.
Testing outer boundary treatments for the Einstein equations
Rinne, Oliver; Scheel, Mark A
2007-01-01
Various methods of treating outer boundaries in numerical relativity are compared using a simple test problem: a Schwarzschild black hole with an outgoing gravitational wave perturbation. Numerical solutions computed using different boundary treatments are compared to a `reference' numerical solution obtained by placing the outer boundary at a very large radius. For each boundary treatment, the full solutions including constraint violations and extracted gravitational waves are compared to those of the reference solution, thereby assessing the reflections caused by the artificial boundary. These tests use a first-order generalized harmonic formulation of the Einstein equations. Constraint-preserving boundary conditions for this system are reviewed, and an improved boundary condition on the gauge degrees of freedom is presented. Alternate boundary conditions evaluated here include freezing the incoming characteristic fields, Sommerfeld boundary conditions, and the constraint-preserving boundary conditions of K...
The Pauli equation with complex boundary conditions
International Nuclear Information System (INIS)
We consider one-dimensional Pauli Hamiltonians in a bounded interval with possibly non-self-adjoint Robin-type boundary conditions. We study the influence of the spin–magnetic interaction on the interplay between the type of boundary conditions and the spectrum. Special attention is paid to PT-symmetric boundary conditions with the physical choice of the time-reversal operator T. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’. (paper)
The Pauli equation with complex boundary conditions
Kochan, D.; Krejčiřík, D.; Novák, R.; Siegl, P.
2012-11-01
We consider one-dimensional Pauli Hamiltonians in a bounded interval with possibly non-self-adjoint Robin-type boundary conditions. We study the influence of the spin-magnetic interaction on the interplay between the type of boundary conditions and the spectrum. Special attention is paid to {PT}-symmetric boundary conditions with the physical choice of the time-reversal operator {T}. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.
A New Algorithm for System of Integral Equations
Directory of Open Access Journals (Sweden)
Abdujabar Rasulov
2014-01-01
Full Text Available We develop a new algorithm to solve the system of integral equations. In this new method no need to use matrix weights. Beacause of it, we reduce computational complexity considerable. Using the new algorithm it is also possible to solve an initial boundary value problem for system of parabolic equations. To verify the efficiency, the results of computational experiments are given.
The open boundary equation (discussion paper)
Diederen, D.; Savenije, H.H.G.; Toffolon, M.
2015-01-01
We present a new equation describing the hydrodynamics in infinitely long tidal channels (i.e., no reflection) under the influence of oceanic forcing. The proposed equation is a simple relationship between partial derivatives of water level and velocity. It is formally derived for a progressive wave
A LOCAL BOUNDARY INTEGRAL EQUATION[1mm］METHOD FOR THE ELASTICITY PROBLEM%弹性力学问题的局部边界积分方程方法
Institute of Scientific and Technical Information of China (English)
龙述尧; 许敬晓
2000-01-01
The basic concept and numerical implementation of a local boundaryintegral equation formulation for solving the elasticity problem havebeen presented in the present paper. It is a new truly meshless method,because the numerical implementation of the method leads to an efficientmeshless discrete model. The concept of a companion solution isintroduced, such that the traction terms would not appear in theintegrals over the local boundary after the modified integral kernel isused for all nodes whose local boundary s falls within theglobal boundary of the given problem; it uses the moving leastsquare approximations, and involves only boundary integration over alocal boundary centered at the node in question. It poses nodifficulties in satisfying essential boundary conditions, and leads to abanded and sparse system matrix. The undependence of the solution onthe size of the integral local boundary provides a great flexibilityin dealing with the numerical model of the elastic plane problems undervarious boundary conditions with arbitrary shapes. Convergence studiesin the numerical examples show that the present method possesses anexcellent rate of convergence and reasonably accurate results for boththe unknown displacement and strain energy, as the originalapproximated trial solutions have nice continuity and smoothness. Thenumerical results also show that using both linear and quadratic basesas well as spline and Gaussian weight functions in approximationfunctions can give quite accurate numerical results. Compared with the conventional boundary element method based on the globalboundary integral equations, the present method is advantageous in thefollowing aspects: i) No boundary and domain element needed to be constructed in the presentmethod, while it is necessary to discretize both the entire domain andits boundary for the conventional boundary element method in general. Thevolume and boundary integrals in the present method are evaluated onlyover small
Institute of Scientific and Technical Information of China (English)
Xiao-jing LIU; Ji-zeng WANG; Xiao-min WANG; You-he ZHOU
2014-01-01
General exact solutions in terms of wavelet expansion are obtained for multi-term time-fractional diffusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial differential equations are converted into time-fractional ordinary differ-ential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-diffusion problems are given to validate the proposed analytical method.
Boundary value problemfor multidimensional fractional advection-dispersion equation
Directory of Open Access Journals (Sweden)
Khasambiev Mokhammad Vakhaevich
2015-05-01
Full Text Available In recent time there is a very great interest in the study of differential equations of fractional order, in which the unknown function is under the symbol of fractional derivative. It is due to the development of the theory of fractional integro-differential theory and application of it in different fields.The fractional integrals and derivatives of fractional integro-differential equations are widely used in modern investigations of theoretical physics, mechanics, and applied mathematics. The fractional calculus is a very powerful tool for describing physical systems, which have a memory and are non-local. Many processes in complex systems have nonlocality and long-time memory. Fractional integral operators and fractional differential operators allow describing some of these properties. The use of the fractional calculus will be helpful for obtaining the dynamical models, in which integro-differential operators describe power long-time memory by time and coordinates, and three-dimensional nonlocality for complex medium and processes.Differential equations of fractional order appear when we use fractal conception in physics of the condensed medium. The transfer, described by the operator with fractional derivatives at a long distance from the sources, leads to other behavior of relatively small concentrations as compared with classic diffusion. This fact redefines the existing ideas about safety, based on the ideas on exponential velocity of damping. Fractional calculus in the fractal theory and the systems with memory have the same importance as the classic analysis in mechanics of continuous medium.In recent years, the application of fractional derivatives for describing and studying the physical processes of stochastic transfer is very popular too. Many problems of filtration of liquids in fractal (high porous medium lead to the need to study boundary value problems for partial differential equations in fractional order.In this paper the
Analysis of Laminar Boundary Layer Equations
Directory of Open Access Journals (Sweden)
R. Yesman
2012-01-01
Full Text Available The paper proposes methodology for analysis and calculation of laminar fluid flow processes in a boundary layer.The presented dependences can be used for practical calculations while power carriers of various application are moving in the channels of heat and power devices.
Boundary Element Method with Non—overlapping Domain Decomposition for Diffusion Equation
Institute of Scientific and Technical Information of China (English)
ZHUJialin; ZHANGTaiping
2002-01-01
A boundary element method based on non-overlapping domain decomposition method to solve the time-dependent diffusion equations is presented.The time-dependent fundamental solution is used in the formulation of boundary integrals and the time integratioin process always restarts from the initial time condition.The process of replacing the interface values,which needs a summation of boundary integrals related to the boundary values at previous time steps can be treated in parallel parallel iterative procedure,Numerical experiments demonstrate that the implementation of the present alogrithm is efficient.
Integrable (2k)-Dimensional Hitchin Equations
Ward, R S
2016-01-01
This letter describes a completely-integrable system of Yang-Mills-Higgs equations which generalizes the Hitchin equations on a Riemann surface to arbitrary k-dimensional complex manifolds. The system arises as a dimensional reduction of a set of integrable Yang-Mills equations in 4k real dimensions. Our integrable system implies other generalizations such as the Simpson equations and the non-abelian Seiberg-Witten equations. Some simple solutions in the k=2 case are described.
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.
Boundary value problems of discrete generalized Emden-Fowler equation
Institute of Scientific and Technical Information of China (English)
YU; Jianshe; GUO; Zhiming
2006-01-01
By using the critical point theory, some sufficient conditions for the existence of the solutions to the boundary value problems of a discrete generalized Emden-Fowler equation are obtained. In a special case, a sharp condition is obtained for the existence of the boundary value problems of the above equation. For a linear case, by the discrete variational theory, a necessary and sufficient condition for the existence, uniqueness and multiplicity of the solutions is also established.
Stochastic integration and differential equations
Protter, Philip E
2003-01-01
It has been 15 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it "a new approach". The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, t...
Numerical Quadrature of Periodic Singular Integral Equations
DEFF Research Database (Denmark)
Krenk, Steen
1978-01-01
This paper presents quadrature formulae for the numerical integration of a singular integral equation with Hilbert kernel. The formulae are based on trigonometric interpolation. By integration a quadrature formula for an integral with a logarithmic singularity is obtained. Finally it is demonstra......This paper presents quadrature formulae for the numerical integration of a singular integral equation with Hilbert kernel. The formulae are based on trigonometric interpolation. By integration a quadrature formula for an integral with a logarithmic singularity is obtained. Finally...
BOUNDARY LAYER AND VANISHING DIFFUSION LIMIT FOR NONLINEAR EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
彭艳
2014-01-01
In this paper, we consider an initial-boundary value problem for some nonlinear evolution equations with damping and diffusion. The main purpose is to investigate the boundary layer effect and the convergence rates as the diffusion parameterαgoes to zero.
A NONLOCAL NONLINEAR BOUNDARY VALUE PROBLEM FOR THE HEAT EQUATIONS
Institute of Scientific and Technical Information of China (English)
YANJINHAI
1996-01-01
The existenoe and limit hehaviour of the solution for a kind of nonloeal noulinear boundary value condition on a part of the boundary is studied for the heat equation, which physicallymeans that the potential is the function of the total flux. When this part of boundary shrinks to a point in a certain way, this condition either results in a Dirac measure or simply disappears in the corresponding problem.
Integral Transform Approach to Generalized Tricomi Equations
Yagdjian, Karen
2014-01-01
We present some integral transform that allows to obtain solutions of the generalized Tricomi equation from solutions of a simpler equation. We used in [13,14],[41]-[46] the particular version of this transform in order to investigate in a unified way several equations such as the linear and semilinear Tricomi equations, Gellerstedt equation, the wave equation in Einstein-de Sitter spacetime, the wave and the Klein-Gordon equations in the de Sitter and anti-de Sitter spacetimes.
Surface integrals approach to solution of some free boundary problems
Directory of Open Access Journals (Sweden)
Igor Malyshev
1988-01-01
Full Text Available Inverse problems in which it is required to determine the coefficients of an equation belong to the important class of ill-posed problems. Among these, of increasing significance, are problems with free boundaries. They can be found in a wide range of disciplines including medicine, materials engineering, control theory, etc. We apply the integral equations techniques, typical for parabolic inverse problems, to the solution of a generalized Stefan problem. The regularization of the corresponding system of nonlinear integral Volterra equations, as well as local existence, uniqueness, continuation of its solution, and several numerical experiments are discussed.
Multiple integral representation for the trigonometric SOS model with domain wall boundaries
Galleas, W
2011-01-01
Using the dynamical Yang-Baxter algebra we derive a functional equation for the partition function of the trigonometric SOS model with domain wall boundary conditions. The solution of the equation is given in terms of a multiple contour integral.
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.
Thermodynamically admissible boundary conditions for the regularized 13 moment equations
Energy Technology Data Exchange (ETDEWEB)
Rana, Anirudh Singh, E-mail: anirudh@uvic.ca [Department of Mechanical and Aerospace Engineering, Gyeongsang National University, Jinju, Gyeongnam 52828 (Korea, Republic of); Struchtrup, Henning, E-mail: struchtr@uvic.ca [Department of Mechanical Engineering, University of Victoria, Victoria, British Columbia V8W 2Y2 (Canada)
2016-02-15
A phenomenological approach to the boundary conditions for linearized R13 equations is derived using the second law of thermodynamics. The phenomenological coefficients appearing in the boundary conditions are calculated by comparing the slip, jump, and thermal creep coefficients with linearized Boltzmann solutions for Maxwell’s accommodation model for different values of the accommodation coefficient. For this, the linearized R13 equations are solved for viscous slip, thermal creep, and temperature jump problems and the results are compared to the solutions of the linearized Boltzmann equation. The influence of different collision models (hard-sphere, Bhatnagar–Gross–Krook, and Maxwell molecules) and accommodation coefficients on the phenomenological coefficients is studied.
LaChapelle, J.
2004-01-01
A path integral is presented that solves a general class of linear second order partial differential equations with Dirichlet/Neumann boundary conditions. Elementary kernels are constructed for both Dirichlet and Neumann boundary conditions. The general solution can be specialized to solve elliptic, parabolic, and hyperbolic partial differential equations with boundary conditions. This extends the well-known path integral solution of the Schr\\"{o}dinger/diffusion equation in unbounded space. ...
On the solution of integral equations with strongly singular kernels
Kaya, A. C.; Erdogan, F.
1987-01-01
Some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m, m greater than or equal 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t-x) sup-m, terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.
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.
Solitons induced by boundary conditions from the Boussinesq equation
Chou, Ru Ling; Chu, C. K.
1990-01-01
The behavior of solitons induced by boundary excitation is investigated at various time-dependent conditions and different unperturbed water depths, using the Korteweg-de Vries (KdV) equation. Then, solitons induced from Boussinesq equations under similar conditions were studied, making it possible to remove the restriction in the KdV equation and to treat soliton head-on collisions (as well as overtaking collisions) and reflections. It is found that the results obtained from the KdV and the Boussinesq equations are in good agreement.
Institute of Scientific and Technical Information of China (English)
任景莉; 葛渭高
2003-01-01
A boundary value problems f or functional differenatial equations, with nonlinear boundary condition, is studied by the theorem of differential inequality. Using new method to construct the upper solution and lower solution, sufficient conditions for the existence of the problems' solution are established. A uniformly valid asymptotic expansions of the solution is also given.
Symmetries of boundary layer equations of power-law fluids of second grade
Institute of Scientific and Technical Information of China (English)
Mehmet Pakdemirli; Yi(g)it Aksoy; Muhammet Y(u)r(u)soy; Chaudry Masood Khalique
2008-01-01
A modified power-law fluid of second grade is considered. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incom-pressible flows, and from which the boundary layer equations are derived. Symmetries of the boundary layer equations are found by using Lie group theory, and then group classifica-tion with respect to power-law index is performed. By using one of the symmetries, namely the scaling symmetry, the partial differential system is transformed into an ordinary differential system, which is numerically integrated under the classical boundary layer conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions.
Periodic solutions to nonlinear equations with oblique boundary conditions
Allergretto, Walter; Papini, Duccio
2012-01-01
We study the existence of positive periodic solutions to nonlinear elliptic and parabolic equations with oblique and dynamical boundary conditions and non-local terms. The results are obtained through fixed point theory, topological degree methods and properties of related linear elliptic problems with natural boundary conditions and possibly non-symmetric principal part. As immediate consequences, we also obtain estimates on the principal eigenvalue for non-symmetric elliptic ...
Traveling waves for a boundary reaction-diffusion equation
Caffarelli, L; Sire, Y
2011-01-01
We prove the existence of a traveling wave solution for a boundary reaction diffusion equation when the reaction term is the combustion nonlinearity with ignition temperature. A key role in the proof is plaid by an explicit formula for traveling wave solutions of a free boundary problem obtained as singular limit for the reaction-diffusion equation (the so-called high energy activation energy limit). This explicit formula, which is interesting in itself, also allows us to get an estimate on the decay at infinity of the traveling wave (which turns out to be faster than the usual exponential decay).
Distribution theory for Schrödinger’s integral equation
Energy Technology Data Exchange (ETDEWEB)
Lange, Rutger-Jan, E-mail: rutger-jan.lange@cantab.net [VU University Amsterdam, 1081 HV Amsterdam (Netherlands)
2015-12-15
Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrödinger’s equation. This paper, in contrast, investigates the integral form of Schrödinger’s equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schrödinger’s integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schrödinger’s differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov’s [J. Math. Anal. Appl. 201(1), 297–323 (1996)] result to hypersurfaces. Second, we derive a new closed-form solution to Schrödinger’s integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schrödinger’s differential equation. Third, we derive boundary conditions for “super-singular” potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schrödinger’s integral equation is a viable tool for studying singular interactions in quantum mechanics.
Coleman-Gurtin type equations with dynamic boundary conditions
Gal, Ciprian G.; Shomberg, Joseph L.
2015-02-01
We present a new formulation and generalization of the classical theory of heat conduction with or without fading memory. As a special case, we investigate the well-posedness of systems which consist of Coleman-Gurtin type equations subject to dynamic boundary conditions, also with memory. Nonlinear terms are defined on the interior of the domain and on the boundary and subject to either classical dissipation assumptions, or to a nonlinear balance condition in the sense of Gal (2012). Additionally, we do not assume that the interior and the boundary share the same memory kernel.
Inequalities applicable to retarded Volterra integral equations
Directory of Open Access Journals (Sweden)
B. G. Pachpatte
2004-12-01
Full Text Available The main objective of this paper is to establish explicit bounds on certain integral inequialities which can be used as tools in the study of certain classes of retarded Volterra integral equations.
Numerical solutions of telegraph equations with the Dirichlet boundary condition
Ashyralyev, Allaberen; Turkcan, Kadriye Tuba; Koksal, Mehmet Emir
2016-08-01
In this study, the Cauchy problem for telegraph equations in a Hilbert space is considered. Stability estimates for the solution of this problem are presented. The third order of accuracy difference scheme is constructed for approximate solutions of the problem. Stability estimates for the solution of this difference scheme are established. As a test problem to support theoretical results, one-dimensional telegraph equation with the Dirichlet boundary condition is considered. Numerical solutions of this equation are obtained by first, second and third order of accuracy difference schemes.
Boundary integral methods for microfluidic problems
Burbidge, Adam
2015-01-01
Microscale experiments of reduced complexity allow one to tease out and examine some of the interesting phenomena that manifest in large hierarchically structured materials which are of general interest across many industries. Recent advances in high speed imaging techniques and post-processing allow experiments to yield small scale information which was previously unavailable, or extremely difficult to obtain. This additional information provides new challenges in terms of theoretical understanding and prediction that requires new tools. We discuss generalised weighted residual numerical methods as a means of solving physically derived systems of PDEs, using the steady Stokes equation as an example. These formulations require integration of arbitrary functions of submanifolds which often will have a lower dimensionality than the parent manifold, leading to cumbersome calculations of the Jacobian determinant. We provide a tensorial view of the transformation, in which the natural element coordinate system is a non-orthogonal frame, and derive an expression for the Jacobian factor in terms of the contravariant metric tensor gij. This approach has the additional advantage that it can be extended to yield the local surface curvature, which will be essential for correct implementation of free surface boundaries.
QUENCHING ON BOUNDARY TO THE NEWTON FILTRATION EQUATION (Ⅰ)
Institute of Scientific and Technical Information of China (English)
段志文; 谢春红; 卢伟明
2003-01-01
This paper discusses the global existence and quenching of the solution to the Newton filtration equation with the nonlinear boundary condition.The authors also discuss the profile of the quenching solution in the quenching time and obtain the quenching rate of the quenching solution.
Positive Solutions for Nonlinear Differential Equations with Periodic Boundary Condition
Directory of Open Access Journals (Sweden)
Shengjun Li
2012-01-01
Full Text Available We study the existence of positive solutions for second-order nonlinear differential equations with nonseparated boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a nonlinear alternative principle of Leray-Schauder. Recent results in the literature are generalized and significantly improved.
An extension of the transpired skin-friction equation to compressible turbulent boundary layers
Silva-Freire, Atila P.
1988-11-01
A skin-friction equation for transpired incompressible turbulent boundary layer, proposed in a previous paper (Silva-Freire, 1988), is extended to compressible flow. The expression derived here is simple and gives more consistent results than the momentum-integral equation. The difficulty with the present formulation, however, is that the wake profile parameter due to injection has to be carefully determined in order to obtain good results.
Institute of Scientific and Technical Information of China (English)
Mo Jiaqi
2007-01-01
A class of nonlinear initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions and using the theory of differential inequalities the asymptotic solution of the initial boundary value problems is studied.
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.
Gravity actions, boundary terms and second-order field equations
International Nuclear Information System (INIS)
The Lovelock action is a natural generalization of the Einstein-Hilbert gravity action in higher (more than four) dimensions. Certain boundary terms have to be added to this action because of various reasons which we briefly review. Dimensional reduction of the Lovelock action leads to a generalized Einstein-Yang-Mills (+ scalar field or σ-models) action in lower dimensions. Since the field equations derived from the Lovelock action are of second order only, the same holds for the dimensionally reduced action. The necessary boundary terms for the latter action are then obtained by dimensional reduction of the boundary terms in the Lovelock action. We consider reduction from n to n-1 dimensions in some detail and obtain generalizations of Horndeski's non-minimally coupled Einstein-Maxwell action with second-order field equations in more than four dimensions. (orig./HSI)
Solving wave equation with spectral methods and nonreflecting boundary conditions
Novák, J; Novak, Jerome; Bonazzola, Silvano
2002-01-01
A multidomain spectral method for solving wave equations is presented. This method relies on the expansion of functions on basis of spherical harmonics $(Y_l^m(\\theta, \\phi))$ for the angular dependence and of Chebyshev polynomials $T_n(x)$ for the radial part. The spherical domains consist of shells surrounding a nucleus and cover the space up to a finite radius $R$ at which boundary conditions are imposed. Time derivatives are estimated using standard finite-differences second order schemes, which are chosen to be implicit to allow for (almost) any size of time-step. Emphasis is put on the implementation of absorbing boundary conditions that allow for the numerical boundary to be completely transparent to the physical wave. This is done using a multipolar expansion of an exact boundary condition for outgoing waves, which is truncated at some point. Using an auxiliary function, which is solution of a wave equation on the sphere defining the outer boundary of the numerical grid, the absorbing boundary conditi...
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...
An integral equation representation approach for valuing Russian options with a finite time horizon
Jeon, Junkee; Han, Heejae; Kim, Hyeonuk; Kang, Myungjoo
2016-07-01
In this paper, we first describe a general solution for the inhomogeneous Black-Scholes partial differential equation with mixed boundary conditions using Mellin transform techniques. Since Russian options with a finite time horizon are usually formulated into the inhomogeneous free-boundary Black-Scholes partial differential equation with a mixed boundary condition, we apply our method to Russian options and derive an integral equation satisfied by Russian options with a finite time horizon. Furthermore, we present some numerical solutions and plots of the integral equation using recursive integration methods and demonstrate the computational accuracy and efficiency of our method compared to other competing approaches.
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.
Algebraic Bethe Ansatz for O(2N) sigma models with integrable diagonal boundaries
Gombor, Tamas
2015-01-01
The finite volume problem of O(2N) sigma models with integrable diagonal boundaries on a finite interval is investigated. The double row transfer matrix is diagonalized by Algebraic Bethe Ansatz. The boundary Bethe Yang equations for the particle rapidities and the accompanying Bethe Ansatz equations are derived.
Asymptotic integration of differential and difference equations
Bodine, Sigrun
2015-01-01
This book presents the theory of asymptotic integration for both linear differential and difference equations. This type of asymptotic analysis is based on some fundamental principles by Norman Levinson. While he applied them to a special class of differential equations, subsequent work has shown that the same principles lead to asymptotic results for much wider classes of differential and also difference equations. After discussing asymptotic integration in a unified approach, this book studies how the application of these methods provides several new insights and frequent improvements to results found in earlier literature. It then continues with a brief introduction to the relatively new field of asymptotic integration for dynamic equations on time scales. Asymptotic Integration of Differential and Difference Equations is a self-contained and clearly structured presentation of some of the most important results in asymptotic integration and the techniques used in this field. It will appeal to researchers i...
Integral equations for the electromagnetic field in dielectrics
Mostowski, Jan; Załuska-Kotur, Magdalena A.
2016-09-01
We study static the electric field and electromagnetic waves in dielectric media. In contrast to the standard approach, we use, formulate and solve integral equations for the field. We discuss the case of an electrostatic field of a point charge placed inside a dielectric; the integral equation approach allows us to find and interpret the dielectric constant in terms of molecular polarizability. Next we discuss propagation of electromagnetic waves using the same integral equation approach. We derive the dispersion relation and find the reflection and transmission coefficients at the boundary between the vacuum and the dielectric. The present approach supplements the standard approach based on macroscopic Maxwell equations and contributes to better a understanding of some electromagnetic effects.
Blow-up estimates for semilinear parabolic systems coupled in an equation and a boundary condition
Institute of Scientific and Technical Information of China (English)
WANG; Mingxin(
2001-01-01
［1］Wang, S., Wang, M. X., Xie, C. H., Reaction-diffusion systems with nonlinear boundary conditions, Z. angew. Math.Phys., 1997, 48(6): 994－1001.［2］Fila, M., Quittner, P., The blow-up rate for a semilinear parabolic system, J. Math. Anal. Appl., 1999, 238: 468－476.［3］Hu, B., Remarks on the blow-up estimate for solutions of the heat equation with a nonlinear boundary condition, Differential Integral Equations, 1996, 9(5): 891－901.［4］Hu, B. , Yin, H. M., The profile near blow-up time for solution of the heat equation with a nonlinear boundary condition,Trans. of Amer. Math. Soc., 1994, 346: 117－135.［5］Amann, H., Parabolic equations and nonlinear boundary conditions, J. of Diff. Eqns., 1988, 72: 201－269.［6］Deng, K., Blow-up rates for parabolic systems, Z. angew. Math. Phys. ,1996, 47: 132－143.［7］Fila, M., Levine, H. A., On critical exponents for a semilinear parabolic system coupled in an equation and a boundary condition, J. Math. Anal. Appl., 1996, 204: 494－521.
Maxwell boundary conditions imply non-Lindblad master equation
Bamba, Motoaki; Imoto, Nobuyuki
2016-09-01
From the Hamiltonian connecting the inside and outside of a Fabry-Pérot cavity, which is derived from the Maxwell boundary conditions at a mirror of the cavity, a master equation of a non-Lindblad form is derived when the cavity embeds matters, although we can transform it to the Lindblad form by performing the rotating-wave approximation to the connecting Hamiltonian. We calculate absorption spectra by these Lindblad and non-Lindblad master equations and also by the Maxwell boundary conditions in the framework of the classical electrodynamics, which we consider the most reliable approach. We found that, compared to the Lindblad master equation, the absorption spectra by the non-Lindblad one agree better with those by the Maxwell boundary conditions. Although the discrepancy is highlighted only in the ultrastrong light-matter interaction regime with a relatively large broadening, the master equation of the non-Lindblad form is preferable rather than of the Lindblad one for pursuing the consistency with the classical electrodynamics.
Maxwell boundary conditions impose non-Lindblad master equation
Bamba, Motoaki
2016-01-01
From the Hamiltonian connecting the inside and outside of an Fabry-Perot cavity, which is derived from the Maxwell boundary conditions at a mirror of the cavity, a master equation of a non-Lindblad form is derived when the cavity embeds matters, although we can transform it to the Lindblad form by performing the rotating-wave approximation to that Hamiltonian. We calculate absorption spectra by these Lindblad and non-Lindblad master equations and also by the Maxwell boundary conditions in framework of the classical electrodynamics, which we consider the most reliable approach. We found that, compared to the Lindblad master equation, the absorption spectra by the non-Lindblad one agree better with those by the Maxwell boundary conditions. Although the discrepancy is highlighted only in the ultra-strong light-matter interaction regime with a relatively large broadening, the master equation of the non-Lindblad form is preferable rather than of the Lindblad one for pursuing the consistency with the classical elec...
Optimal control problems for impulsive systems with integral boundary conditions
Directory of Open Access Journals (Sweden)
Allaberen Ashyralyev
2013-03-01
Full Text Available In this article, the optimal control problem is considered when the state of the system is described by the impulsive differential equations with integral boundary conditions. Applying the Banach contraction principle the existence and uniqueness of the solution is proved for the corresponding boundary problem by the fixed admissible control. The first and second variation of the functional is calculated. Various necessary conditions of optimality of the first and second order are obtained by the help of the variation of the controls.
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.
Dual integral equations with Fox's H-function kernel
Directory of Open Access Journals (Sweden)
R. N. Kalia
1984-01-01
Full Text Available The dual integral equations involving Bessel function kernels were first considered by Weber in 1873. The problem comprised of finding potential of an electrified disc which belongs to a general category of mixed boundary value problems. Titchmarsh gave the formal solution using Wiener-Hopf procedure. We use this direct method as improvised by Busbridge to solve a class of dual integral equations which can be reduced to other known kernels by particularizing the parameters in the Fox's H-function.
A class of two-dimensional dual integral equations and its application
Institute of Scientific and Technical Information of China (English)
FAN Tian-you; SUN Zhu-feng
2007-01-01
Because exact analytic solution is not available, we use double expansion and boundary collocation to construct an approximate solution for a class of two-dimensional dual integral equations in mathematical physics. The integral equations by this procedure are reduced to infinite algebraic equations. The accuracy of the solution lies in the boundary collocation technique. The application of which for some complicated initialboundary value problems in solid mechanics indicates the method is powerful.
New integrability case for the Riccati equation
Mak, M K
2012-01-01
A new integrability condition of the Riccati equation $dy/dx=a(x)+b(x)y+c(x)y^{2}$ is presented. By introducing an auxiliary equation depending on a generating function $f(x)$, the general solution of the Riccati equation can be obtained if the coefficients $a(x)$, $b(x)$, $c(x)$, and the function $f(x)$ satisfy a particular constraint. The validity and reliability of the method are tested by obtaining the general solutions of some Riccati type differential equations. Some applications of the integrability conditions for the case of the damped harmonic oscillator with time dependent frequency, and for solitonic wave, are briefly discussed.
PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS INVOLVING PETTIS INTEGRAL
Institute of Scientific and Technical Information of China (English)
Hussein A.H. Salem
2011-01-01
In this article, we investigate the existence of Pseudo solutions for some frac- tional order boundary value problem with integral boundary conditions in the Banach space of continuous function equipped with its weak topology. The class of such problems constitute a very interesting and important class of problems. They include two, three, multi-point and nonlocal boundary-value problems as special cases. In our investigation, the right hand side of the above problem is assumed to be Pettis integrable function. To encompass the full scope of this article, we give an example illustrating the main result.
Sanderse, B.; Verstappen, R. W. C. P.; Koren, B.
2014-01-01
Harlow and Welch [Phys. Fluids 8 (1965) 2182-2189] introduced a discretization method for the incompressible Navier-Stokes equations conserving the secondary quantities kinetic energy and vorticity, besides the primary quantities mass and momentum. This method was extended to fourth order accuracy by several researchers [25,14,21]. In this paper we propose a new consistent boundary treatment for this method, which is such that continuous integration-by-parts identities (including boundary contributions) are mimicked in a discrete sense. In this way kinetic energy is exactly conserved even in case of non-zero tangential boundary conditions. We show that requiring energy conservation at the boundary conflicts with order of accuracy conditions, and that the global accuracy of the fourth order method is limited to second order in the presence of boundaries. We indicate how non-uniform grids can be employed to obtain full fourth order accuracy.
Counting master integrals: Integration by parts vs. functional equations
Kniehl, Bernd A
2016-01-01
We illustrate the usefulness of functional equations in establishing relationships between master integrals under the integration-by-parts reduction procedure by considering a certain two-loop propagator-type diagram as an example.
Counting master integrals. Integration by parts vs. functional equations
International Nuclear Information System (INIS)
We illustrate the usefulness of functional equations in establishing relationships between master integrals under the integration-by-parts reduction procedure by considering a certain two-loop propagator-type diagram as an example.
Counting master integrals. Integration by parts vs. functional equations
Energy Technology Data Exchange (ETDEWEB)
Kniehl, Bernd A.; Tarasov, Oleg V. [Hamburg Univ. (Germany). II. Inst. fuer Theoretische Physik
2016-01-15
We illustrate the usefulness of functional equations in establishing relationships between master integrals under the integration-by-parts reduction procedure by considering a certain two-loop propagator-type diagram as an example.
A boundary matching micro/macro decomposition for kinetic equations
Lemou, Mohammed
2010-01-01
We introduce a new micro/macro decomposition of collisional kinetic equations which naturally incorporates the exact space boundary conditions. The idea is to write the distribution fonction $f$ in all its domain as the sum of a Maxwellian adapted to the boundary (which is not the usual Maxwellian associated with $f$) and a reminder kinetic part. This Maxwellian is defined such that its 'incoming' velocity moments coincide with the 'incoming' velocity moments of the distribution function. Important consequences of this strategy are the following. i) No artificial boundary condition is needed in the micro/macro models and the exact boundary condition on $f$ is naturally transposed to the macro part of the model. ii) It provides a new class of the so-called 'Asymptotic preserving' (AP) numerical schemes: such schemes are consistent with the original kinetic equation for all fixed positive value of the Knudsen number $\\eps$, and if $\\eps \\to 0 $ with fixed numerical parameters then these schemes degenerate into ...
THE COUPLING OF NATURAL BOUNDARY ELEMENT AND FINITE ELEMENT METHOD FOR 2D HYPERBOLIC EQUATIONS
Institute of Scientific and Technical Information of China (English)
De-hao Yu; Qi-kui Du
2003-01-01
In this paper, we investigate the coupling of natural boundary element and finite ele-ment methods of exterior initial boundary value problems for hyperbolic equations. Thegoverning equation is first discretized in time, leading to a time-step scheme, where anexterior elliptic problem has to be solved in each time step. Second, a circular artifi-in an unbounded domain is transformed into the nonlocal boundary value problem in abounded subdomain. And the natural integral equation and the Poisson integral formulaare obtained in the infinite domain Ω2 outside circle of radius R. The coupled variationalformulation is given. Only the function itself, not its normal derivative at artificial bound-and the boundary element stiffness matrix has a few different elements. Such a coupledmethod is superior to the one based on direct boundary element method. This paper dis-cusses finite element discretization for variational problem and its corresponding numericaltechnique, and the convergence for the numerical solutions. Finally, the numerical exampleis presented to illustrate feasibility and efficiency of this method.
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.
Linear integral transformations and hierarchies of integrable nonlinear evolution equations
Nijhoff, Frank W.
1988-07-01
Integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations. These are (Riemann-Hilbert type of) integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an (indefinite) parameter k. The potential matrices in these equations obey a set of nonlinear recursion relations, leading to a heirarchy of nonlinear partial differential equations. In decreasing order the same equations give rise to a “reciprocal” hierarchy, associated with Heisenberg ferromagnet type of equations. Central in the treatment is an embedding of the hierarchy into an infinite-matrix structure, which is constructed on the basis of the integral equations. In terms of this infinite-matrix structure the equations governing the hierarchies become quite simple. Furthermore, it leads in a straightforward way to various generalizations, such as to other types of linear spectral problems, multicomponent system and lattice equations. Generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment. Finally, some results on conserved densities and the Hamiltonian structure are briefly discussed.
Explicit Integration of Friedmann's Equation with Nonlinear Equations of State
Chen, Shouxin; Yang, Yisong
2015-01-01
This paper is a continuation of our earlier study on the integrability of the Friedmann equations in the light of the Chebyshev theorem. Our main focus will be on a series of important, yet not previously touched, problems when the equation of state for the perfect-fluid universe is nonlinear. These include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born--Infeld, and two-fluid models. We show that some of these may be integrated using Chebyshev's result while other are out of reach by the theorem but may be integrated explicitly by other methods. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution. For example, in the Chaplygin gas universe, it is seen that, as far as there is a tiny presence of nonlinear matter, linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics ...
A Nonlinear Singularly Perturbed Problem for Reaction Diffusion Equations with Boundary Perturbation
Institute of Scientific and Technical Information of China (English)
Jia-qi Mo; Wan-tao Lin
2005-01-01
A nonlinear singularly perturbed problems for reaction diffusion equation with boundary perturbation is considered. Under suitable conditions, the asymptotic behavior of solution for the initial boundary value problems of reaction diffusion equations is studied using the theory of differential inequalities.
Path Integral for the Dirac Equation
Polonyi, Janos
1998-01-01
A c-number path integral representation is constructed for the solution of the Dirac equation. The integration is over the real trajectories in the continuous three-space and other two canonical pairs of compact variables controlling the spin and the chirality flips.
Path Integral for Relativistic Equations of Motion
Gosselin, Pierre; Polonyi, Janos
1997-01-01
A non-Grassmanian path integral representation is given for the solution of the Klein-Gordon and the Dirac equations. The trajectories of the path integral are rendered differentiable by the relativistic corrections. The nonrelativistic limit is briefly discussed from the point of view of the renormalization group.
A Collocation Method for Volterra Integral Equations
Kolk, Marek
2010-09-01
We propose a piecewise polynomial collocation method for solving linear Volterra integral equations of the second kind with logarithmic kernels which, in addition to a diagonal singularity, may have a singularity at the initial point of the interval of integration. An attainable order of the convergence of the method is studied. We illustrate our results with a numerical example.
Integrability of Lie Systems Through Riccati Equations
Cariñena, José F.; de Lucas, Javier
Integrability conditions for Lie systems are related to reduction or transformation processes. We here analyse a geometric method to construct integrability conditions for Riccati equations following these approaches. This approach provides us with a unified geometrical viewpoint that allows us to analyse some previous works on the topic and explain new properties. Moreover, this new approach can be straightforwardly generalised to describe integrability conditions for any Lie system. Finally, we show the usefulness of our treatment in order to study the problem of the linearisability of Riccati equations.
Integrability of Lie systems through Riccati equations
Cariñena, José F
2010-01-01
Integrability conditions for Lie systems are related to reduction or transformation processes. We here analyse a geometric method to construct integrability conditions for Riccati equations following these approaches. This approach provides us with a unified geometrical viewpoint that allows us to analyse some previous works on the topic and explain new properties. Moreover, this new approach can be straightforwardly generalised to describe integrability conditions for any Lie system. Finally, we show the usefulness of our treatment in order to study the problem of the linearisability of Riccati equations.
A Numerical Method for 1-D Parabolic Equation with Nonlocal Boundary Conditions
Directory of Open Access Journals (Sweden)
M. Tadi
2014-01-01
Full Text Available This paper is concerned with a local method for the solution of one-dimensional parabolic equation with nonlocal boundary conditions. The method uses a coordinate transformation. After the coordinate transformation, it is then possible to obtain exact solutions for the resulting equations in terms of the local variables. These exact solutions are in terms of constants of integration that are unknown. By imposing the given boundary conditions and smoothness requirements for the solution, it is possible to furnish a set of linearly independent conditions that can be used to solve for the constants of integration. A number of examples are used to study the applicability of the method. In particular, three nonlinear problems are used to show the novelty of the method.
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.
Periodic and Boundary Value Problems for Second Order Differential Equations
Indian Academy of Sciences (India)
Nikolaos S Papageorgiou; Francesca Papalini
2001-02-01
In this paper we study second order scalar differential equations with Sturm–Liouville and periodic boundary conditions. The vector field (, , ) is Caratheodory and in some instances the continuity condition on or is replaced by a monotonicity type hypothesis. Using the method of upper and lower solutions as well as truncation and penalization techniques, we show the existence of solutions and extremal solutions in the order interval determined by the upper and lower solutions. Also we establish some properties of the solutions and of the set they form.
On a third order parabolic equation with a nonlocal boundary condition
Directory of Open Access Journals (Sweden)
Abdelfatah Bouziani
2000-01-01
Full Text Available In this paper we demonstrate the existence, uniqueness and continuous dependence of a strong solution upon the data, for a mixed problem which combine classical boundary conditions and an integral condition, such as the total mass, flux or energy, for a third order parabolic equation. We present a functional analysis method based on an a priori estimate and on the density of the range of the operator generated by the studied problem.
Variational integrators for nonvariational partial differential equations
Kraus, Michael; Maj, Omar
2015-08-01
Variational integrators for Lagrangian dynamical systems provide a systematic way to derive geometric numerical methods. These methods preserve a discrete multisymplectic form as well as momenta associated to symmetries of the Lagrangian via Noether's theorem. An inevitable prerequisite for the derivation of variational integrators is the existence of a variational formulation for the considered problem. Even though for a large class of systems this requirement is fulfilled, there are many interesting examples which do not belong to this class, e.g., equations of advection-diffusion type frequently encountered in fluid dynamics or plasma physics. On the other hand, it is always possible to embed an arbitrary dynamical system into a larger Lagrangian system using the method of formal (or adjoint) Lagrangians. We investigate the application of the variational integrator method to formal Lagrangians, and thereby extend the application domain of variational integrators to include potentially all dynamical systems. The theory is supported by physically relevant examples, such as the advection equation and the vorticity equation, and numerically verified. Remarkably, the integrator for the vorticity equation combines Arakawa's discretisation of the Poisson brackets with a symplectic time stepping scheme in a fully covariant way such that the discrete energy is exactly preserved. In the presentation of the results, we try to make the geometric framework of variational integrators accessible to non specialists.
Adaptive integral equation methods in transport theory
International Nuclear Information System (INIS)
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
Spectral integration of linear boundary value problems
Viswanath, Divakar
2012-01-01
Spectral integration is a method for solving linear boundary value problems which uses the Chebyshev series representation of functions to avoid the numerical discretization of derivatives. It is occasionally attributed to Zebib (J. of Computational Physics vol. 53 (1984), p. 443-455) and more often to Greengard (SIAM J. on Numerical Analysis vol. 28 (1991), p. 1071-1080). Its advantage is believed to be its relative immunity to errors that arise when nearby grid points are used to approximate derivatives. In this paper, we reformulate the method of spectral integration by changing it in four different ways. The changes consist of a more convenient integral formulation, a different way to treat and interpret boundary conditions, treatment of higher order problems in factored form, and the use of piecewise Chebyshev grid points. Our formulation of spectral integration is more flexible and powerful as show by its ability to solve a problem that would otherwise take 8192 grid points using only 96 grid points. So...
NATURAL BOUNDARY INTEGRAL METHOD AND ITS NEW DEVELOPMENT
Institute of Scientific and Technical Information of China (English)
De-hao Yu
2004-01-01
In this paper, the natural boundary integral method, and some related methods, including coupling method of the natural boundary elements and finite elements, which is also called DtN method or the method with exact artificial boundary conditions, domain decomposition methods based on the natural boundary reduction, and the adaptive boundary element method with hyper-singular a posteriori error estimates, are discussed.
Inequalities for differential and integral equations
Ames, William F
1997-01-01
Inequalities for Differential and Integral Equations has long been needed; it contains material which is hard to find in other books. Written by a major contributor to the field, this comprehensive resource contains many inequalities which have only recently appeared in the literature and which can be used as powerful tools in the development of applications in the theory of new classes of differential and integral equations. For researchers working in this area, it will be a valuable source of reference and inspiration. It could also be used as the text for an advanced graduate course.Key Features* Covers a variety of linear and nonlinear inequalities which find widespread applications in the theory of various classes of differential and integral equations* Contains many inequalities which have only recently appeared in literature and cannot yet be found in other books* Provides a valuable reference to engineers and graduate students
New integral equation for simple fluids
Kang, Hong Seok; Ree, Francis H.
1995-09-01
We present a new integral equation for the radial distribution function of classical fluids. It employs the bridge function for a short-range repulsive reference system which was used earlier in our dense fluid perturbation theory. The bridge function is evaluated using Ballone et al.'s closure relation. Applications of the integral equation to the Lennard-Jones and inverse nth-power (n=12, 9, 6, and 4) repulsive systems show that it can predict thermodynamic and structural properties in close agreement with results from computer simulations and the reference-hypernetted-chain equation. We also discuss thermodynamic consistency tests on the new equation and comparisons with the integral equations of Rogers and Young and of Zerah and Hansen. The present equation has no parameter to adjust. This unique feature offers a significant advantage as it eliminates a time-consuming search to optimize such parameters appearing in other theories. It permits practical applications needing complex intermolecular potentials and for multicomponent systems.
Algorithms For Integrating Nonlinear Differential Equations
Freed, A. D.; Walker, K. P.
1994-01-01
Improved algorithms developed for use in numerical integration of systems of nonhomogenous, nonlinear, first-order, ordinary differential equations. In comparison with integration algorithms, these algorithms offer greater stability and accuracy. Several asymptotically correct, thereby enabling retention of stability and accuracy when large increments of independent variable used. Accuracies attainable demonstrated by applying them to systems of nonlinear, first-order, differential equations that arise in study of viscoplastic behavior, spread of acquired immune-deficiency syndrome (AIDS) virus and predator/prey populations.
A Generic Length-scale Equation For Second-order Turbulence Models of Oceanic Boundary Layers
Umlauf, L.; Burchard, H.
A generic transport equation for a generalized length-scale in second-order turbulence closure models for geophysical boundary layers is suggested. This variable consists of the products of powers of the turbulent kinetic energy, k, and the integral length-scale, l. The new approach generalizes traditional second-order models used in geophysical boundary layer modelling, e.g. the Mellor-Yamada model and the k- model, which, however, can be recovered as special cases. It is demonstrated how this new model can be calibrated with measurements in some typical geophysical boundary layer flows. As an example, the generic model is applied to the uppermost oceanic boundary layer directly influenced by the effects of breaking surface waves. Recent measurements show that in this layer the classical law of the wall is invalid, since there turbulence is dominated by turbulent transport of TKE from above, and not by shear-production. A widely accepted approach to describe the wave-affected layer with a one-equation turbulence model was suggested by Craig and Banner (1994). Here, some deficien- cies of their solutions are pointed out and a generalization of their ideas for the case of two-equation models is suggested. Direct comparison with very recently obtained measurements of the dissipation rate, , in the wave-affected boundary layer with com- puted results clearly demonstrate that only the generic two-equation model yields cor- rect predictions for the profiles of and the turbulent length scale, l. Also, the pre- dicted velocity profiles in the wave-affected layer, important e.g. for the interpretation of surface drifter experiments, are reproduced correctly only by the generic model. Implementation and computational costs of the generic model are comparable with traditonal two-equation models.
Institute of Scientific and Technical Information of China (English)
Gui-Qiang Chen; Dan Osborne; Zhongmin Qian
2009-01-01
We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in RN with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-fiat boundary. We observe that, under the nonhomogeneons boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition (without the incompressibility condition), which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in RN(n≥3) with nonhomogeneous vorticity boundary condition converge in L2 to the corresponding Euler equations satisfying the kinematic condition.
PREFACE: Symmetries and Integrability of Difference Equations
Doliwa, Adam; Korhonen, Risto; Lafortune, Stéphane
2007-10-01
The notion of integrability was first introduced in the 19th century in the context of classical mechanics with the definition of Liouville integrability for Hamiltonian flows. Since then, several notions of integrability have been introduced for partial and ordinary differential equations. Closely related to integrability theory is the symmetry analysis of nonlinear evolution equations. Symmetry analysis takes advantage of the Lie group structure of a given equation to study its properties. Together, integrability theory and symmetry analysis provide the main method by which nonlinear evolution equations can be solved explicitly. Difference equations (DE), like differential equations, are important in numerous fields of science and have a wide variety of applications in such areas as mathematical physics, computer visualization, numerical analysis, mathematical biology, economics, combinatorics, and quantum field theory. It is thus crucial to develop tools to study and solve DEs. While the theory of symmetry and integrability for differential equations is now largely well-established, this is not yet the case for discrete equations. Although over recent years there has been significant progress in the development of a complete analytic theory of difference equations, further tools are still needed to fully understand, for instance, the symmetries, asymptotics and the singularity structure of difference equations. The series of SIDE meetings on Symmetries and Integrability of Difference Equations started in 1994. Its goal is to provide a platform for an international and interdisciplinary communication for researchers working in areas associated with integrable discrete systems, such as classical and quantum physics, computer science and numerical analysis, mathematical biology and economics, discrete geometry and combinatorics, theory of special functions, etc. The previous SIDE meetings took place in Estérel near Montréal, Canada (1994), at the University of
The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential
Liu, Shuangqian; Yang, Xiongfeng
2016-08-01
Boundary effects are central to the dynamics of the dilute particles governed by the Boltzmann equation. In this paper, we study both the diffuse reflection and the specular reflection boundary value problems for the Boltzmann equation with a soft potential, in which the collision kernel is ruled by the inverse power law. For the diffuse reflection boundary condition, based on an L 2 argument and its interplay with intricate {L^∞} analysis for the linearized Boltzmann equation, we first establish the global existence and then obtain the exponential decay in {L^∞} space for the nonlinear Boltzmann equation in general classes of bounded domain. It turns out that the zero lower bound of the collision frequency and the singularity of the collision kernel lead to some new difficulties for achieving the a priori {L^∞} estimates and time decay rates of the solution. In the course of the proof, we capture some new properties of the probability integrals along the stochastic cycles and improve the {L^2-L^∞} theory to give a more direct approach to overcome those difficulties. As to the specular reflection condition, our key contribution is to develop a new time-velocity weighted {L^∞} theory so that we could deal with the greater difficulties stemming from the complicated velocity relations among the specular cycles and the zero lower bound of the collision frequency. From this new point, we are also able to prove that the solutions of the linearized Boltzmann equation tend to equilibrium exponentially in {L^∞} space with the aid of the L 2 theory and a bootstrap argument. These methods, in the latter case, can be applied to the Boltzmann equation with soft potential for all other types of boundary condition.
Boundary integral method applied in chaotic quantum billiards
Li, B; Li, Baowen; Robnik, Marko
1995-01-01
The boundary integral method (BIM) is a formulation of Helmholtz equation in the form of an integral equation suitable for numerical discretization to solve the quantum billiard. This paper is an extensive numerical survey of BIM in a variety of quantum billiards, integrable (circle, rectangle), KAM systems (Robnik billiard) and fully chaotic (ergodic, such as stadium, Sinai billiard and cardioid billiard). On the theoretical side we point out some serious flaws in the derivation of BIM in the literature and show how the final formula (which nevertheless was correct) should be derived in a sound way and we also argue that a simple minded application of BIM in nonconvex geometries presents serious difficulties or even fails. On the numerical side we have analyzed the scaling of the averaged absolute value of the systematic error \\Delta E of the eigenenergy in units of mean level spacing with the density of discretization (b = number of numerical nodes on the boundary within one de Broglie wavelength), and we f...
Path Integral Methods for Stochastic Differential Equations
Chow, Carson C.; Buice, Michael A.
2015-01-01
Stochastic differential equations (SDEs) have multiple applications in mathematical neuroscience and are notoriously difficult. Here, we give a self-contained pedagogical review of perturbative field theoretic and path integral methods to calculate moments of the probability density function of SDEs. The methods can be extended to high dimensional systems such as networks of coupled neurons and even deterministic systems with quenched disorder.
Nonparametric (smoothed) likelihood and integral equations
Groeneboom, Piet
2012-01-01
We show that there is an intimate connection between the theory of nonparametric (smoothed) maximum likelihood estimators for certain inverse problems and integral equations. This is illustrated by estimators for interval censoring and deconvolution problems. We also discuss the asymptotic efficiency of the MLE for smooth functionals in these models.
Three New Integrable Hierarchies of Equations
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
A general Lie algebra Vs and the corresponding loop algebra ～Vs are constructed, from which the linear isospectral Lax pairs are established, whose compatibility presents the zero curvature equation. As its application, a new Lax integrable hierarchy containing two parameters is worked out. It is not Liouville-integrable, however, its two reduced systems are Liouville-integrable, whose Hamiltonian structures are derived by making use of the quadratic-form identity and the γ formula (i.e. the computational formula on the constant γ appeared in the trace identity and the quadratic-form identity).
THREE POINT BOUNDARY VALUE PROBLEMS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
Mujeeb ur Rehman; Rahmat Ali Khan; Naseer Ahmad Asif
2011-01-01
In this paper,we study existence and uniqueness of solutions to nonlinear three point boundary value problems for fractional differential equation of the type cDδ0+u(t) =f(t,u(t),cDσ0+u(t)),t ∈[0,T],u(0) =αu(η),u(T) =βu(η),where1 ＜δ＜2,0＜σ＜ 1,α,β∈R,η∈(0,T),αη(1-β)+(1-α)(T-βη) ≠0 and cDoδ+,cDσ0+ are the Caputo fractional derivatives.We use Schauder fixed point theorem and contraction mapping principle to obtain existence and uniqueness results.Examples are also included to show the applicability of our results.
Lectures on differential equations for Feynman integrals
International Nuclear Information System (INIS)
Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations (DE). These lectures give a review of these developments, while not assuming any prior knowledge of the subject. After an introduction to DE for Feynman integrals, we point out how they can be simplified using algorithms available in the mathematical literature. We discuss how this is related to a recent conjecture for a canonical form of the equations. We also discuss a complementary approach that is based on properties of the space–time loop integrands, and explain how the ideas of leading singularities and d-log representations can be used to find an optimal basis for the DE. Finally, as an application of these ideas we show how single-scale integrals can be bootstrapped using the Drinfeld associator of a DE. (topical review)
Directory of Open Access Journals (Sweden)
Obidjon Kh. Abdullaev
2016-06-01
Full Text Available In this work, we study the existence and uniqueness of solutions to non-local boundary value problems with integral gluing condition. Mixed type equations (parabolic-hyperbolic involving the Caputo fractional derivative have loaded parts in Riemann-Liouville integrals. Thus we use the method of integral energy to prove uniqueness, and the method of integral equations to prove existence.
Transparent boundary conditions for the wave equation in one dimension and for a Dirac-like equation
Directory of Open Access Journals (Sweden)
M. Pilar Velasco
2015-11-01
Full Text Available We present a method to achieve transparent boundary conditions for the one-dimensional wave equation, and show its numerical implementation using a finite-difference method. We also present an alternative method for building the same transparent boundary conditions using a Dirac-like equation and a Spinor-like formalism. Finally, we extend our method to the three-dimensional wave equation with radial symmetry.
Nonlocal singular problem with integral condition for a second-order parabolic equation
Directory of Open Access Journals (Sweden)
Ahmed Lakhdar Marhoune
2015-03-01
Full Text Available We prove the existence and uniqueness of a strong solution for a parabolic singular equation in which we combine Dirichlet with integral boundary conditions given only on parts of the boundary. The proof uses a priori estimate and the density of the range of the operator generated by the problem considered.
Accurate computation of Galerkin double surface integrals in the 3-D boundary element method
Adelman, Ross; Duraiswami, Ramani
2015-01-01
Many boundary element integral equation kernels are based on the Green's functions of the Laplace and Helmholtz equations in three dimensions. These include, for example, the Laplace, Helmholtz, elasticity, Stokes, and Maxwell's equations. Integral equation formulations lead to more compact, but dense linear systems. These dense systems are often solved iteratively via Krylov subspace methods, which may be accelerated via the fast multipole method. There are advantages to Galerkin formulations for such integral equations, as they treat problems associated with kernel singularity, and lead to symmetric and better conditioned matrices. However, the Galerkin method requires each entry in the system matrix to be created via the computation of a double surface integral over one or more pairs of triangles. There are a number of semi-analytical methods to treat these integrals, which all have some issues, and are discussed in this paper. We present novel methods to compute all the integrals that arise in Galerkin fo...
Boundary conditions for the Pauli equation: application to photodetachment of Cs-.
Bahrim, C; Fabrikant, I I; Thumm, U
2001-09-17
We formulate the boundary conditions near the atomic nucleus for solving the Pauli equation, based on the analytic solution of the Dirac equation for a Coulomb potential. We then integrate the Pauli equation using an effective potential that is adjusted to reproduce Dirac R-matrix scattering phase shifts, and find the (3)P(o)(1) resonance contribution to the photodetachment cross section of Cs-. Our photodetachment cross sections agree with recent experiments by Scheer et al. [Phys. Rev. Lett. 80, 684 (1998)] after tuning the resonance position by 2.4 meV. We also provide angle-differential photodetachment cross sections and the corresponding asymmetry parameter beta near the Cs(6s) threshold. PMID:11580504
A boundary integral formalism for stochastic ray tracing in billiards
Chappell, David J
2014-01-01
Determining the flow of rays or particles driven by a force or velocity field is fundamental to modelling many physical processes, including weather forecasting and the simulation of molecular dynamics. High frequency wave energy distributions can also be approximated using flow or transport equations. Applications arise in underwater and room acoustics, vibro-acoustics, seismology, electromagnetics, quantum mechanics and in producing computer generated imagery. In many practical applications, the driving field is not known exactly and the dynamics are determined only up to a degree of uncertainty. This paper presents a boundary integral framework for propagating flows including uncertainties, which is shown to systematically interpolate between a deterministic and a completely random description of the trajectory propagation. A simple but efficient discretisation approach is applied to model uncertain billiard dynamics in an integrable rectangular domain.
A boundary integral formalism for stochastic ray tracing in billiards
Energy Technology Data Exchange (ETDEWEB)
Chappell, David J. [School of Science and Technology, Nottingham Trent University, Clifton Campus, Nottingham NG11 8NS (United Kingdom); Tanner, Gregor [School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD (United Kingdom)
2014-12-15
Determining the flow of rays or non-interacting particles driven by a force or velocity field is fundamental to modelling many physical processes. These include particle flows arising in fluid mechanics and ray flows arising in the geometrical optics limit of linear wave equations. In many practical applications, the driving field is not known exactly and the dynamics are determined only up to a degree of uncertainty. This paper presents a boundary integral framework for propagating flows including uncertainties, which is shown to systematically interpolate between a deterministic and a completely random description of the trajectory propagation. A simple but efficient discretisation approach is applied to model uncertain billiard dynamics in an integrable rectangular domain.
Noncontinuous data boundary value problems for Schr(o)dinger equation in Lipschitz domains
Institute of Scientific and Technical Information of China (English)
TAO Xiangxing
2006-01-01
The noncontinuous data boundary value problems for Schr(o)dinger equations in Lipschitz domains and its progress are pointed out in this paper.Particularly,the Lp boundary value problems with p＞1,and Hp boundary value problems with p＜1 have been studied.Some open problems about the Besov-Sobolev and Orlicz boundary value problems are given.
Spatial Interpolation Methods for Integrating Newton's Equation
Gueron, Shay; Shalloway, David
1996-11-01
Numerical integration of Newton's equation in multiple dimensions plays an important role in many fields such as biochemistry and astrophysics. Currently, some of the most important practical questions in these areas cannot be addressed because the large dimensionality of the variable space and complexity of the required force evaluations precludes integration over sufficiently large time intervals. Improving the efficiency of algorithms for this purpose is therefore of great importance. Standard numerical integration schemes (e.g., leap-frog and Runge-Kutta) ignore the special structure of Newton's equation that, for conservative systems, constrains the force to be the gradient of a scalar potential. We propose a new class of "spatial interpolation" (SI) integrators that exploit this property by interpolating the force in space rather than (as with standard methods) in time. Since the force is usually a smoother function of space than of time, this can improve algorithmic efficiency and accuracy. In particular, an SI integrator solves the one- and two-dimensional harmonic oscillators exactly with one force evaluation per step. A simple type of time-reversible SI algorithm is described and tested. Significantly improved performance is achieved on one- and multi-dimensional benchmark problems.
Institute of Scientific and Technical Information of China (English)
Jia-qi Mo; Wan-tao Lin
2006-01-01
A class of nonlinear singularly perturbed problems for reaction diffusion equations with boundary perturbation are considered. Under suitable conditions, the asymptotic behavior of solution for the initial boundary value problems is studied using the theory of differential inequalities.
Institute of Scientific and Technical Information of China (English)
OuyangCheng; MoJiaqi
2005-01-01
The nonlinear singularly perturbed problems for reaction diffusion equations with boundary perturbation are considered. Under suitable conditions, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems is studied.
LIMIT BEHAVIOUR OF SOLUTIONS TO EQUIVALUED SURFACE BOUNDARY VALUE PROBLEM FOR PARABOLIC EQUATIONS
Institute of Scientific and Technical Information of China (English)
LI Fengquan
2002-01-01
In this paper, we discuss the limit behaviour of solutions to equivalued surface boundary value problem for parabolic equations when the equivalued surface boundary shrinks to a point and the space dimension of the domain is two or more.
A Priori Estimates for Solutions of Boundary Value Problems for Fractional-Order Equations
Alikhanov, A A
2011-01-01
We consider boundary value problems of the first and third kind for the diffusionwave equation. By using the method of energy inequalities, we find a priori estimates for the solutions of these boundary value problems.
An initial-boundary value problem for three-dimensional Zakharov-Kuznetsov equation
Faminskii, Andrei V.
2016-02-01
An initial-boundary value problem with homogeneous Dirichlet boundary conditions for three-dimensional Zakharov-Kuznetsov equation is considered. Results on global existence, uniqueness and large-time decay of weak solutions in certain weighted spaces are established.
SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR ELLIPTIC EQUATION WITH A CURVE OF TURNING POINT
Institute of Scientific and Technical Information of China (English)
MoJiaqi
2002-01-01
The singularly perturbed elliptic equation boundary value problem with a curve of turning point is considered. Using the method of multiple scales and the comparison theorem,the asymptotic behavior of solution for the boundary value problem is studied.
SOLVING INTEGRAL EQUATIONS WITH LOGARITHMIC KERNEL BY USING PERIODIC QUASI-WAVELET
Institute of Scientific and Technical Information of China (English)
Han-lin Chen; Si-long Peng
2000-01-01
In solving integral equations with logarithmic kernel which arises from the boundary integral equation reformulation of some boundary value problems for the two dimensional Helmholtz equation, we combine the Galerkin method with Beylkin's ([2]) approach, series of dense and nonsymmetric matrices may appear if we use traditional method. By appealing the so-called periodic quasi-wavelet (PQW in abbr.) ([5]), some of these matrices become diagonal, therefore we can find a algorithm with only O(K(m)2) arithmetic operations where m is the highest level. The Galerkin approximation has a polynomial rate of convergence.
Recursion operators, conservation laws, and integrability conditions for difference equations
Mikhailov, A. V.; Wang, Jing Ping; Xenitidis, P.
2011-04-01
We attempt to propose an algebraic approach to the theory of integrable difference equations. We define the concept of a recursion operator for difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. As in the case of partial differential equations, these canonical densities can serve as integrability conditions for difference equations. We obtain the recursion operators for the Viallet equation and all the Adler-Bobenko-Suris equations.
Monotone iterative technique for fractional differential equations with periodic boundary conditions
Directory of Open Access Journals (Sweden)
J. D. Ramírez
2009-01-01
Full Text Available In this paper we develop Monotone Method using upper and lower solutions for fractional differential equations with periodic boundary conditions. Initially we develop a comparison result and prove that the solution of the linear fractional differential equation with periodic boundary condition exists and is unique. Using this we develop iterates which converge uniformly monotonically to minimal and maximal solutions of the nonlinear fractional differential equations with periodic boundary conditions in the weighted norm.
Monotone iterative technique for fractional differential equations with periodic boundary conditions
J. D. Ramírez; A. S. Vatsala
2009-01-01
In this paper we develop Monotone Method using upper and lower solutions for fractional differential equations with periodic boundary conditions. Initially we develop a comparison result and prove that the solution of the linear fractional differential equation with periodic boundary condition exists and is unique. Using this we develop iterates which converge uniformly monotonically to minimal and maximal solutions of the nonlinear fractional differential equations with periodic boundary con...
Quano, Y H
2001-01-01
The $A^{(1)}_{n-1}$-face model with boundary reflection is considered on the basis of the boundary CTM bootstrap. We construct the fused boundary Boltzmann weights to determine the normalization factor. We derive difference equations of the quantum Knizhnik-Zamolodchikov type for correlation functions of the present model. The simplest difference equations are solved the in the case of the free boundary condition.
Energy Technology Data Exchange (ETDEWEB)
Pingenot, J; Jandhyala, V
2007-03-01
This report summarizes the work performed for Lawrence Livermore National Laboratory (LLNL) at the University of Washington between September 2004 and May 2006. This project studied fast solvers and stability for time domain integral equations (TDIE), especially as applied to radiating boundary for a massively parallel FEM solver.
Institute of Scientific and Technical Information of China (English)
XU Xi-Xiang; ZHANG Yu-Feng
2004-01-01
A discrete matrix spectral problem and the associated hierarchy of Lax integrable lattice equations are presented, and it is shown that the resulting Lax integrable lattice equations are all Liouville integrable discrete Hamiltonian systems. A new integrable symplectic map is given by binary Bargmann .constraint of the resulting hierarchy.Finally, an infinite set of conservation laws is given for the resulting hierarchy.
Integrated watershed planning across jurisdictional boundaries
Watts, A. W.; Roseen, R.; Stacey, P.; Bourdeau, R.
2014-12-01
We will present the foundation for an Coastal Watershed Integrated Plan for three communities in southern New Hampshire. Small communities are often challenged by complex regulatory requirements and limited resources, but are wary of perceived risks in engaging in collaborative projects with other communities. Potential concerns include loss of control, lack of resources to engage in collaboration, technical complexity, and unclear benefits. This project explores a multi-town subwatershed application of integrated planning across jurisdictional boundaries that addresses some of today's highest priority water quality issues: wastewater treatment plant upgrades for nutrient removal; green infrastructure stormwater management for developing and re-developing areas; and regional monitoring of ecosystem indicators in support of adaptive management to achieve nutrient reduction and other water quality goals in local and downstream waters. The project outcome is a collaboratively-developed inter-municipal integrated plan, and a monitoring framework to support cross jurisdictional planning and assess attainment of water quality management goals. This research project has several primary components: 1) assessment of initial conditions, including both the pollutant load inputs and the political, economic and regulatory status within each community, 2) a pollutant load model for point and non-point sources, 3) multi-criteria evaluation of load reduction alternatives 4) a watershed management plan optimized for each community, and for Subwatersheds combining multiple communities. The final plan will quantify the financial and other benefits/drawbacks to each community for both inter municipal and individual pollution control approaches. We will discuss both the technical and collaborative aspects of the work, with lessons learned regarding science to action, incorporation of social, economic and water quality assessment parameters, and stakeholder/researcher interaction.
Institute of Scientific and Technical Information of China (English)
A.S. BERDYSHEV; A. CABADA; B.Kh. TURMETOV
2014-01-01
This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.
Fokas, A. S.; De Lillo, S.
2014-03-01
So-called inverse scattering provides a powerful method for analyzing the initial value problem for a large class of nonlinear evolution partial differential equations which are called integrable. In the late 1990s, the first author, motivated by inverse scattering, introduced a new method for analyzing boundary value problems. This method provides a unified treatment for linear, linearizable and integrable nonlinear partial differential equations. Here, this method, which is often referred to as the unified transform, is illustrated for the following concrete cases: the heat equation on the half-line; the nonlinear Schrödinger equation on the half-line; Burger's equation on the half-line; and Burger's equation on a moving boundary.
Bounded solutions for fuzzy differential and integral equations
Energy Technology Data Exchange (ETDEWEB)
Nieto, Juan J. [Departamento de Analisis Matematico Facultad de Matematicas Universidad de Santiago de Compostela, 15782 (Spain)] e-mail: amnieto@usc.es; Rodriguez-Lopez, Rosana [Departamento de Analisis Matematico Facultad de Matematicas Universidad de Santiago de Compostela, 15782 (Spain)] e-mail: amrosana@usc.es
2006-03-01
We find sufficient conditions for the boundness of every solution of first-order fuzzy differential equations as well as certain fuzzy integral equations. Our results are based on several theorems concerning crisp differential and integral inequalities.
Path Integral and Solutions of the Constraint Equations: The Case of Reducible Gauge Theories
Ferraro, R.; Henneaux, M.; Puchin, M.
1994-01-01
It is shown that the BRST path integral for reducible gauge theories, with appropriate boundary conditions on the ghosts, is a solution of the constraint equations. This is done by relating the BRST path integral to the kernel of the evolution operator projected on the physical subspace.
Singular Integral Equations with Cosecant Kernel in Solutions with Singularities of High Order
Institute of Scientific and Technical Information of China (English)
HAN Hui-li; DU Jin-yuan
2005-01-01
We have discussed and solved the boundary value problem with period 2aπ and the singular integral equation with kernel csc t-t0/a in solution having singularities of high order, where the smooth contour of integration is in the strip 0＜Rez＜aπ.
Energy Technology Data Exchange (ETDEWEB)
Rosnitskiy, P., E-mail: pavrosni@yandex.ru; Yuldashev, P., E-mail: petr@acs366.phys.msu.ru; Khokhlova, V., E-mail: vera@acs366.phys.msu.ru [Physics Faculty, Moscow State University, Leninskie Gory, 119991 Moscow (Russian Federation)
2015-10-28
An equivalent source model was proposed as a boundary condition to the nonlinear parabolic Khokhlov-Zabolotskaya (KZ) equation to simulate high intensity focused ultrasound (HIFU) fields generated by medical ultrasound transducers with the shape of a spherical shell. The boundary condition was set in the initial plane; the aperture, the focal distance, and the initial pressure of the source were chosen based on the best match of the axial pressure amplitude and phase distributions in the Rayleigh integral analytic solution for a spherical transducer and the linear parabolic approximation solution for the equivalent source. Analytic expressions for the equivalent source parameters were derived. It was shown that the proposed approach allowed us to transfer the boundary condition from the spherical surface to the plane and to achieve a very good match between the linear field solutions of the parabolic and full diffraction models even for highly focused sources with F-number less than unity. The proposed method can be further used to expand the capabilities of the KZ nonlinear parabolic equation for efficient modeling of HIFU fields generated by strongly focused sources.
Evolution equations: Frobenius integrability, conservation laws and travelling waves
International Nuclear Information System (INIS)
We give new results concerning the Frobenius integrability and solution of evolution equations admitting travelling wave solutions. In particular, we give a powerful result which explains the extraordinary integrability of some of these equations. We also discuss ‘local’ conservations laws for evolution equations in general and demonstrate all the results for the Korteweg–de Vries equation. (paper)
Funnel Equations and Multivalued Integration Problems for Control Synthesis
Kurzhanski, A.B.; Nikonov, O.I.
1989-01-01
This paper indicates a sequence of evolutionary "funnel equations" with set-valued solutions which are crucial for the construction of respective feed-back control strategies along the schemes introduced by N.N. Krasovskii. The integration of these funnel equations leads to a sequence of multivalued integrals that generalize some of those introduced earlier (Auman's integral, the Convolution integral, Pontryagin's Alternated integral).
Inverse Lax-Wendroff procedure for numerical boundary conditions of convection-diffusion equations
Lu, Jianfang; Fang, Jinwei; Tan, Sirui; Shu, Chi-Wang; Zhang, Mengping
2016-07-01
We consider numerical boundary conditions for high order finite difference schemes for solving convection-diffusion equations on arbitrary geometry. The two main difficulties for numerical boundary conditions in such situations are: (1) the wide stencil of the high order finite difference operator requires special treatment for a few ghost points near the boundary; (2) the physical boundary may not coincide with grid points in a Cartesian mesh and may intersect with the mesh in an arbitrary fashion. For purely convection equations, the so-called inverse Lax-Wendroff procedure [28], in which we convert the normal derivatives into the time derivatives and tangential derivatives along the physical boundary by using the equations, has been quite successful. In this paper, we extend this methodology to convection-diffusion equations. It turns out that this extension is non-trivial, because totally different boundary treatments are needed for the diffusion-dominated and the convection-dominated regimes. We design a careful combination of the boundary treatments for the two regimes and obtain a stable and accurate boundary condition for general convection-diffusion equations. We provide extensive numerical tests for one- and two-dimensional problems involving both scalar equations and systems, including the compressible Navier-Stokes equations, to demonstrate the good performance of our numerical boundary conditions.
Shu, Jian-Jun
2014-01-01
A very simple and accurate numerical method which is applicable to systems of differentio-integral equations with quite general boundary conditions has been devised. Although the basic idea of this method stems from the Keller Box method, it solves the problem of systems of differential equations involving integral operators not previously considered by the Keller Box method. Two main preparatory stages are required: (i) a merging procedure for differential equations and conditions without integral operators and; (ii) a reduction procedure for differential equations and conditions with integral operators. The differencing processes are effectively simplified by means of the unit-step function. The nonlinear difference equations are solved by Newton method using an efficient block arrow-like matrix factorization technique. As an example of the application of this method, the systems of equations for combined gravity body force and forced convection in laminar film condensation can be solved for prescribed valu...
Explicit expressions for three-dimensional boundary integrals in linear elasticity
Energy Technology Data Exchange (ETDEWEB)
Nintcheu Fata, Sylvain [ORNL
2011-01-01
On employing isoparametric, piecewise linear shape functions over a flat triangle, exact formulae are derived for all surface potentials involved in the numerical treatment of three-dimensional singular and hyper-singular boundary integral equations in linear elasticity. These formulae are valid for an arbitrary source point in space and are represented as analytical expressions along the edges of the integration triangle. They can be employed to solve integral equations defined on triangulated surfaces via a collocation method or may be utilized as analytical expressions for the inner integrals in a Galerkin technique. A numerical example involving a unit triangle and a source point located at various distances above it, as well as sample problems solved by a collocation boundary element method for the Lame equation are included to validate the proposed formulae.
Initial-boundary value problems for a class of nonlinear thermoelastic plate equations
Institute of Scientific and Technical Information of China (English)
Zhang Jian-Wen; Rong Xiao-Liang; Wu Run-Heng
2009-01-01
This paper studies initial-boundary value problems for a class of nonlinear thermoelastic plate equations. Under some certain initial data and boundary conditions,it obtains an existence and uniqueness theorem of global weak solutions of the nonlinear thermoelstic plate equations,by means of the Galerkin method. Moreover,it also proves the existence of strong and classical solutions.
REGULARITY THEORY FOR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS WITH NEUMANN BOUNDARY CONDITIONS
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The objective of this paper is to consider the theory of regularity of systems of partial differential equations with Neumann boundary conditions. It complements previous works of the authors for the Dirichlet case. This type of problem is motivated by stochastic differential games. The Neumann case corresponds to stochastic differential equations with reflection on boundary of the domain.
Well-posed initial-boundary value problems for the Zakharov-Kuznetsov equation
Directory of Open Access Journals (Sweden)
Andrei V. Faminskii
2008-09-01
Full Text Available This paper deals with non-homogeneous initial-boundary value problems for the Zakharov-Kuznetsov equation, which is one of the variants of multidimensional generalizations of the Korteweg-de Vries equation. Results on local and global well-posedness are established in a scale of Sobolev-type spaces under natural assumptions on initial and boundary data.
A Cartesian Embedded Boundary Method for the Compressible Navier-Stokes Equations
Energy Technology Data Exchange (ETDEWEB)
Kupiainen, M; Sjogreen, B
2008-03-21
We here generalize the embedded boundary method that was developed for boundary discretizations of the wave equation in second order formulation in [6] and for the Euler equations of compressible fluid flow in [11], to the compressible Navier-Stokes equations. We describe the method and we implement it on a parallel computer. The implementation is tested for accuracy and correctness. The ability of the embedded boundary technique to resolve boundary layers is investigated by computing skin-friction profiles along the surfaces of the embedded objects. The accuracy is assessed by comparing the computed skin-friction profiles with those obtained by a body fitted discretization.
Adomian Method for Solving Fuzzy Fredholm-Volterra Integral Equations
M. Barkhordari Ahmadi; M. MOSLEH; M. Otadi
2013-01-01
In this paper, Adomian method has been applied to approximate the solution of fuzzy volterra-fredholm integral equation. That, by using parametric form of fuzzy numbers, a fuzzy volterra-fredholm integral equation has been converted to a system of volterra-fredholm integral equation in crisp case. Finally, the method is explained with illustrative examples.
Adomian Method for Solving Fuzzy Fredholm-Volterra Integral Equations
Directory of Open Access Journals (Sweden)
M. Barkhordari Ahmadi
2013-09-01
Full Text Available In this paper, Adomian method has been applied to approximate the solution of fuzzy volterra-fredholm integral equation. That, by using parametric form of fuzzy numbers, a fuzzy volterra-fredholm integral equation has been converted to a system of volterra-fredholm integral equation in crisp case. Finally, the method is explained with illustrative examples.
On a problem for wave equation with data on the whole boundary
Yessirkegenov, Nurgissa
2016-08-01
In this paper we propose a new formulation of boundary value problem for a one-dimensional wave equation in a rectangular domain in which boundary conditions are given on the whole boundary. We prove the well-posedness of boundary value problem in the classical and generalized senses. In order to substantiate the well-posedness of this problem it is necessary to have an effective representation of the general solution of the problem. In this direction we obtain a convenient representation of the general solution for the wave equation in a rectangular domain based on d'Alembert classical formula. The constructed general solution automatically satisfies the boundary conditions by a spatial variable. Further, by setting different boundary conditions according to temporary variable, we get some functional or functional-differential equations. Thus, the proof of the well-posedness of the formulated problem is reduced to question of the existence and uniqueness of solutions of the corresponding functional equations.
Atluri, Satya N.; Shen, Shengping
2002-01-01
In this paper, a very simple method is used to derive the weakly singular traction boundary integral equation based on the integral relationships for displacement gradients. The concept of the MLPG method is employed to solve the integral equations, especially those arising in solid mechanics. A moving Least Squares (MLS) interpolation is selected to approximate the trial functions in this paper. Five boundary integral Solution methods are introduced: direct solution method; displacement boundary-value problem; traction boundary-value problem; mixed boundary-value problem; and boundary variational principle. Based on the local weak form of the BIE, four different nodal-based local test functions are selected, leading to four different MLPG methods for each BIE solution method. These methods combine the advantages of the MLPG method and the boundary element method.
Boundary Conditions for 2D Boussinesq-type Wave-Current Interaction Equations
Directory of Open Access Journals (Sweden)
Mera M.
2011-01-01
Full Text Available This research focuses on the development of a set of two-dimensional boundary conditions for specific governing equations. The governing equations are existing Boussinesqtype equations which is capable of simulating wave-current interaction. The present boundary conditions consist of for waves only case and for currents only case. To simulate wave-current interaction, the two kinds of the present boundary conditions are then combined. A numerical model based on both the existing governing equations and the present boundary conditions is applied to simulation of currents only and of wave-current interaction propagating over a basin with a submerged shoal. The results of the numerical model show that the present boundary conditions go well with the existing Boussinesq-type wave-current interaction equations.
Partial differential equations IX elliptic boundary value problems
Egorov, Yu; Shubin, M
1997-01-01
This EMS volume gives an overview of the modern theory of elliptic boundary value problems. The contribution by M.S. Agranovich is devoted to differential elliptic boundary problems, mainly in smooth bounded domains, and their spectral properties. This article continues his contribution to EMS 63. The contribution by A. Brenner and E. Shargorodsky concerns the theory of boundary value problems for elliptic pseudodifferential operators. Problems both with and without the transmission property, as well as parameter-dependent problems are considered. The article by B. Plamenevskij deals with general differential elliptic boundary value problems in domains with singularities.
Boundary conditions for the Einstein-Christoffel formulation of Einstein's equations
Arnold, Douglas N.; Nicolae Tarfulea
2007-01-01
Specifying boundary conditions continues to be a challenge in numerical relativity in order to obtain a long time convergent numerical simulation of Einstein's equations in domains with artificial boundaries. In this paper, we address this problem for the Einstein--Christoffel (EC) symmetric hyperbolic formulation of Einstein's equations linearized around flat spacetime. First, we prescribe simple boundary conditions that make the problem well posed and preserve the constraints. Next, we indi...
A Kind of Discrete Non-Reflecting Boundary Conditions for Varieties of Wave Equations
Institute of Scientific and Technical Information of China (English)
Xiu-min Shao; Zhi-ling Lan
2002-01-01
In this paper, a new kind of discrete non-reflecting boundary conditions is developed. It can be used for a variety of wave equations such as the acoustic wave equation, the isotropic and anisotropic elastic wave equations and the equations for wave propagation in multi-phase media and so on. In this kind of boundary conditions, the composition of all artificial reflected waves, but not the individual reflected ones, is considered and eliminated. Thus, it has a uniform formula for different wave equations. The velocity CA of the composed reflected wave is determined in the way to make the reflection coefficients minimal, the value of which depends on equations. In this paper, the construction of the boundary conditions is illustrated and CA is found, numerical results are presented to illustrate the effectiveness of the boundary conditions.
BOUNDARY INTEGRAL FORMULA OF ELASTIC PROBLEMS IN CIRCLE PLANE
Institute of Scientific and Technical Information of China (English)
DONG Zheng-zhu; LI Shun-cai; YU De-hao
2005-01-01
By bianalytic functions, the boundary integral formula of the stress function for the elastic problem in a circle plane is developed. But this integral formula includes a strongly singular integral and can not be directly calculated. After the stress function is expounded to Fourier series, making use of some formulas in generalized functions to the convolutions, the boundary integral formula which does not include strongly singular integral is derived further. Then the stress function can be got simply by the integration of the values of the stress function and its derivative on the boundary. Some examples are given. It shows that the boundary integral formula of the stress function for the elastic problem is convenient.
Boundary singularities and characteristics of Hamilton–Jacobi equation
Melikyan, A.; Olsder, G.J.
2010-01-01
Boundary-value problems for first order PDEs are locally considered, when the classical sufficient condition for the solution existence does not hold, but a solution still exists, possibly defined on one or both sides of the boundary surface. We note three situations when such a surface (locally) ar
Ivanyshyn Yaman, Olha; Le Louër, Frédérique
2016-09-01
This paper deals with the material derivative analysis of the boundary integral operators arising from the scattering theory of time-harmonic electromagnetic waves and its application to inverse problems. We present new results using the Piola transform of the boundary parametrisation to transport the integral operators on a fixed reference boundary. The transported integral operators are infinitely differentiable with respect to the parametrisations and simplified expressions of the material derivatives are obtained. Using these results, we extend a nonlinear integral equations approach developed for solving acoustic inverse obstacle scattering problems to electromagnetism. The inverse problem is formulated as a pair of nonlinear and ill-posed integral equations for the unknown boundary representing the boundary condition and the measurements, for which the iteratively regularized Gauss-Newton method can be applied. The algorithm has the interesting feature that it avoids the numerous numerical solution of boundary value problems at each iteration step. Numerical experiments are presented in the special case of star-shaped obstacles.
Porosity of Free Boundaries in the Obstacle Problem for Quasilinear Elliptic Equations
Indian Academy of Sciences (India)
Jun Zheng; Zhihua Zhang; Peihao Zhao
2013-08-01
In this paper, we establish growth rate of solutions near free boundaries in the identical zero obstacle problem for quasilinear elliptic equations. As a result, we obtain porosity of free boundaries, which is naturally an extension of the previous works by Karp et al. (J. Diff. Equ. 164 (2000) 110–117) for -Laplacian equations, and by Zheng and Zhang (J. Shaanxi Normal Univ. 40(2) (2012) 11–13, 18) for -Laplacian type equations.
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.
International Nuclear Information System (INIS)
The 3D MHD free boundary equilibrium NEMEC code combines the fixed boundary VMEC code with the vacuum NESTOR code. In NESTOR a Neumann problem is solved for a single valued potential Φ by an integral equation technique satisfying the condition rvec BV· rvec n≡(rvec B0+∇Φ)· rvec n=0 on the plasma-vacuum boundary (rvec B0=field produced by external currents and the total toroidal plasma current It, rvec n=exterior normal to the boundary). The only quantity required for the free boundary equilibrium computation is the value of BV2/2 at the boundary. Here, an equivalent method for computing the vacuum field rvec BV is considered which is particularly suited for the case It≠0 and makes use of the fact that this field can be generated by the external current jext and by a surface current rvec j on the plasma vacuum boundary. The difficulty inherent to the Green's function method is the numerical integration of the singular Green's function. This is solved by regularizing the kernel: an analytically integrable function with the same singularity is subtracted and its integral is added to the integral equation
Boundary Layer Equations and Lie Group Analysis of a Sisko Fluid
Directory of Open Access Journals (Sweden)
Gözde Sarı
2012-01-01
Full Text Available Boundary layer equations are derived for the Sisko fluid. Using Lie group theory, a symmetry analysis of the equations is performed. A partial differential system is transferred to an ordinary differential system via symmetries. Resulting equations are numerically solved. Effects of non-Newtonian parameters on the solutions are discussed.
KdV equation under periodic boundary conditions and its perturbations
Huang, Guan; Kuksin, Sergei
2013-01-01
In this paper we discuss properties of the KdV equation under periodic boundary conditions, especially those which are important to study perturbations of the equation. Next we review what is known now about long-time behaviour of solutions for perturbed KdV equations.
The KdV equation under periodic boundary conditions and its perturbations
Guan, Huang; Kuksin, Sergei
2014-09-01
In this paper we discuss properties of the Korteweg-de Vries (KdV) equation under periodic boundary conditions, especially those which are important for studying perturbations of the equation. We then review what is known about the long-time behaviour of solutions for perturbed KdV equations.
Edwards, S.; Reuther, J.; Chattot, J. J.
The objective of this paper is to present a control theory approach for the design of airfoils in the presence of viscous compressible flows. A coupled system of the integral boundary layer and the Euler equations is solved to provide rapid flow simulations. An adjoint approach consistent with the complete coupled state equations is employed to obtain the sensitivities needed to drive a numerical optimization algorithm. Design to a target pressure distribution is demonstrated on an RAE 2822 airfoil at transonic speeds.
BOUNDARY VALUE PROBLEM TO DYNAMIC EQUATION ON TIME SCALE
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
In this paper we consider a nonlinear first-order boundary value problem on a time scale. The existence results of three positive solutions are obtained using fixed point theorems. Finally,examples are presented to illustrate the main results.
Integrability of two coupled Kadomtsev–Petviashvili equations
Indian Academy of Sciences (India)
Abdul-Majid Wazwaz
2011-08-01
The integrability of two coupled KP equations is studied. The simpliﬁed Hereman form of Hirota’s bilinear method is used to examine the integrability of each coupled equation. Multiplesoliton solutions and multiple singular soliton solutions are formally derived for each coupled KdV equation.
Tatsii, R. M.; Pazen, O. Yu.
2016-03-01
A constructive scheme for the construction of a solution of a mixed problem for the heat conduction equation with piecewise-continuous coefficients coordinate-dependent in the final interval is suggested and validated in the present work. The boundary conditions are assumed to be most general. The scheme is based on: the reduction method, the concept of quasi-derivatives, the currently accepted theory of the systems of linear differential equations, the Fourier method, and the modified method of eigenfunctions. The method based on this scheme should be related to direct exact methods of solving mixed problems that do not employ the procedures of constructing Green's functions or integral transformations. Here the theorem of eigenfunction expansion is adapted for the case of coefficients that have discontinuity points of the 1st kind. The results obtained can be used, for example, in investigating the process of heat transfer in a multilayer slab under conditions of ideal thermal contact between the layers. A particular case of piecewise-continuous coefficients is considered. A numerical example of calculation of a temperature field in a real four-layer building slab under boundary conditions of the 3rd kind (conditions of convective heat transfer) that model the phenomenon of fire near one of the external surfaces is given.
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
This paper deals with boundary value problems for linear uniformly elliptic systems. First the general linear uniformly elliptic system of the first order equations is reduced to complex form, and then the compound boundary value problem for the complex equations of the first order is discussed. The approximate solutions of the boundary value problem are found by the variation-difference method, and the error estimates for the approximate solutions are derived.Finally the approximate method of the oblique derivative problem for linear uniformly elliptic equations of the second or der is introduced.
Local absorbing boundary conditions for nonlinear wave equation on unbounded domain.
Li, Hongwei; Wu, Xiaonan; Zhang, Jiwei
2011-09-01
The numerical solution of the nonlinear wave equation on unbounded spatial domain is considered. The artificial boundary method is introduced to reduce the nonlinear problem on unbounded spatial domain to an initial boundary value problem on a bounded domain. Using the unified approach, which is based on the operator splitting method, we construct the efficient nonlinear local absorbing boundary conditions for the nonlinear wave equation, and give the stability analysis of the resulting boundary conditions. Finally, several numerical examples are given to demonstrate the effectiveness of our method.
Toward a theory of integrable hyperbolic equations of third order
International Nuclear Information System (INIS)
Examples are considered of integrable third order hyperbolic equations with two independent variables. In particular, an equation is found which admits as evolutionary symmetries the Krichever–Novikov equation and the modified Landau–Lifshitz system. The problem of the choice of dynamical variables for the hyperbolic equations is discussed. (paper)
Towards the theory of integrable hyperbolic equations of third order
Adler, V. E.; Shabat, A. B.
2012-01-01
The examples are considered of integrable hyperbolic equations of third order with two independent variables. In particular, an equation is found which admits as evolutionary symmetries the Krichever--Novikov equation and the modified Landau--Lifshitz system. The problem of choice of dynamical variables for the hyperbolic equations is discussed.
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.
Levi, Decio; Olver, Peter; Thomova, Zora; Winternitz, Pavel
2009-11-01
The concept of integrability was introduced in classical mechanics in the 19th century for finite dimensional continuous Hamiltonian systems. It was extended to certain classes of nonlinear differential equations in the second half of the 20th century with the discovery of the inverse scattering transform and the birth of soliton theory. Also at the end of the 19th century Lie group theory was invented as a powerful tool for obtaining exact analytical solutions of large classes of differential equations. Together, Lie group theory and integrability theory in its most general sense provide the main tools for solving nonlinear differential equations. Like differential equations, difference equations play an important role in physics and other sciences. They occur very naturally in the description of phenomena that are genuinely discrete. Indeed, they may actually be more fundamental than differential equations if space-time is actually discrete at very short distances. On the other hand, even when treating continuous phenomena described by differential equations it is very often necessary to resort to numerical methods. This involves a discretization of the differential equation, i.e. a replacement of the differential equation by a difference one. Given the well developed and understood techniques of symmetry and integrability for differential equations a natural question to ask is whether it is possible to develop similar techniques for difference equations. The aim is, on one hand, to obtain powerful methods for solving `integrable' difference equations and to establish practical integrability criteria, telling us when the methods are applicable. On the other hand, Lie group methods can be adapted to solve difference equations analytically. Finally, integrability and symmetry methods can be combined with numerical methods to obtain improved numerical solutions of differential equations. The origin of the SIDE meetings goes back to the early 1990s and the first
Difference equations and cluster algebras I: Poisson bracket for integrable difference equations
Inoue, Rei
2010-01-01
We introduce the cluster algebraic formulation of the integrable difference equations, the discrete Lotka-Volterra equation and the discrete Liouville equation, from the view point of the general T-system and Y-system. We also study the Poisson structure for the cluster algebra, and give the associated Poisson bracket for the two difference equations.
Directory of Open Access Journals (Sweden)
Abdelfatah Bouziani
2003-01-01
Full Text Available This paper deals with weak solution in weighted Sobolev spaces, of three-point boundary value problems which combine Dirichlet and integral conditions, for linear and quasilinear parabolic equations in a domain with curved lateral boundaries. We, firstly, prove the existence, uniqueness, and continuous dependence of the solution for the linear equation. Next, analogous results are established for the quasilinear problem, using an iterative process based on results obtained for the linear problem.
Functional derivatives, Schr\\"{o}dinger equations, and Feynman integration
Dynin, Alexander
2007-01-01
Schr\\"{o}dinger equations in \\emph{functional derivatives} are solved via quantized Galerkin limit of antinormal functional Feynman integrals for Schr\\"{o}dinger equations in \\emph{partial derivatives
THE COLLOCATION METHODS FOR SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNELS
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
This paper applies the singular integral operators,singular quadrature operators and discretization matrices associated withsingular integral equations with Cauchy kernels, which are established in [1],to give a unified framework for various collocation methods of numericalsolutions of singular integral equations with Cauchy kernels. Under theframework, the coincidence of the direct quadrature method and the indirectquadrature method is very simple and obvious.
Solving Abel integral equations of first kind via fractional calculus
Directory of Open Access Journals (Sweden)
Salman Jahanshahi
2015-04-01
Full Text Available We give a new method for numerically solving Abel integral equations of first kind. An estimation for the error is obtained. The method is based on approximations of fractional integrals and Caputo derivatives. Using trapezoidal rule and Computer Algebra System Maple, the exact and approximation values of three Abel integral equations are found, illustrating the effectiveness of the proposed approach.
INITIAL BOUNDARY VALUE PROBLEM FOR A DAMPED NONLINEAR HYPERBOLIC EQUATION
Institute of Scientific and Technical Information of China (English)
陈国旺
2003-01-01
In the paper, the existence and uniqueness of the generalized global solution and the classical global solution of the initial boundary value problems for the nonlinear hyperbolic equationare proved by Galerkin method and the sufficient conditions of blow-up of solution in finite time are given.
Integrability of the Inozemtsev Spin Chain with Open Boundary Conditions
Institute of Scientific and Technical Information of China (English)
XUE Wei-Xing; WANG Yu-Peng
2004-01-01
@@ A long range interacting quantum spin chain with open boundary conditions is proposed. By constructing the reflection spin-Dunkl operators, the integrability of the model is proven. The model is therefore exactly solvable via the asymptotic Bethe ansatz method and falls into the universal class of the usual Heisenberg spin chain with boundary fields.
Non-singular field-only surface integral equations for electromagnetic scattering
Klaseboer, Evert; Chan, Derek Y C
2016-01-01
A boundary integral formulation of electromagnetics that involves only the components of E and H is derived without the use of surface currents that appear in the classical PMCHWT formulation. The kernels of the boundary integral equations for E and H are non-singular so that all field quantities at the surface can be determined to high precision and also geometries with closely spaced surfaces present no numerical difficulties. Quadratic elements can readily be used to represent the surfaces so that the surface integrals can be calculated to higher numerical precision than using planar elements for the same numbers of degrees of freedom.
A CLASS OF SINGULARLY PERTURBED INITIAL BOUNDARY PROBLEM FOR REACTION DIFFUSION EQUATION
Institute of Scientific and Technical Information of China (English)
Xie Feng
2003-01-01
The singularly perturbed initial boundary value problem for a class of reaction diffusion equation isconsidered. Under appropriate conditions, the existence-uniqueness and the asymptotic behavior of the solu-tion are showed by using the fixed-point theorem.
Otelbaev, Mukhtarbay; Koshanov, Bakytbek D.
2016-08-01
This paper describes the correct narrowing of the Navier-Stokes equations in a stationary three-dimensional cube and clarified the correct formulation of the boundary conditions for the pressure in the environment.
Asymptotic behavior of solutions to nonlinear parabolic equation with nonlinear boundary conditions
Directory of Open Access Journals (Sweden)
Diabate Nabongo
2008-01-01
Full Text Available We show that solutions of a nonlinear parabolic equation of second order with nonlinear boundary conditions approach zero as t approaches infinity. Also, under additional assumptions, the solutions behave as a function determined here.
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.
Blow-up estimates for semilinear parabolic systems coupled in an equation and a boundary condition
Institute of Scientific and Technical Information of China (English)
王明新
2001-01-01
This paper deals with the blow-up rate estimates of solutions for semilinear parabolic systems coupled in an equation and a boundary condition. The upper and lower bounds of blow-up rates have been obtained.
On solvability of some boundary value problems for a biharmonic equation with periodic conditions
Karachik, Valery V.; Massanov, Saparbay K.; Turmetov, Batirkhan Kh.
2016-08-01
In the paper we study questions about solvability of some boundary value problems with periodic conditions for an inhomogeneous biharmonic equation. The exact conditions for solvability of the problems are found.
Periodic solutions of a non-linear wave equation with homogeneous boundary conditions
Energy Technology Data Exchange (ETDEWEB)
Rudakov, I A [M.V. Lomonosov Moscow State University, Moscow (Russian Federation)
2006-02-28
We prove the existence of time-periodic solutions of a non-linear wave equation with homogeneous boundary conditions. The non-linear term either has polynomial growth or satisfies a 'non-resonance' condition.
Institute of Scientific and Technical Information of China (English)
LiHongyu; SunJingxian
2005-01-01
By using topological method, we study a class of boundary value problem for a system of nonlinear ordinary differential equations. Under suitable conditions,we prove the existence of positive solution of the problem.
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.
THE ARTIFICIAL BOUNDARY CONDITION FOR EXTERIOR OSEEN EQUATION IN 2-D SPACE
Institute of Scientific and Technical Information of China (English)
Chun-xiong Zheng; Hou-de Han
2002-01-01
A finite element method for the solution of Oseen equation in exterior domain is proposed. In this method, a circular artificial boundary is introduced to make the computational domain finite. Then, the exact relation between the normal stress and the prescribed velocity field on the artificial boundary can be obtained analytically. This relation can serve as an boundary condition for the boundary value problem defined on the finite domain bounded by the artificial boundary. Numerical experiment is presented to demonstrate the performance of the method.
Exponential integrators for the incompressible Navier-Stokes equations.
Energy Technology Data Exchange (ETDEWEB)
Newman, Christopher K.
2004-07-01
We provide an algorithm and analysis of a high order projection scheme for time integration of the incompressible Navier-Stokes equations (NSE). The method is based on a projection onto the subspace of divergence-free (incompressible) functions interleaved with a Krylov-based exponential time integration (KBEI). These time integration methods provide a high order accurate, stable approach with many of the advantages of explicit methods, and can reduce the computational resources over conventional methods. The method is scalable in the sense that the computational costs grow linearly with problem size. Exponential integrators, used typically to solve systems of ODEs, utilize matrix vector products of the exponential of the Jacobian on a vector. For large systems, this product can be approximated efficiently by Krylov subspace methods. However, in contrast to explicit methods, KBEIs are not restricted by the time step. While implicit methods require a solution of a linear system with the Jacobian, KBEIs only require matrix vector products of the Jacobian. Furthermore, these methods are based on linearization, so there is no non-linear system solve at each time step. Differential-algebraic equations (DAEs) are ordinary differential equations (ODEs) subject to algebraic constraints. The discretized NSE constitute a system of DAEs, where the incompressibility condition is the algebraic constraint. Exponential integrators can be extended to DAEs with linear constraints imposed via a projection onto the constraint manifold. This results in a projected ODE that is integrated by a KBEI. In this approach, the Krylov subspace satisfies the constraint, hence the solution at the advanced time step automatically satisfies the constraint as well. For the NSE, the projection onto the constraint is typically achieved by a projection induced by the L{sup 2} inner product. We examine this L{sup 2} projection and an H{sup 1} projection induced by the H{sup 1} semi-inner product. The H
Energy Technology Data Exchange (ETDEWEB)
Mancas, Stefan C. [Department of Mathematics, Embry–Riddle Aeronautical University, Daytona Beach, FL 32114-3900 (United States); Rosu, Haret C., E-mail: hcr@ipicyt.edu.mx [IPICYT, Instituto Potosino de Investigacion Cientifica y Tecnologica, Apdo Postal 3-74 Tangamanga, 78231 San Luis Potosí, SLP (Mexico)
2013-09-02
We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in their first-kind form have only cubic and quadratic terms. Then, employing an old integrability criterion due to Chiellini, we introduce the corresponding integrable dissipative equations. For illustration, we present the cases of some integrable dissipative Fisher, nonlinear pendulum, and Burgers–Huxley type equations which are obtained in this way and can be of interest in applications. We also show how to obtain Abel solutions directly from the factorization of second order nonlinear equations.
Institute of Scientific and Technical Information of China (English)
SU XIN-WEI
2011-01-01
This paper is devoted to study the existence and uniqueness of solutions to a boundary value problem of nonlinear fractional differential equation with impulsive effects. The arguments are based upon Schauder and Banach fixed-point theorems. We improve and generalize the results presented in [B. Ahmad, S. Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Analysis: Hybrid Systems, 3(2009), 251258].
Sobolev type equations of time-fractional order with periodical boundary conditions
Plekhanova, Marina
2016-08-01
The existence of a unique local solution for a class of time-fractional Sobolev type partial differential equations endowed by the Cauchy initial conditions and periodical with respect to every spatial variable boundary conditions on a parallelepiped is proved. General results are applied to study of the unique solvability for the initial boundary value problem to Benjamin-Bona-Mahony-Burgers and Allair partial differential equations.
Existence of Solutions for p-Laplace Equations Subjected to Neumann Boundary Value Problem
Institute of Scientific and Technical Information of China (English)
HU Zhi-gang; RUI Wen-juan; LIU Wen-bing
2006-01-01
The existence of solutions for one dimensional p-Laplace equation (φp(u'))'=f(t,u,u') with t∈(0,1) and φp(s)=│s│p-2s,s≠0 subjected to Neumann boundary value problem at u'(0)=0,u'(1)=0. By using the degree theory, the sufficient conditions of the existence of solutions for p-Laplace equation subjected to Neumann boundary value condition are established.
Calculation of transonic flows using an extended integral equation method
Nixon, D.
1976-01-01
An extended integral equation method for transonic flows is developed. In the extended integral equation method velocities in the flow field are calculated in addition to values on the aerofoil surface, in contrast with the less accurate 'standard' integral equation method in which only surface velocities are calculated. The results obtained for aerofoils in subcritical flow and in supercritical flow when shock waves are present compare satisfactorily with the results of recent finite difference methods.
Exact solution of the open XXZ chain with general integrable boundary terms at roots of unity
Murgan, R; Shi, C; Murgan, Rajan; Nepomechie, Rafael I.; Shi, Chi
2006-01-01
We propose a Bethe-Ansatz-type solution of the open spin-1/2 integrable XXZ quantum spin chain with general integrable boundary terms and bulk anisotropy values i \\pi/(p+1), where p is a positive integer. All six boundary parameters are arbitrary, and need not satisfy any constraint. The solution is in terms of generalized T - Q equations, having more than one Q function. We find numerical evidence that this solution gives the complete set of 2^N transfer matrix eigenvalues, where N is the number of spins.
High Order Numerical Solution of Integral Transport Equation in Slab Geometry
Institute of Scientific and Technical Information of China (English)
沈智军; 袁光伟; 沈隆钧
2002-01-01
@@ There are some common numerical methods for solving neutron transport equation, which including the well-known discrete ordinates method, PN approximation and integral transport methods[1]. There exists certain singularities in the solution of transport equation near the boundary and interface[2]. It gives rise to the difficulty in the construction of high order accurate numerical methods. The numerical solution obtained by now can not attain the second order convergent accuracy[3,4].
Transformations of Heun's equation and its integral relations
El-Jaick, Léa Jaccoud
2010-01-01
For each variable transformation which preserves the form of Heun's equation we find a transformation which leaves invariant the form of the equation for the kernels of integral relations among solutions of the former equation. This enables us to generate new kernels for the Heun equation, given by single hypergeometric functions (Lambe-Ward-type kernels) and by products of two hypergeometric functions (Erd\\'elyi-type). Such kernels, by a limiting process, afford new kernels for the confluent Heun equation as well.
Picard-Fuchs Equations, Hauptmoduls and Integrable Systems
Harnad, J.
1999-01-01
The Schwarzian equations satisfied by certain Hauptmoduls (i.e., uniformizing functions for Riemann surfaces of genus zero) are derived from the Picard-Fuchs equations for families of elliptic curves and associated surfaces. The inhomogeneous Picard-Fuchs equations associated to elliptic integrals with varying endpoints are derived and used to determine solutions of equations that are algebraically related to a class of Painlev\\'e VI equations.
ICM: an Integrated Compartment Method for numerically solving partial differential equations
International Nuclear Information System (INIS)
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
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.
Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions
Grunau, H.-Ch.; Sweers, G.
2001-01-01
Cranston, Fabes and Zhao ([26], [5]) established the uniform bound sup x; y 2 x 6= y R G1;n (x; z)G1;n (z; y) dz G1;n (x; y) M < 1; (1) where G1;n (x; y) is the Green function for the Laplacian - with Dirichlet boundary conditions on a Lipschitz domain - Rn with n 3 (see [27] for n = 2). This
Institute of Scientific and Technical Information of China (English)
Sun Hye PARK
2014-01-01
In this paper, we investigate the influence of boundary dissipation on the de-cay property of solutions for a transmission problem of Kirchhoff type wave equation with boundary memory condition. By introducing suitable energy and Lyapunov functionals, we establish a general decay estimate for the energy, which depends on the behavior of relaxation function.
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.
The immersed boundary method for the (2D) incompressible Navier-Stokes equations
Meûlen, R.J.R. vander
2006-01-01
Immersed Boundary Methods (IBMs) are a class of methods in Computational Fluid Dynamics where the grids do not conform to the shape of the body. Instead they employ Cartesian meshes and alternative ways to incorporate the boundary conditions in the (discrete) governing equations. The simple grids an
Finite-Volume Analysis for the Cahn-Hilliard equation with Dynamic boundary conditions
Nabet, Flore
2014-01-01
This work is devoted to the convergence analysis of a finite-volume approximation of the 2D Cahn-Hilliard equation with dynamic boundary conditions. The method that we propose couples a 2d-finite-volume method in a bounded, smooth domain and a 1d-finite-volume method on its boundary. We prove convergence of the sequence of approximate solutions.
Daveau, Christian
2007-01-01
We consider for the time-dependent Maxwell's equations the inverse problem of identifying locations and certain properties of small electromagnetic inhomogeneities in a homogeneous background medium from dynamic measurements of the tangential component of the magnetic field on the boundary (or a part of the boundary) of a domain.
A Kind of Boundary Element Methods for Boundary Value Problem of Helmholtz Equation
Institute of Scientific and Technical Information of China (English)
张然; 姜正义; 马富明
2004-01-01
Problems for electromagnetic scattering are of significant importance in many areas of technology. In this paper we discuss the scattering problem of electromagnetic wave incident by using boundary element method associated with splines. The problem is modelled by a boundary value problem for the Helmholtz eouation
Partial differential equations of mathematical physics and integral equations
Guenther, Ronald B
1996-01-01
This book was written to help mathematics students and those in the physical sciences learn modern mathematical techniques for setting up and analyzing problems. The mathematics used is rigorous, but not overwhelming, while the authors carefully model physical situations, emphasizing feedback among a beginning model, physical experiments, mathematical predictions, and the subsequent refinement and reevaluation of the physical model itself. Chapter 1 begins with a discussion of various physical problems and equations that play a central role in applications. The following chapters take up the t
The BV solution of the parabolic equation with degeneracy on the boundary
Directory of Open Access Journals (Sweden)
Zhan Huashui
2016-12-01
Full Text Available Consider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.
GLOBAL SOLUTIONS TO AN INITIAL BOUNDARY VALUE PROBLEM FOR THE MULLINS EQUATION
Institute of Scientific and Technical Information of China (English)
Hans-Dieter Alber; Zhu Peicheng
2007-01-01
In this article we study the global existence of solutions to an initial boundary value problem for the Mullins equation which describes the groove development at the grain boundaries of a heated polycrystal, both the Dirichlet and the Neumann boundary conditions are considered. For the classical solution we also investigate the large time behavior, it is proved that the solution converges to a constant, in the L∞(Ω)-norm, as time tends to infinity.
RADIATION BOUNDARY CONDITIONS FOR MAXWELL'S EQUATIONS: A REVIEW OF ACCURATE TIME-DOMAIN FORMULATIONS
Institute of Scientific and Technical Information of China (English)
Thomas Hagstrom; Stephen Lau
2007-01-01
We review time-domain formulations of radiation boundary conditions for Maxwell's equations, focusing on methods which can deliver arbitrary accuracy at acceptable computational cost. Examples include fast evaluations of nonlocal conditions on symmetric and general boundaries, methods based on identifying and evaluating equivalent sources, and local approximations such as the perfectly matched layer and sequences of local boundary conditions. Complexity estimates are derived to assess work and storage requirements as a function of wavelength and simulation time.
Moving boundary problems for the Harry Dym equation and its reciprocal associates
Rogers, Colin
2015-12-01
Moving boundary problems of generalised Stefan type are considered for the Harry Dym equation via a Painlevé II symmetry reduction. Exact solutions of such nonlinear boundary value problems are obtained in terms of Yablonski-Vorob'ev polynomials corresponding to an infinite sequence of values of the Painlevé II parameter. The action of two kinds of reciprocal transformation on the moving boundary problems is described.
Sliding periodic boundary conditions for lattice Boltzmann and lattice kinetic equations
Adhikari, R.; Desplat, J. -C.; Stratford, K.
2005-01-01
We present a method to impose linear shear flow in discrete-velocity kinetic models of hydrodynamics through the use of sliding periodic boundary conditions. Our method is derived by an explicit coarse-graining of the Lees-Edwards boundary conditions for Couette flow in molecular dynamics, followed by a projection of the resulting equations onto the subspace spanned by the discrete velocities of the lattice Boltzmann method. The boundary conditions are obtained without resort to perturbative ...
Lopes Filho, Milton C.; Nussenzveig Lopes, Helena J.; Titi, Edriss S.; Zang, Aibin
2015-06-01
The second-grade fluid equations are a model for viscoelastic fluids, with two parameters: α > 0, corresponding to the elastic response, and , corresponding to viscosity. Formally setting these parameters to 0 reduces the equations to the incompressible Euler equations of ideal fluid flow. In this article we study the limits of solutions of the second-grade fluid system, in a smooth, bounded, two-dimensional domain with no-slip boundary conditions. This class of problems interpolates between the Euler- α model (), for which the authors recently proved convergence to the solution of the incompressible Euler equations, and the Navier-Stokes case ( α = 0), for which the vanishing viscosity limit is an important open problem. We prove three results. First, we establish convergence of the solutions of the second-grade model to those of the Euler equations provided , as α → 0, extending the main result in (Lopes Filho et al., Physica D 292(293):51-61, 2015). Second, we prove equivalence between convergence (of the second-grade fluid equations to the Euler equations) and vanishing of the energy dissipation in a suitably thin region near the boundary, in the asymptotic regime , as α → 0. This amounts to a convergence criterion similar to the well-known Kato criterion for the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations. Finally, we obtain an extension of Kato's classical criterion to the second-grade fluid model, valid if , as . The proof of all these results relies on energy estimates and boundary correctors, following the original idea by Kato.
On a Time-Fractional Integrodifferential Equation via Three-Point Boundary Value Conditions
Directory of Open Access Journals (Sweden)
Dumitru Baleanu
2015-01-01
Full Text Available The existence and the uniqueness theorems play a crucial role prior to finding the numerical solutions of the fractional differential equations describing the models corresponding to the real world applications. In this paper, we study the existence of solutions for a time-fractional integrodifferential equation via three-point boundary value conditions.
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...
On The Stabilization of the Linear Kawahara Equation with Periodic Boundary Conditions
Directory of Open Access Journals (Sweden)
Patricia N. da Silva
2015-03-01
Full Text Available We study the stabilization of global solutions of the linear Kawahara equation (K with periodic boundary conditions under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using separation of variables, the Ingham inequality, multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K model.
On The Stabilization of the Linear Kawahara Equation with Periodic Boundary Conditions
Patricia N. da Silva; Carlos F. Vasconcellos
2015-01-01
We study the stabilization of global solutions of the linear Kawahara equation (K) with periodic boundary conditions under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using separation of variables, the Ingham inequality, multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model.
Strang-type preconditioners for solving fractional diffusion equations by boundary value methods
Gu, Xian-Ming; Huang, Ting-Zhu; Zhao, Xi-Le; Li, Hou-Biao; Li, Liang
2015-01-01
The finite difference scheme with the shifted Grünwarld formula is employed to semi-discrete the fractional diffusion equations. This spatial discretization can reduce to the large system of ordinary differential equations (ODEs) with initial values. Recently, boundary value method (BVM) was develop
A finite element-boundary integral method for cavities in a circular cylinder
Kempel, Leo C.; Volakis, John L.
1992-01-01
Conformal antenna arrays offer many cost and weight advantages over conventional antenna systems. However, due to a lack of rigorous mathematical models for conformal antenna arrays, antenna designers resort to measurement and planar antenna concepts for designing non-planar conformal antennas. Recently, we have found the finite element-boundary integral method to be very successful in modeling large planar arrays of arbitrary composition in a metallic plane. We extend this formulation to conformal arrays on large metallic cylinders. In this report, we develop the mathematical formulation. In particular, we discuss the shape functions, the resulting finite elements and the boundary integral equations, and the solution of the conformal finite element-boundary integral system. Some validation results are presented and we further show how this formulation can be applied with minimal computational and memory resources.
Financial integration in Europe : Evidence from Euler equation tests
Lemmen, J.J.G.; Eijffinger, S.C.W.
1995-01-01
This paper applies Obstfeld's Euler equation tests to assess the degree of financial integration in the European Union. In addition, we design a new Euler equation test which is intimately related to Obstfeld's Euler equation tests. Using data from the latest Penn World Table (Mark 6), we arrive at
The solution of integral equation of plasma diffusion problems
International Nuclear Information System (INIS)
Two topics which concern with the solution of one dimensional diffusion equation are discussed in this paper. The solution of the one dimensional diffusion equation with one initial condition is discussed in the first topic, and the solution of the same equation with two initial conditions is discussed in the second topic. Both solutions are presented in integral forms. (author)
Boundary element method for the solution of the diffusion equation in cylindrical symmetry
International Nuclear Information System (INIS)
Equations for the solution of the diffusion equation in plane Cartesian geometry with the Boundary Element method was derived. The equation for the axi-symmetric case were set and included in the computer program. The results were compared to those obtained by the Finite Difference method. Comparing the results some advantages of the proposed method can be observed, with implications on the multidimensional problems. (author)
Jiang, Song; Ju, Qiangchang; Li, Fucai
2010-01-01
This paper is concerned with the incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. It is rigorously shown that the weak solutions of the compressible magnetohydrodynamic equations converge to the strong solution of the viscous or inviscid incompressible magnetohydrodynamic equations as long as the latter exists both for the well-prepared initial data and general initial data. Furthermore, the convergence rates are also obtained in the ca...
THE INITIAL BOUNDARY VALUE PROBLEM FOR QUASI-LINEAR SCHR(O)DINGER-POISSON EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
In this article, the author studies the initial-(Dirichlet) boundary problem for a high-field version of the Schr(o)dinger-Poisson equations, which include a nonlinear term in the Poisson equation corresponding to a field-dependent dielectric constant and an effective potential in the Schrodinger equations on the unit cube. A global existence and uniqueness is established for a solution to this problem.
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.
Galagusz, Ryan; Nave, Jean-Christophe
2015-01-01
We present a high order, Fourier penalty method for the Maxwell's equations in the vicinity of perfect electric conductor boundary conditions. The approach relies on extending the irregular non-periodic domain of the equations to a regular periodic domain by removing the exact boundary conditions and introducing an analytic forcing term in the extended domain. The forcing, or penalty term is chosen to systematically enforce the boundary conditions to high order in the penalty parameter, which then allows for higher order numerical methods. We present an efficient numerical method for constructing the penalty term, and discretize the resulting equations using a Fourier spectral method. We demonstrate convergence orders of up to 3.5 for the one dimensional Maxwell's equations, and show that the numerical method does not suffer from dispersion (or pollution) errors. We also illustrate the approach in two dimensions and demonstrate convergence orders of 2.5 for transverse magnetic modes and 1.5 for the transverse...
Energy Technology Data Exchange (ETDEWEB)
Zhu, Changjiang; Duan, Renjun [Laboratory of Nonlinear Analysis, Department of Mathematics, Central China Normal University, Wuhan 430079, People' s Republic of China (China)
2003-02-28
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.
THE DIFFERENTIAL INTEGRAL EQUATIONS ON SMOOTH CLOSED ORIENTABLE MANIFOLDS
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
Using integration by parts and Stokes' formula, the authors give a new definition of Hadamard principal value of higher order singular integrals with Bochner-Martinelli kernel on smooth closed orientable manifolds in Cn. The Plemelj formula and composite formula of higher order singular integral arc obtained. Differential integral equations on smooth closed orientable manifolds are treated by using the composite formula.
Mixed initial-boundary value problem for equations of motion of Kelvin-Voigt fluids
Baranovskii, E. S.
2016-07-01
The initial-boundary value problem for equations of motion of Kelvin-Voigt fluids with mixed boundary conditions is studied. The no-slip condition is used on some portion of the boundary, while the impermeability condition and the tangential component of the surface force field are specified on the rest of the boundary. The global-in-time existence of a weak solution is proved. It is shown that the solution is unique and depends continuously on the field of external forces, the field of surface forces, and initial data.
Alekseev, Gennady
2016-04-01
We consider the boundary value problem for stationary magnetohydrodynamic equations of electrically and heat conducting fluid under inhomogeneous mixed boundary conditions for electromagnetic field and temperature and Dirichlet condition for the velocity. The problem describes the thermoelectromagnetic flow of a viscous fluid in 3D bounded domain with the boundary consisting of several parts with different thermo- and electrophysical properties. The global solvability of the boundary value problem is proved and the apriori estimates of the solution are derived. The sufficient conditions on the data are established which provide a local uniqueness of the solution.
Institute of Scientific and Technical Information of China (English)
MO Jiaqi; WANG Hui; LIN Wantao
2006-01-01
The perturbed boundary undercurrent is an exceptional event in the tropical atmosphere and ocean. It is a complicated nonlinear system. Its appearance badly affects not only natural conditions such as climate and environment,but also global economic development and human living, and brings about many calamities. Thus there is very attractive study on its rules in the international academic circles. Many scholars made more studies on its local and whole behaviors using different methods, such as self-anamnestic principle, Fokker-Plank Equation method, higher order singular pedigree and predictable study, rapid change on boundary, indeterminate adaptive control, multi-cogradient method and so on. Nonlinear perturbed theory and approximate method are very attractive studies in the international academic circles. Many scholars considered a class of nonlinear problems for the ordinary differential equation, the reaction diffusion equations, the boundary value of elliptic equation, the initial boundary value of hyperbolic equation, the shock layer solution of nonlinear equation and so on. In this paper, a class of perturbed mechanism for the western boundary undercurrents in the equator Pacific is considered. Under suitable conditions, using a homotopic mapping theory and method, we obtain a simple and rapid arbitrary order approximate solution for the corresponding nonlinear system. For example, a special case shows that using the homotopic mapping method, there is a high accuracy for the computed value.It is also provided from the results that the solution for homotopic mapping solving method can be used for analyzing operator for perturbed mechanism of western boundary undercurrents in the equator Pacific.
Shape integral method for magnetospheric shapes. [boundary layer calculations
Michel, F. C.
1979-01-01
A method is developed for calculating the shape of any magnetopause to arbitrarily high precision. The method uses an integral equation which is evaluated for a trial shape. The resulting values of the integral equation as a function of auxiliary variables indicate how close one is to the desired solution. A variational method can then be used to improve the trial shape. Some potential applications are briefly mentioned.
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.
Calculation of unsteady transonic flows using the integral equation method
Nixon, D.
1978-01-01
The basic integral equations for a harmonically oscillating airfoil in a transonic flow with shock waves are derived; the reduced frequency is assumed to be small. The problems associated with shock wave motion are treated using a strained coordinate system. The integral equation is linear and consists of both line integrals and surface integrals over the flow field which are evaluated by quadrature. This leads to a set of linear algebraic equations that can be solved directly. The shock motion is obtained explicitly by enforcing the condition that the flow is continuous except at a shock wave. Results obtained for both lifting and nonlifting oscillatory flows agree satisfactorily with other accurate results.
Energy Technology Data Exchange (ETDEWEB)
Quano, Yas-Hiro [Department of Medical Electronics, Suzuka University of Medical Science, Kishioka-cho, Suzuka (Japan)]. E-mail: quanoy@suzuka-u.ac.jp
2001-10-19
The A{sup (1)}{sub n-1}-face model with boundary reflection is considered on the basis of the boundary corner transfer matrix bootstrap. We construct the fused boundary Boltzmann weights to determine the normalization factor. We derive difference equations of the quantum Knizhnik-Zamolodchikov type for correlation functions of the boundary model. The simplest difference equations are solved in the case of the free boundary condition. (author)
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.
A device adaptive inflow boundary condition for Wigner equations of quantum transport
Energy Technology Data Exchange (ETDEWEB)
Jiang, Haiyan [Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081 (China); Lu, Tiao [HEDPS and CAPT, LMAM and School of Mathematical Sciences, Peking University, Beijing 100871 (China); Cai, Wei, E-mail: wcai@uncc.edu [Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223-0001 (United States)
2014-02-01
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.
Spatial integration of boundaries in a 3D virtual environment.
Bouchekioua, Youcef; Miller, Holly C; Craddock, Paul; Blaisdell, Aaron P; Molet, Mikael
2013-10-01
Prior research, using two- and three-dimensional environments, has found that when both human and nonhuman animals independently acquire two associations between landmarks with a common landmark (e.g., LM1-LM2 and LM2-LM3), each with its own spatial relationship, they behave as if the two unique LMs have a known spatial relationship despite their never having been paired. Seemingly, they have integrated the two associations to create a third association with its own spatial relationship (LM1-LM3). Using sensory preconditioning (Experiment 1) and second-order conditioning (Experiment 2) procedures, we found that human participants integrated information about the boundaries of pathways to locate a goal within a three-dimensional virtual environment in the absence of any relevant landmarks. Spatial integration depended on the participant experiencing a common boundary feature with which to link the pathways. These results suggest that the principles of associative learning also apply to the boundaries of an environment.
Analytical Nonlocal Electrostatics Using Eigenfunction Expansions of Boundary-Integral Operators
Bardhan, Jaydeep P; Brune, Peter R
2012-01-01
In this paper, we present an analytical solution to nonlocal continuum electrostatics for an arbitrary charge distribution in a spherical solute. Our approach relies on two key steps: (1) re-formulating the PDE problem using boundary-integral equations, and (2) diagonalizing the boundary-integral operators using the fact their eigenfunctions are the surface spherical harmonics. To introduce this uncommon approach for analytical calculations in separable geometries, we rederive Kirkwood's classic results for a protein surrounded concentrically by a pure-water ion-exclusion layer and then a dilute electrolyte (modeled with the linearized Poisson--Boltzmann equation). Our main result, however, is an analytical method for calculating the reaction potential in a protein embedded in a nonlocal-dielectric solvent, the Lorentz model studied by Dogonadze and Kornyshev. The analytical method enables biophysicists to study the new nonlocal theory in a simple, computationally fast way; an open-source MATLAB implementatio...
Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras
Energy Technology Data Exchange (ETDEWEB)
Grahovski, Georgi G., E-mail: G.Grahovski@leeds.ac.uk [Department of Applied Mathematics, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT (United Kingdom); Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chausee, Sofia 1784 (Bulgaria); Mikhailov, Alexander V., E-mail: A.V.Mikhailov@leeds.ac.uk [Department of Applied Mathematics, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT (United Kingdom)
2013-12-17
Integrable discretisations for a class of coupled (super) nonlinear Schrödinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra. Elementary Darboux transformations are constructed. As a result, Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained. The compatibility (Bianchi commutativity) of these Darboux transformations leads to integrable Grassmann generalisations of the difference Toda and NLS equations. The resulting systems will have discrete Lax representations provided by the set of two consistent elementary Darboux transformations. For the two discrete systems obtained, initial value and initial-boundary problems are formulated.
Directory of Open Access Journals (Sweden)
Bashir Ahmad
2010-01-01
Full Text Available We study a Dirichlet boundary value problem for Langevin equation involving two fractional orders. Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. However, ordinary Langevin equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractal medium, numerous generalizations of Langevin equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Langevin equation. This gives rise to the fractional Langevin equation with a single index. Recently, a new type of Langevin equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskii's fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space.
Master equations and the theory of stochastic path integrals
Weber, Markus F
2016-01-01
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. We discuss analytical and numerical methods for the solution of master equations, keeping our focus on methods that are applicable even when stochastic fluctuations are strong. The reviewed methods include the generating function technique and the Poisson representation, as well as novel ways of mapping the forward and backward master equations onto linear partial differential equations (PDEs). Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE obeyed by the generating function. After outlining these methods, we solve the derived PDEs in terms of two path integrals. The path integrals provide distinct exact representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Furthermore, we review a method for the approxima...
Implementation of higher-order absorbing boundary conditions for the Einstein equations
Rinne, Oliver; Scheel, Mark A; Pfeiffer, Harald P
2008-01-01
We present an implementation of absorbing boundary conditions for the Einstein equations based on the recent work of Buchman and Sarbach. In this paper, we assume that spacetime may be linearized about Minkowski space close to the outer boundary, which is taken to be a coordinate sphere. We reformulate the boundary conditions as conditions on the gauge-invariant Regge-Wheeler-Zerilli scalars. Higher-order radial derivatives are eliminated by rewriting the boundary conditions as a system of ODEs for a set of auxiliary variables intrinsic to the boundary. From these we construct boundary data for a set of well-posed constraint-preserving boundary conditions for the Einstein equations in a first-order generalized harmonic formulation. This construction has direct applications to outer boundary conditions in simulations of isolated systems (e.g., binary black holes) as well as to the problem of Cauchy-perturbative matching. As a test problem for our numerical implementation, we consider linearized multipolar grav...
POSITIVE SOLUTIONS OF A NONLINEAR THREE-POINT EIGENVALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS
Directory of Open Access Journals (Sweden)
FAOUZI HADDOUCHI
2015-11-01
Full Text Available In this paper, we study the existence of positive solutions of a three-point integral boundary value problem (BVP for the following second-order differential equation u''(t + \\lambda a(tf(u(t = 0; 0 0 is a parameter, 0 <\\eta < 1, 0 <\\alpha < 1/{\\eta}. . By using the properties of the Green's function and Krasnoselskii's fixed point theorem on cones, the eigenvalue intervals of the nonlinear boundary value problem are considered, some sufficient conditions for the existence of at least one positive solutions are established.
Feshchenko, R M
2016-01-01
In this paper exact 1D transparent boundary conditions (TBC) for the 2D parabolic wave equation with a linear or a quadratic dependence of the dielectric permittivity on the transversal coordinate are reported. Unlike the previously derived TBCs they contain only elementary functions. The obtained boundary conditions can be used to numerically solve the 2D parabolic equation describing the propagation of light in weakly bent optical waveguides and fibers including waveguides with variable curvature. They also are useful when solving the equivalent 1D Schr\\"odinger equation with a potential depending linearly or quadratically on the coordinate. The prospects and problems of discretization of the derived transparent boundary conditions are discussed.
Morales, Mayckol; Herrera, William J
2015-01-01
We find solutions of Laplace's equation with specific boundary conditions (in which such solutions take either the value zero or unity in each surface) using a generic curvilinear system of coordinates. Such purely geometrical solutions (that we shall call Basic Harmonic Functions BHF's) are utilized to obtain a more general class of solutions for Laplace's equation, in which the functions take arbitrary constant values on the boundaries. On the other hand, the BHF's are also used to obtain the capacitance of many electrostatic configurations of conductors. This method of finding solutions of Laplace's equation and capacitances with multiple symmetries is particularly simple, owing to the fact that the method of separation of variables becomes much simpler under the boundary conditions that lead to the BHF's. Examples of application in complex symmetries are given. Then, configurations of succesive embedding of conductors are also examined. In addition, expressions for electric fields between two conductors a...
Feng, Zhaosheng
Many physical phenomena can be described by nonlinear models. The last few decades have seen an enormous growth of the applicability of nonlinear models and of the development of related nonlinear concepts. This has been driven by modern computer power as well as by the discovery of new mathematical techniques, which include two contrasting themes: (i) the theory of dynamical systems, most popularly associated with the study of chaos, and (ii) the theory of integrable systems associated, among other things, with the study of solitons. In this dissertation, we study two nonlinear models. One is the 1-dimensional vibrating string satisfying wtt - wxx = 0 with van der Pol boundary conditions. We formulate the problem into an equivalent first order Hyperbolic system, and use the method of characteristics to derive a nonlinear reflection relation caused by the nonlinear boundary conditions. Thus, the problem is reduced to the discrete iteration problem of the type un+1 = F( un). Periodic solutions are investigated, an invariant interval for the Abel equation is studied, and numerical simulations and visualizations with different coefficients are illustrated. The other model is the Korteweg-de Vries-Burgers (KdVB) equation. In this dissertation, we proposed two new approaches: One is what we currently call First Integral Method, which is based on the ring theory of commutative algebra. Applying the Hilbert-Nullstellensatz, we reduce the KdVB equation to a first-order integrable ordinary differential equation. The other approach is called the Coordinate Transformation Method, which involves a series of variable transformations. Some new results on the traveling wave solution are established by using these two methods, which not only are more general than the existing ones in the previous literature, but also indicate that some corresponding solutions presented in the literature contain errors. We clarify the errors and instead give a refined result.
An energy absorbing far-field boundary condition for the elastic wave equation
Energy Technology Data Exchange (ETDEWEB)
Petersson, N A; Sjogreen, B
2008-07-15
The authors present an energy absorbing non-reflecting boundary condition of Clayton-Engquist type for the elastic wave equation together with a discretization which is stable for any ratio of compressional to shear wave speed. They prove stability for a second order accurate finite-difference discretization of the elastic wave equation in three space dimensions together with a discretization of the proposed non-reflecting boundary condition. The stability proof is based on a discrete energy estimate and is valid for heterogeneous materials. The proof includes all six boundaries of the computational domain where special discretizations are needed at the edges and corners. The stability proof holds also when a free surface boundary condition is imposed on some sides of the computational domain.
Désilles, Anya; Frankowska, Hélène
2013-01-01
International audience Abstract. A solution of the initial-boundary value problem on the strip (0,1) × [0, 1] for scalar conservation laws with strictly convex flux can be obtained by considering gradients of the unique solution V to an associated Hamilton-Jacobi equation (with appropriately defined initial and boundary conditions). It was shown in Frankowska (2010) that V can be expressed as the minimum of three value functions arising in calculus of variations problems that, in turn, can...
Exact Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls
Institute of Scientific and Technical Information of China (English)
Tatsien LI; Bopeng RAO
2013-01-01
In this paper,the exact synchronization for a coupled system of wave equations with Dirichlet boundary controls and some related concepts are introduced.By means of the exact null controllability of a reduced coupled system,under certain conditions of compatibility,the exact synchronization,the exact synchronization by groups,and the exact null controllability and synchronization by groups are all realized by suitable boundary controls.
Quenching rates for heat equations with coupled singular nonlinear boundary flux
Institute of Scientific and Technical Information of China (English)
2008-01-01
We study finite time quenching for heat equations coupled via singular nonlinear bound-ary flux. A criterion is proposed to identify the simultaneous and non-simultaneous quenchings. In particular, three kinds of simultaneous quenching rates are obtained for different nonlinear exponent re-gions and appropriate initial data. This extends an original work by Pablo, Quir′os and Rossi for a heat system with coupled inner absorption terms subject to homogeneous Neumann boundary conditions.
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.
Equations for a laminar boundary layer of a dilated liquid in a transverse magnetic field
Energy Technology Data Exchange (ETDEWEB)
Samokhin, V.N.
1984-01-01
A system of equations is examined which describes the magnetohydrodynamic boundary layer of a dilated liquid in a transverse magnetic field. The self modeling problem with an exponential law of change in the speed of the external stream and magnetic induction is studied. Localization of the perturbation in the liquid speed in the boundary layer is established and the change in the properties of the solution associated with this is shown.
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
In this paper,authors discuss the numerical methods of general discontinuous boundary value problems for elliptic complex equations of first order.They first give the well posedness of general discontinuous boundary value problems,reduce the discontinuousboundary value problems to a variation problem,and then find the numerical solutions ofabove problem by the finite element method.Finally authors give some error-estimates of the foregoing numerical solutions.
The Explicit Solutions of Riccati Equation by Integral Series
Yan, Yimin
2010-01-01
The paper aims at exactly solving the linear differential equation and the matrix Riccati equation with variable coefficients. Starting with the simplest structure of them, this article promotes the exponential function by introducing two maps with integral series: $\\mathcal{E}(X)$ and $\\mathcal{F}(X)$, which extend the important properties of exponential function: convergence, reversibility, and the relationship of determinant. Then,the article shows two approaches to the Riccati equation solving: (1)one is the Simplified Way: summed up the Riccati equation into the simplest form $\\frac{\\partial}{\\partial x}W+WPW-Q=0$, and gets the accurate solution;(2) the other is Matrix Way: directly deal with the general Riccati equation, and transform into the matrix linear differential equation. And then the solution could be further expresses with the elementary form with the particular solution. At the end, we can see that Riccati equation solving is somehow a particular case of linear differential equation when view...
A forced fractional Schrödinger equation with a Neumann boundary condition
Esquivel, L.; Kaikina, Elena I.
2016-07-01
We study the initial-boundary value problem for the nonlinear fractional Schrödinger equation {ut+i(uxx+12π∫0∞sign(x-y)|x-y|12uy( y)dy)+i|u|2u=0, t>0, x>0u(x,0)=u0(x), x>0,ux(0,t)=h(t), t>0. We prove the global-in-time existence of solutions for a nonlinear fractional Schrödinger equation with inhomogeneous Neumann boundary conditions. We are also interested in the study of the asymptotic behaviour of the solutions.
A numerical method for the elliptic Monge-Amp\\`ere equation with transport boundary conditions
Froese, Brittany D
2011-01-01
The problem of optimal mass transport arises in numerous applications including image registration, mesh generation, reflector design, and astrophysics. One approach to solving this problem is via the Monge-Amp\\`ere equation. While recent years have seen much work in the development of numerical methods for solving this equation, very little has been done on the implementation of the transport boundary conditions. In this paper, we propose a method for solving the transport problem by iteratively solving a Monge-Amp\\`ere equation with Neumann boundary conditions. We present a new discretization for the equation, which converges to the viscosity solution. The resulting system is solved efficiently with Newton's method. We provide several challenging computational examples that demonstrate the effectiveness and efficiency ($O(M)-O(M^{1.3})$ time) of the proposed method.
Institute of Scientific and Technical Information of China (English)
MO Jia-qi
2008-01-01
In this paper, a class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation are considered under suitable conditions. Firstly, by dint of the regular perturbation method, the outer solution of the original problem is obtained. Secondly, by using the stretched variable and the expansion theory of power series the initial layer of the solution is constructed. And then, by using the theory of differential inequalities, the asymptotic behavior of the solution for the initial boundary value problems is studied. Finally, using some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.
Institute of Scientific and Technical Information of China (English)
Jingsun Yao; Jiaqi Mo
2005-01-01
The nonlinear nonlocal singularly perturbed initial boundary value problems for reaction diffusion equations with a boundary perturbation is considered. Under suitable conditions, the outer solution of the original problem is obtained. Using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed. And then using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems is studied. Finally the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.
Institute of Scientific and Technical Information of China (English)
MO Jia-qi; WANG Hui; LIN Wan-tao
2005-01-01
A class of nonlinear nonlocal for singularly perturbed Robin initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions, first, the outer solution of the original problem was obtained. Secondly, using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer was constructed. Finally, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems was studied, and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation were discussed.
Institute of Scientific and Technical Information of China (English)
Wei Gong; Ningning Yan
2009-01-01
In this paper.we discuss the a posteriori error estimate of the finite element approximation for the boundary control problems governed by the parabolic partial differential equations.Three different a posteriori error estimators are provided for the parabolic boundary control problems with the observations of the distributed state.the boundary state and the final state.It is proven that these estimators are reliable bounds of the finite element approximation errors,which can be used as the indicators of the mesh refinement in adaptive finite element methods.
Second-order boundary estimates for solutions to singular elliptic equations in borderline cases
Directory of Open Access Journals (Sweden)
Claudia Anedda
2011-04-01
Full Text Available Let $Omegasubset R^N$ be a bounded smooth domain. We investigate the effect of the mean curvature of the boundary $partialOmega$ on the behaviour of the solution to the homogeneous Dirichlet boundary value problem for the equation $Delta u+f(u=0$. Under appropriate growth conditions on $f(t$ as $t$ approaches zero, we find asymptotic expansions up to the second order of the solution in terms of the distance from $x$ to the boundary $partialOmega$.
The Focusing NLS Equation on the Half-Line with Periodic Boundary Conditions
Kamvissis, S.; Fokas, A. S.
2007-01-01
We consider the Dirichlet problem for the focusing NLS equation on the half-line, with given Schwartz initial data and boundary data $q(0,t)$ equal to an exponentially decaying perturbation $u(t)$ of the periodic boundary data $ a e^{2i\\omega t + i \\epsilon}$ at $x=0.$ It is known from PDE theory that this is a well-posed problem (for fixed initial data and fixed $u$). On the other hand, the associated inverse scattering transform formalism involves the Neumann boundary value for $x=0$. Thus ...
Shenavar, Hossein
2016-03-01
We impose Neumann boundary condition to solve cosmological perturbation equations and we derive a modified Friedmann equation and a new lensing equation. To check the new lensing equation and the value of Neumann constant, a sample that contains ten strong lensing systems is surveyed. Except for one lens, masses of the other lenses are found to be within the constrains of the observational data. Furthermore, we argue that by using the concept of geometrodynamic clocks it is possible to modify the equation of motion of massive particles too. Also, a sample that includes 101 HSB and LSB galaxies is used to re-estimate the value of the Neumann constant and we found that this value is consistent with the prior evaluation from Friedmann and lensing equations. Finally, the growth of structure is studied by a Newtonian approach which resulted in a more rapid rate of the structure formation in matter dominated era.
Huimin Yu
2012-01-01
The asymptotic behavior (as well as the global existence) of classical solutions to the 3D compressible Euler equations are considered. For polytropic perfect gas $(P(\\rho )={P}_{0}{\\rho }^{\\gamma })$ , time asymptotically, it has been proved by Pan and Zhao (2009) that linear damping and slip boundary effect make the density satisfying the porous medium equation and the momentum obeying the classical Darcy's law. In this paper, we use a more general method and extend this resu...
Strang-type preconditioners for solving fractional diffusion equations by boundary value methods
Gu, Xian-Ming; Huang, Ting-Zhu; Zhao, Xi-Le; Li, Hou-Biao; Li, Liang
2013-01-01
The finite difference scheme with the shifted Gr\\"{u}nwarld formula is employed to semi-discrete the fractional diffusion equations. This spatial discretization can reduce to the large system of ordinary differential equations (ODEs) with initial values. Recently, boundary value method (BVM) was developed as a popular algorithm for solving large systems of ODEs. This method requires the solutions of one or more nonsymmetric, large and sparse linear systems. In this paper, the GMRES method wit...
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 ...
Eddy Current Analysis of Thin Metal Container in Induction Heating by Line Integral Equations
Fujita, Hagino; Ishibashi, Kazuhisa
In recent years, induction-heating cookers have been disseminated explosively. It is wished to commercialize flexible and disposable food containers that are available for induction heating. In order to develop a good quality food container that is heated moderately, it is necessary to analyze accurately eddy currents induced in a thin metal plate. The integral equation method is widely used for solving induction-heating problems. If the plate thickness approaches zero, the surface integral equations on the upper and lower plate surfaces tend to become the same and the equations become ill conditioned. In this paper, firstly, we derive line integral equations from the boundary integral equations on the assumption that the electromagnetic fields in metal are attenuated rapidly compared with those along the metal surface. Next, so as to test validity of the line integral equations, we solve the eddy current induced in a thin metal container in induction heating and obtain power density given to the container and impedance characteristics of the heating coil. We compare computed results with those by FEM.
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.
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.
跨共振的周期-积分边值问题%Periodic-Integral Boundary Value Problems across Resonance
Institute of Scientific and Technical Information of China (English)
宋新; 杨雪
2011-01-01
研究二阶微分方程周期-积分边值问题,应用最优控制理论给出了跨多个共振情形下的二阶微分方程周期-积分边值问题唯一可解的最优条件.%The periodic-integral boundary value problems for second order differential equations were considered. On the basis of optimal control theory method, we gave an optimal condition of the unique solvability to the periodic-integral boundary value problems for second order differential equations across multiple resonance.
Pan, Wenxiao; Bao, Jie; Tartakovsky, Alexandre
2013-11-01
A Continuous Boundary Force (CBF) method was developed for implementing Robin (Navier) boundary condition (BC) that can describe no-slip or slip conditions (slip length from zero to infinity) at the fluid-solid interface. In the CBF method the Robin BC is replaced by a homogeneous Neumann BC and an additional volumetric source term in the governing momentum equation. The formulation is derived based on an approximation of the sharp boundary with a diffuse interface of finite thickness, across which the BC is reformulated by means of a smoothed characteristic function. The CBF method is easy to be implemented in Lagrangian particle-based methods. We first implemented it in smoothed particle hydrodynamics (SPH) to solve numerically the Navier-Stokes equations subject to spatial-independent or dependent Robin BC in two and three dimensions. The numerical accuracy and convergence is examined through comparisons with the corresponding finite difference or finite element solutions. The CBF method is further implemented in smoothed dissipative particle dynamics (SDPD), a mesoscale scheme, for modeling slip flows commonly existent in micro/nano channels and microfluidic devices. The authors acknowledge the funding support by the ASCR Program of the Office of Science, U.S. Department of Energy.
Generalised Fourier integral operator methods for hyperbolic equations with singularities
Garetto, Claudia; Oberguggenberger, Michael
2011-01-01
This article addresses linear hyperbolic partial differential equations and pseudodifferential equations with strongly singular coefficients and data, modelled as members of algebras of generalised functions. We employ the recently developed theory of generalised Fourier integral operators to construct parametrices for the solutions and to describe propagation of singularities in this setting. As required tools, the construction of generalised solutions to eikonal and transport equations is g...
Kernel approximation for solving few-body integral equations
Christie, I.; Eyre, D.
1986-06-01
This paper investigates an approximate method for solving integral equations that arise in few-body problems. The method is to replace the kernel by a degenerate kernel defined on a finite dimensional subspace of piecewise Lagrange polynomials. Numerical accuracy of the method is tested by solving the two-body Lippmann-Schwinger equation with non-separable potentials, and the three-body Amado-Lovelace equation with separable two-body potentials.
DUAL INTEGRAL EQUATIONS INVOLVING LEGENDRE FUNCTIONS IN DISTRIBUTION SPACES
Directory of Open Access Journals (Sweden)
P. K. BANERJI, DESHNA LOONKER
2010-11-01
Full Text Available In this paper we use the Mehler-Fock transformation to obtain thesolution of dual integral equations involving Legendre functions. The solutionso obtained is proved to be distributional because they satisfy properties ofdistribution space.
A geometric approach to integrability conditions for Riccati equations
Directory of Open Access Journals (Sweden)
Arturo Ramos
2007-09-01
Full Text Available Several instances of integrable Riccati equations are analyzed from the geometric perspective of the theory of Lie systems. This provides us a unifying viewpoint for previous approaches.
Integral conditions in the theory of the Beltrami equations
Ryazanov, V; Yakubov, E
2010-01-01
It is shown that many recent and new results on the existence of ACL homeomorphic solutions for the degenerate Beltrami equations with integral constraints follow from our extension of the well--known Lehto existence theorem.
LETTER TO THE EDITOR: Deconstructing an integrable lattice equation
Ramani, A.; Joshi, N.; Grammaticos, B.; Tamizhmani, T.
2006-02-01
We show that an integrable lattice equation, obtained by J Hietarinta using the 'consistency around a cube' method without the tetrahedron assumption, is indeed solvable by linearization. We also present its nonautonomous extension.
Riccati equation-based generalization of Dawson's integral function
Messina, R; Messina, A; Napoli, A
2007-01-01
A new generalization of Dawson's integral function based on the link between a Riccati nonlinear differential equation and a second-order ordinary differential equation is reported. The MacLaurin expansion of this generalized function is built up and to this end an explicit formula for a generic cofactor of a triangular matrix is deduced.
Sloss, J. M.; Kranzler, S. K.
1972-01-01
The equivalence of a considered integral equation form with an infinite system of linear equations is proved, and the localization of the eigenvalues of the infinite system is expressed. Error estimates are derived, and the problems of finding upper bounds and lower bounds for the eigenvalues are solved simultaneously.
Kleinert, H.; Zatloukal, V.
2015-01-01
The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.
Positive and Nontrivial Solutions for the Urysohn Integral Equation
Institute of Scientific and Technical Information of China (English)
Daniel FRANCO; Gennaro INFANTE; Donal O'REGAN
2006-01-01
We establish new criteria for the existence of either positive or nonzero solutions of the Urysohn integral equation. We also discuss the existence of an interval of positive eigenvalues and sufficient conditions for the existence of at least a positive eigenvalue with a nonzero or positive eigenfunction for the Urysohn integral operator. Among others, we employ techniques based on fixed point index theory for compact maps, which are new for this type of equation.
Regularized boundary integral representations for dislocations and cracks in smart media
International Nuclear Information System (INIS)
This paper presents a complete set of singularity-reduced boundary integral relations for isolated discontinuities embedded in three-dimensional infinite media. The development is carried out within a broad context that allows the treatment of a well-known class of smart media such as linear piezoelectric, linear piezomagnetic and linear piezoelectromagnetic materials. In addition, resulting boundary integral representations are applicable to general discontinuities of arbitrary geometry and possessing a general jump distribution. The latter aspect allows the treatment of two special kinds of discontinuities: dislocations and cracks. The most attractive feature of the current development is that all integral relations for field quantities such as state variables and their gradients, the body flux, and the generalized interaction energy produced by dislocations are expressed only in terms of line integrals over the dislocation loops and, for cracks, the key governing boundary integral equation is established in a symmetric weak form and contains only weakly singular kernels of O(1/r). Results for the former case are fundamental and useful in the context of dislocation mechanics and modeling while the resulting weakly singular, weak form integral equation constitutes a basis for the development of a well-known numerical technique, called a symmetric Galerkin boundary element method (SGBEM), for analysis of cracked bodies. The weakly singular nature of such an integral equation allows low order interpolations to be used in the numerical approximation. The key ingredient for achieving such development of integral representations is the use of certain special decompositions in the derivative-transferring process via Stokes's theorem. Existence of such decompositions is ensured by a careful consideration of the singularity nature of the kernels, and a particular solution of the weakly singular functions involved is obtained by solving a system of partial differential
Rubbab, Qammar; Mirza, Itrat Abbas; Qureshi, M. Zubair Akbar
2016-07-01
The time-fractional advection-diffusion equation with Caputo-Fabrizio fractional derivatives (fractional derivatives without singular kernel) is considered under the time-dependent emissions on the boundary and the first order chemical reaction. The non-dimensional problem is formulated by using suitable dimensionless variables and the fundamental solutions to the Dirichlet problem for the fractional advection-diffusion equation are determined using the integral transforms technique. The fundamental solutions for the ordinary advection-diffusion equation, fractional and ordinary diffusion equation are obtained as limiting cases of the previous model. Using Duhamel's principle, the analytical solutions to the Dirichlet problem with time-dependent boundary pulses have been obtained. The influence of the fractional parameter and of the drift parameter on the solute concentration in various spatial positions was analyzed by numerical calculations. It is found that the variation of the fractional parameter has a significant effect on the solute concentration, namely, the memory effects lead to the retardation of the mass transport.
Dirac equation for particles with arbitrary half-integral spin
Guseinov, I. I.
2011-11-01
The sets of ? -component irreducible and Clifford algebraic Hermitian and unitary matrices through the two-component Pauli matrices are suggested, where s = 1/2, 3/2, 5/2, … . Using these matrix sets, the eigenvalues of which are ? , the ? -component generalized Dirac equation for a description of arbitrary half-integral spin particles is constructed. In accordance with the correspondence principle, the generalized Dirac equation suggested arises from the condition of relativistic invariance. This equation is reduced to the sets of two-component matrix equations the number of which is equal to ? . The new relativistic invariant equation of motion leads to an equation of continuity with a positive-definite probability density and also to the Klein-Gordon equation. This relativistic equation is causal in the presence of an external electromagnetic field interaction. It is shown that, in the case of nonrelativistic limit, the relativistic equation presented is reduced to the Pauli equation describing the motion of half-integral spin particle in the electromagnetic field.
SOME BOUNDARY VALUE PROBLEMS FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS OF SECOND ORDER
Institute of Scientific and Technical Information of China (English)
Wen Guochun
2007-01-01
The present article deals with some boundary value problems for nonlinear elliptic equations with degenerate rank 0 including the oblique derivative problem. Firstly the formulation and estimates of solutions of the oblique derivative problem are given, and then by the above estimates and the method of parameter extension, the existence of solutions of the above problem is proved. In this article, the complex analytic method is used, namely the corresponding problem for degenerate elliptic complex equations of first order is firstly discussed, afterwards the above problem for the degenerate elliptic equations of second order is solved.
Vanishing viscosity limits for the degenerate lake equations with Navier boundary conditions
Jiu, Quansen; Wu, Jiahong
2011-01-01
The paper is concerned with the vanishing viscosity limit of the two-dimensional degenerate viscous lake equations when the Navier slip conditions are prescribed on the impermeable boundary of a simply connected bounded regular domain. When the initial vorticity is in the Lebesgue space $L^q$ with $2equations possess a unique global solution and the solution converges to a corresponding weak solution of the inviscid lake equations. In the special case when the vorticity is in $L^\\infty$, an explicit convergence rate is obtained.
Accurate numerical resolution of transients in initial-boundary value problems for the heat equation
Flyer, N
2003-01-01
If the initial and boundary data for a PDE do not obey an infinite set of compatibility conditions, singularities will arise in the solution at the corners of the initial time-space domain. For dissipative equations, such as the 1-D heat equation or 1-D convection-diffusion equations, the impacts of these singularities are short lived. However, they can cause a very severe loss of numerical accuracy if we are interested in transient solutions. The phenomenon has been described earlier from a theoretical standpoint. Here, we illustrate it graphically and present a simple remedy which, with only little extra cost and effort, restores full numerical accuracy.
Institute of Scientific and Technical Information of China (English)
Yaohong LI; Xiaoyan ZHANG
2013-01-01
In this paper,we consider boundary value problems for systems of nonlinear thirdorder differential equations.By applying the fixed point theorems of cone expansion and compression of norm type and Leggett-Williams fixed point theorem,the existence of multiple positive solutions is obtained.As application,we give some examples to demonstrate our results.
Weak versus strong no-slip boundary conditions for the Navier-Stokes equations
Abbas, Qaisar; Nordström, Jan
2010-01-01
Original Publication:Qaisar Abbas and Jan Nordström, Weak versus strong no-slip boundary conditions for the Navier-Stokes equations, 2010, Engineering Applications of Computational Fluid Mechanics, (4), 29-38. Copyright: Engineering Applications of Computational Fluid Mechanics
Geometry and a priori estimates for free boundary problems of the Euler's equation
Shatah, Jalal; Zeng, Chongchun
2006-01-01
In this paper we derive estimates to the free boundary problem for the Euler equation with surface tension, and without surface tension provided the Rayleigh-Taylor sign condition holds. We prove that as the surface tension tends to zero, when the Rayleigh-Taylor condition is satisfied, solutions converge to the Euler flow with zero surface tension.
Directory of Open Access Journals (Sweden)
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.
SUPERCONVERGENCE AND A POSTERIORI ERROR ESTIMATES FOR BOUNDARY CONTROL GOVERNED BY STOKES EQUATIONS
Institute of Scientific and Technical Information of China (English)
Hui-po Liu; Ning-ning Yan
2006-01-01
In this paper, the superconvergence results are derived for a class of boundary control problems governed by Stokes equations. We derive superconvergence results for both the control and the state approximation. Base on superconvergence results, we obtain asymptotically exact a posteriori error estimates.
Positive Solutions of Singular Boundary Value Problem of Negative Exponent Emden–Fowler Equation
Indian Academy of Sciences (India)
Yuxia Wang; Xiyu Liu
2003-05-01
This paper investigates the existence of positive solutions of a singular boundary value problem with negative exponent similar to standard Emden–Fowler equation. A necessary and sufficient condition for the existence of [0, 1] positive solutions as well as 1[0, 1] positive solutions is given by means of the method of lower and upper solutions with the Schauder fixed point theorem.
Solvability of Boundary Value Problem at Resonance for Third-Order Functional Differential Equations
Indian Academy of Sciences (India)
Pinghua Yang; Zengji Du; Weigao Ge
2008-05-01
This paper is devoted to the study of boundary value problem of third-order functional differential equations. We obtain some existence results for the problem at resonance under the condition that the nonlinear terms is bounded or generally unbounded. In this paper we mainly use the topological degree theory.
Institute of Scientific and Technical Information of China (English)
Zhao Zengqin
2007-01-01
By using the upper and lower solutions method and fixed point theory, we investigate a class of fourth - order singular differential equations with the Sturm - Liouville Boundary conditions. Some sufficient conditions are obtained for the existence of C2 [ 0, 1 ] positive solutions and C3 [ 0, 1 ] positive solutions.
Institute of Scientific and Technical Information of China (English)
Yepeng Xing; Qiong Wang; Valery G. Romanovski
2009-01-01
We prove several new comparison results and develop the monotone iterative tech-nique to show the existence of extremal solutions to a kind of periodic boundary value problem (PBVP) for nonlinear integro-differential equation of mixed type on time scales.
Non-perturbative solution of free-convective boundary-layer equation by Adomian decomposition method
Energy Technology Data Exchange (ETDEWEB)
Kechil, Seripah Awang [Department of Mathematics, Universiti Teknologi MARA, 40450 Shah Alam Selangor (Malaysia); Hashim, Ishak [School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor (Malaysia)]. E-mail: ishak_h@ukm.my
2007-03-19
A free-convective boundary layer flow modeled by a system of nonlinear ordinary differential equations is considered. The system is solved using the Adomian decomposition method (ADM) which yields an analytic solution in the form of a rapidly convergent infinite series with easily computable terms. The analytical solutions and the pertinent features of the illustrations show the efficiency of the method.
Initial boundary value problems of nonlinear wave equations in an exterior domain
International Nuclear Information System (INIS)
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
Burgers equation with no-flux boundary conditions and its application for complete fluid separation
Watanabe, Shinya; Matsumoto, Sohei; Higurashi, Tomohiro; Ono, Naoki
2016-09-01
Burgers equation in a one-dimensional bounded domain with no-flux boundary conditions at both ends is proven to be exactly solvable. Cole-Hopf transformation converts not only the governing equation to the heat equation with an extra damping but also the nonlinear mixed boundary conditions to Dirichlet boundary conditions. The average of the solution v bar is conserved. Consequently, from an arbitrary initial condition, solutions converge to the equilibrium solution which is unique for the given v bar. The problem arises naturally as a continuum limit of a network of certain micro-devices. Each micro-device imperfectly separates a target fluid component from a mixture of more than one component, and its input-output concentration relationships are modeled by a pair of quadratic maps. The solvability of the initial boundary value problem is used to demonstrate that such a network acts as an ideal macro-separator, separating out the target component almost completely. Another network is also proposed which leads to a modified Burgers equation with a nonlinear diffusion coefficient.
Institute of Scientific and Technical Information of China (English)
KANG Ping; YAO Jianli
2009-01-01
In this paper, we investigate the existence of symmetric solutions of singular nonlocal boundary value problems for systems of differential equations. Our analysis relies on a nonlinear alternative of Leray - schauder type. Our results presented here unify, generalize and significantly improve many known results in the literature.
Inertial Manifolds for the 3D Cahn-Hilliard Equations with Periodic Boundary Conditions
Kostianko, A; Zelik, S.
2014-01-01
The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using the proper extension of the so-called spatial averaging principle introduced by G. Sell and J. Mallet-Paret. Moreover, the extra regularity of this manifold is also obtained.
Quasi-linear dynamics in nonlinear Schr\\" odinger equation with periodic boundary conditions
Erdogan, M. Burak; Zharnitsky, Vadim
2007-01-01
It is shown that a large subset of initial data with finite energy ($L^2$ norm)evolves nearly linearly in nonlinear Schr\\" odinger equation with periodic boundary conditions. These new solutions are not perturbations of the known ones such as solitons, semiclassical or weakly linear solutions.
Rudakov, I. A.
2015-10-01
We prove the existence of a countable family of time-periodic solutions of the quasilinear equation of beam vibrations with homogeneous boundary conditions and time-periodic right-hand side in the case when the non-linear term has power growth.
Institute of Scientific and Technical Information of China (English)
熊岳山; 韦永康
2001-01-01
The sediment reaction and diffusion equation with generalized initial and boundary condition is studied. By using Laplace transform and Jordan lemma , an analytical solution is got, which is an extension of analytical solution provided by Cheng Kwokming James ( only diffusion was considered in analytical solution of Cheng ). Some problems arisen in the computation of analytical solution formula are also analysed.
NONTRIVIAL SOLUTIONS TO SINGULAR BOUNDARY VALUE PROBLEMS FOR FOURTH-ORDER DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
The singular boundary value problems for fourth-order differential equations are considered under some conditions concerning the first eigenvalues of the relevant linear operators. Sufficient conditions which guarantee the existence of nontrivial solutions are obtained. We use the topological degree to prove our main results.
THREE-POINT BOUNDARY VALUE PROBLEM FOR p-LAPLACIAN DIFFERENTIAL EQUATION AT RESONANCE
Institute of Scientific and Technical Information of China (English)
Minggang Zong; Wenyi Cai
2009-01-01
By topological degree theory, the three-point boundary value problem for p-Laplacian differential equation at resonance is studied. Some new results on the existence of so-lutions are obtained, which improve and extend some known ones in the previous literatures.
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...
On an initial-boundary value problem for the nonlinear Schrödinger equation
Directory of Open Access Journals (Sweden)
Herbert Gajewski
1979-01-01
Full Text Available We study an initial-boundary value problem for the nonlinear Schrödinger equation, a simple mathematical model for the interaction between electromagnetic waves and a plasma layer. We prove a global existence and uniqueness theorem and establish a Galerkin method for solving numerically the problem.
Directory of Open Access Journals (Sweden)
Thanin Sitthiwirattham
2013-01-01
Full Text Available We study a new class of three-point boundary value problems of nonlinear second-order q-difference equations. Our problems contain different numbers of q in derivatives and integrals. By using a variety of fixed point theorems (such as Banach’s contraction principle, Boyd and Wong fixed point theorem for nonlinear contractions, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative and Leray-Schauder degree theory, some new existence and uniqueness results are obtained. Illustrative examples are also presented.
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.
Gal, Ciprian G.; Warma, Mahamadi
2016-08-01
We investigate the long term behavior in terms of finite dimensional global and exponential attractors, as time goes to infinity, of solutions to a semilinear reaction-diffusion equation on non-smooth domains subject to nonlocal Robin boundary conditions, characterized by the presence of fractional diffusion on the boundary. Our results are of general character and apply to a large class of irregular domains, including domains whose boundary is Hölder continuous and domains which have fractal-like geometry. In addition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence, and dimension, for the well-known reaction-diffusion equation in smooth domains, the framework we develop also makes possible a number of new results for all diffusion models in other non-smooth settings.
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.
SINGULAR INTEGRAL EQUATIONS WITH COSECANT KERNEL IN SOLUTIONS WITH SINGULARITIES OF ORDER ONE
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
In this article, periodic Riemann boundary value problem with period 2aπ along closed smooth contours is discussed, and thensingular integral equation with kernel csc t-t0/a along closed smooth contours restricted in the strip 0 ＜ Rez ＜ aπ is discussed.Finally, the solutions with singularities of order one for the above two problems are discussed.
Bracken, Paul
2010-04-01
A system of evolution equations can be developed from the structure equations for a submanifold embedded in a three-dimensional space. It is seen how these same equations can be obtained from a generalized matrix Lax pair provided a single constraint equation is imposed. This can be done in Euclidean space as well as in Minkowski space. The integrable systems which result from this process can be thought of as generalizing the SO(3) and SO(2,1) Lax pairs which have been studied previously.
A Hierarchy of Integrable Lattice Soliton Equations and New Integrable Symplectic Map
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
Starting from a discrete spectral problem, a hierarchy of integrable lattice soliton equations is derived. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses discrete bi-Hamiltonian structure.A new integrable symplectic map and finite-dimensional integrable systems are given by nonlinearization method. The binary Bargmann constraint gives rise to a B(a)cklund transformation for the resulting integrable lattice equations. At last, conservation laws of the hierarchy are presented.
Fringe integral equation method for a truncated grounded dielectric slab
DEFF Research Database (Denmark)
Jørgensen, Erik; Maci, S.; Toccafondi, A.
2001-01-01
is applied to the case of an electric line source located at the air-dielectric interface of the slab. Numerical results are compared with those calculated by a physical optics approach and by an alternative solution, in which the integral equation is constructed from the field continuity through an aperture......The problem of scattering by a semi-infinite grounded dielectric slab illuminated by an arbitrary incident TMz polarized electric field is studied by solving a new set of “fringe” integral equations (F-IEs), whose functional unknowns are physically associated to the wave diffraction processes...... occurring at the truncation. The F-IEs are obtained by subtracting from the surface/surface integral equations pertinent to the truncated slab, an auxiliary set of equations obtained for the canonical problem of an infinite grounded slab illuminated by the same source. The F-IEs are solved by the method...
High-Order Accurate Solutions to the Helmholtz Equation in the Presence of Boundary Singularities
Britt, Darrell Steven, Jr.
Problems of time-harmonic wave propagation arise in important fields of study such as geological surveying, radar detection/evasion, and aircraft design. These often involve highfrequency waves, which demand high-order methods to mitigate the dispersion error. We propose a high-order method for computing solutions to the variable-coefficient inhomogeneous Helmholtz equation in two dimensions on domains bounded by piecewise smooth curves of arbitrary shape with a finite number of boundary singularities at known locations. We utilize compact finite difference (FD) schemes on regular structured grids to achieve highorder accuracy due to their efficiency and simplicity, as well as the capability to approximate variable-coefficient differential operators. In this work, a 4th-order compact FD scheme for the variable-coefficient Helmholtz equation on a Cartesian grid in 2D is derived and tested. The well known limitation of finite differences is that they lose accuracy when the boundary curve does not coincide with the discretization grid, which is a severe restriction on the geometry of the computational domain. Therefore, the algorithm presented in this work combines high-order FD schemes with the method of difference potentials (DP), which retains the efficiency of FD while allowing for boundary shapes that are not aligned with the grid without sacrificing the accuracy of the FD scheme. Additionally, the theory of DP allows for the universal treatment of the boundary conditions. One of the significant contributions of this work is the development of an implementation that accommodates general boundary conditions (BCs). In particular, Robin BCs with discontinuous coefficients are studied, for which we introduce a piecewise parameterization of the boundary curve. Problems with discontinuities in the boundary data itself are also studied. We observe that the design convergence rate suffers whenever the solution loses regularity due to the boundary conditions. This is
Directory of Open Access Journals (Sweden)
J.-S. Chen
2011-04-01
Full Text Available This study presents a generalized analytical solution for one-dimensional solute transport in finite spatial domain subject to arbitrary time-dependent inlet boundary condition. The governing equation includes terms accounting for advection, hydrodynamic dispersion, linear equilibrium sorption and first order decay processes. The generalized analytical solution is derived by using the Laplace transform with respect to time and the generalized integral transform technique with respect to the spatial coordinate. Several special cases are presented and compared to illustrate the robustness of the derived generalized analytical solution. Result shows an excellent agreement. The analytical solutions of the special cases derived in this study have practical applications. Moreover, the derived generalized solution which consists an integral representation is evaluated by the numerical integration to extend its usage. The developed generalized solution offers a convenient tool for further development of analytical solution of specified time-dependent inlet boundary conditions or numerical evaluation of the concentration field for arbitrary time-dependent inlet boundary problem.
Convolution integral equation of Fredholm type with certain trancedental functions
Directory of Open Access Journals (Sweden)
V. B. L. Chaurasia
2008-09-01
Full Text Available The aim of this paper is to establish a solution of a certain class of convolution integral equation of Fredholm type whose kernel involve certain product of special function by using Riemann-Liouville and Weyl fractional integral operators. Some interesting particular cases are also considered.
The Boundary Value Problem for Elliptic Equation in the Corner Domain
Zhidkov, E P
2000-01-01
This work is devoted to the studies of the solution behavior of the boundary value problem for a nonlinear elliptic equation in the corner domain. The formulation of the boundary value problem arises in magnitostatics when finding the magnetic field distribution by the method of two scalar potentials in the domain comprising ferromagnetic and vacuum. The problem nonlinearity is stipulated by the dependence of the medium properties (magnetic permeability) on the solution to be found. In connection with that the solution of such a problem has to be found by numerical methods, a question arises about the behavior of the boundary value problem solution around the angular point of the ferromagnetic. This work shows that if the magnetic permeability function meets certain requirments, the corresponding solution of the boundary value problem will have a limited gradient.
The second boundary value problem for equations of viscoelastic diffusion in polymers
Vorotnikov, Dmitry A
2009-01-01
The classical approach to diffusion processes is based on Fick's law that the flux is proportional to the concentration gradient. Various phenomena occurring during propagation of penetrating liquids in polymers show that this type of diffusion exhibits anomalous behavior and contradicts the just mentioned law. However, they can be explained in the framework of non-Fickian diffusion theories based on viscoelasticity of polymers. Initial-boundary value problems for viscoelastic diffusion equations have been studied by several authors. Most of the studies are devoted to the Dirichlet BVP (the concentration is given on the boundary of the domain). In this chapter we study the second BVP, i.e. when the normal component of the concentration flux is prescribed on the boundary, which is more realistic in many physical situations. We establish existence of weak solutions to this problem. We suggest some conditions on the coefficients and boundary data under which all the solutions tend to the homogeneous state as tim...
Institute of Scientific and Technical Information of China (English)
丁皓江; 江爱民
2003-01-01
To obtain the fundamental solutions for computation of magneto-electro-elastic media by the boundary element method, the general solutions in the case of distinct eigenvalues are derived and expressed in five harmonic functions from the governing equations and the strict differential operator theorem. On the basis of these general solutions, the fundamental solution of infinite magneto-electro-elastic solid are obtained with the method of trial-and-error. Finally, the boundary integral formulation is derived and the corresponding boundary element method program is implemented to perform two numerical calculations(a column under uni-axial tension, uniform electric displacement or uniform magnetic induction, an annular plate simply-supported on outer and inner surfaces under axial loads). The numerical results agree well with the analytical ones.
A GPU-accelerated Direct-sum Boundary Integral Poisson-Boltzmann Solver
Geng, Weihua
2013-01-01
In this paper, we present a GPU-accelerated direct-sum boundary integral method to solve the linear Poisson-Boltzmann (PB) equation. In our method, a well-posed boundary integral formulation is used to ensure the fast convergence of Krylov subspace based linear algebraic solver such as the GMRES. The molecular surfaces are discretized with flat triangles and centroid collocation. To speed up our method, we take advantage of the parallel nature of the boundary integral formulation and parallelize the schemes within CUDA shared memory architecture on GPU. The schemes use only $11N+6N_c$ size-of-double device memory for a biomolecule with $N$ triangular surface elements and $N_c$ partial charges. Numerical tests of these schemes show well-maintained accuracy and fast convergence. The GPU implementation using one GPU card (Nvidia Tesla M2070) achieves 120-150X speed-up to the implementation using one CPU (Intel L5640 2.27GHz). With our approach, solving PB equations on well-discretized molecular surfaces with up ...
Algebraic Integrability of Lotka-Volterra equations in three dimensions
Constandinides, Kyriacos
2009-01-01
We examine the algebraic complete integrability of Lotka-Volterra equations in three dimensions. We restrict our attention to Lotka-Volterra systems defined by a skew symmetric matrix. We obtain a complete classification of such systems. The classification is obtained using Painleve analysis and more specifically by the use of Kowalevski exponents. The imposition of certain integrality conditions on the Kowalevski exponents gives necessary conditions for the algebraic integrability of the corresponding systems. We also show that the conditions are sufficient.
$A_n^{(1)}$ affine Toda field theories with integrable boundary conditions revisited
Doikou, Anastasia
2008-01-01
Generic classically integrable boundary conditions for the $A_{n}^{(1)}$ affine Toda field theories (ATFT) are investigated. The present analysis relies primarily on the underlying algebra, defined by the classical version of the reflection equation. We use as a prototype example the first non-trivial model of the hierarchy i.e. the $A_2^{(1)}$ ATFT, however our results may be generalized for any $A_{n}^{(1)}$ ($n>1$). We assume here two distinct types of boundary conditions called some times soliton preserving (SP), and soliton non-preserving (SNP) associated to two distinct algebras, i.e. the reflection algebra and the ($q$) twisted Yangian respectively. The boundary local integrals of motion are then systematically extracted from the asymptotic expansion of the associated transfer matrix. In the case of SNP boundary conditions we recover previously known results. The other type of boundary conditions (SP), associated to the reflection algebra, are novel in this context and lead to a different set of conser...
Effect of the Lower Boundary Position of the Fourier Equation on the Soil Energy Balance
Institute of Scientific and Technical Information of China (English)
孙菽芬; 张霞
2004-01-01
In this study, the effect of the lower boundary position selection for the Fourier equation on heat transfer and energy balance in soil is evaluated. A detailed numerical study shows that the proper position of the lower boundary is critical when solving the Fourier equation by using zero heat flux as the lower boundary condition. Since the position defines the capacity of soil as a heat sink or source, which absorbs and stores radiation energy from the sky in summer and then releases the energy to the atmosphere in winter, and regulates the deep soil temperature distribution, the depth of the position greatly influences the heat balance within the soil as well as the interaction between the soil and the atmosphere. Based on physical reasoning and the results of numerical simulation, the proper depth of the position should be equal to approximately 3 times of the annual heat wave damping depth. For most soils, the proper lower boundary depth for the Fourier equation should be around 8 m to 15 m, depending on soil texture.
Om, D.; Childs, M. E.
1983-01-01
An approximate integral viscous-inviscid interaction method is presented for calculating the development of a turbulent boundary layer subjected to a normal shock wave induced adverse pressure gradient in an internal axisymmetric flow. The inflow conditions and the downstream pressure are provided for the computation. In the supersonic region of shock pressure rise, the Prandtl-Meyer function is used to couple the viscous and inviscid flows. An analytical model for the coupling process is postulated and appropriate equations are defined. Downstream of the sonic point, one-dimensional inviscid flow is assumed for coupling with the viscous flow. The turbulent boundary layer is calculated using Green's integral lag-entrainment method. Comparisons of the solutions with the experimental data are made for interactions which are unseparated, near separation and separated. For comparison purposes, solutions to the time-dependent, mass-averaged, Navier-Stokes equations incorporating a two-equation, Wilcox-Rubesin turbulence model are also shown. The computed results from the integral method show good agreement with experimental data for unseparated interactions and reasonable agreement with the trend of the viscous effects when the interaction becomes increasingly separated.
Biondini, Gino; Fagerstrom, Emily; Prinari, Barbara
2016-10-01
We formulate the inverse scattering transform (IST) for the defocusing nonlinear Schrödinger (NLS) equation with fully asymmetric non-zero boundary conditions (i.e., when the limiting values of the solution at space infinities have different non-zero moduli). The theory is formulated without making use of Riemann surfaces, and instead by dealing explicitly with the branched nature of the eigenvalues of the associated scattering problem. For the direct problem, we give explicit single-valued definitions of the Jost eigenfunctions and scattering coefficients over the whole complex plane, and we characterize their discontinuous behavior across the branch cut arising from the square root behavior of the corresponding eigenvalues. We pose the inverse problem as a Riemann-Hilbert Problem on an open contour, and we reduce the problem to a standard set of linear integral equations. Finally, for comparison purposes, we present the single-sheet, branch cut formulation of the inverse scattering transform for the initial value problem with symmetric (equimodular) non-zero boundary conditions, as well as for the initial value problem with one-sided non-zero boundary conditions, and we also briefly describe the formulation of the inverse scattering transform when a different choice is made for the location of the branch cuts.
The numerical solution of the boundary inverse problem for a parabolic equation
Vasil'ev, V. V.; Vasilyeva, M. V.; Kardashevsky, A. M.
2016-10-01
Boundary inverse problems occupy an important place among the inverse problems of mathematical physics. They are connected with the problems of diagnosis, when additional measurements on one of the borders or inside the computational domain are necessary to restore the boundary regime in the other border, inaccessible to direct measurements. The boundary inverse problems belong to a class of conditionally correct problems, and therefore, their numerical solution requires the development of special computational algorithms. The paper deals with the solution of the boundary inverse problem for one-dimensional second-order parabolic equations, consisting in the restoration of boundary regime according to measurements inside the computational domain. For the numerical solution of the inverse problem it is proposed to use an analogue of a computational algorithm, proposed and developed to meet the challenges of identification of the right side of the parabolic equations in the works P.N.Vabishchevich and his students based on a special decomposition of solving the problem at each temporal layer. We present and discuss the results of a computational experiment conducted on model problems with quasi-solutions, including with random errors in the input data.
Energy Technology Data Exchange (ETDEWEB)
Kempka, S.N.; Strickland, J.H.; Glass, M.W.; Peery, J.S. [Sandia National Labs., Albuquerque, NM (United States); Ingber, M.S. [Univ. of New Mexico, Albuquerque, NM (United States)
1995-04-01
formulation to satisfy velocity boundary conditions for the vorticity form of the incompressible, viscous fluid momentum equations is presented. The tangential and normal components of the velocity boundary condition are satisfied simultaneously by creating vorticity adjacent to boundaries. The newly created vorticity is determined using a kinematical formulation which is a generalization of Helmholtz` decomposition of a vector field. Though it has not been generally recognized, these formulations resolve the over-specification issue associated with creating voracity to satisfy velocity boundary conditions. The generalized decomposition has not been widely used, apparently due to a lack of a useful physical interpretation. An analysis is presented which shows that the generalized decomposition has a relatively simple physical interpretation which facilitates its numerical implementation. The implementation of the generalized decomposition is discussed in detail. As an example the flow in a two-dimensional lid-driven cavity is simulated. The solution technique is based on a Lagrangian transport algorithm in the hydrocode ALEGRA. ALEGRA`s Lagrangian transport algorithm has been modified to solve the vorticity transport equation and the generalized decomposition, thus providing a new, accurate method to simulate incompressible flows. This numerical implementation and the new boundary condition formulation allow vorticity-based formulations to be used in a wider range of engineering problems.
Zhu, Changjiang; Duan, Renjun
2003-02-01
This paper is concerned with the existence and uniqueness of the entropy solution to the initial boundary value problem for the inviscid Burgers equation \\left\\{\\begin{array}{@{}l@{\\qquad}l@{}} u_t+\\big(\\frac{u^2}{2}\\big)_x=0 x\\gt0\\quad t\\gt0\\\\ u(x,0)=u_0(x) x\\geq0\\\\ u(0,t)=0 t\\geq0. \\end{array}\\right. 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.
A finite element-boundary integral method for conformal antenna arrays on a circular cylinder
Kempel, Leo C.; Volakis, John L.
1992-01-01
Conformal antenna arrays offer many cost and weight advantages over conventional antenna systems. In the past, antenna designers have had to resort to expensive measurements in order to develop a conformal array design. This was due to the lack of rigorous mathematical models for conformal antenna arrays. As a result, the design of conformal arrays was primarily based on planar antenna design concepts. Recently, we have found the finite element-boundary integral method to be very successful in modeling large planar arrays of arbitrary composition in a metallic plane. We are extending this formulation to conformal arrays on large metallic cylinders. In doing so, we will develop a mathematical formulation. In particular, we discuss the finite element equations, the shape elements, and the boundary integral evaluation. It is shown how this formulation can be applied with minimal computation and memory requirements.
Faminskii, Andrei V.
2013-01-01
An initial-boundary value problem in a strip with homogeneous Diriclet boundary conditions for two-dimensional generalized Zakharov-Kuznetsov equation is considered. In particular, dissipative and absorbing degenerate terms can be supplemented to the original Zakharov-Kuznetsov equation. Results on global existence, uniqueness and long-time decay of weak silutions are established.
Belkina, T. A.; Konyukhova, N. B.; Kurochkin, S. V.
2012-10-01
A singular boundary value problem for a second-order linear integrodifferential equation with Volterra and non-Volterra integral operators is formulated and analyzed. The equation is defined on ℝ+, has a weak singularity at zero and a strong singularity at infinity, and depends on several positive parameters. Under natural constraints on the coefficients of the equation, existence and uniqueness theorems for this problem with given limit boundary conditions at singular points are proved, asymptotic representations of the solution are given, and an algorithm for its numerical determination is described. Numerical computations are performed and their interpretation is given. The problem arises in the study of the survival probability of an insurance company over infinite time (as a function of its initial surplus) in a dynamic insurance model that is a modification of the classical Cramer-Lundberg model with a stochastic process rate of premium under a certain investment strategy in the financial market. A comparative analysis of the results with those produced by the model with deterministic premiums is given.
Free boundary value problems for a class of generalized diffusion equation
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The transport behavior of free boundary value problems for a class of generalized diffusion equations was studied. Suitable similarity transformations were used to convert the problems into a class of singular nonlinear two-point boundary value problems and similarity solutions were numerical presented for different representations of heat conduction function, convection function, heat flux function, and power law parameters by utilizing the shooting technique. The results revealed the flux transfer mechanism and the character as well as the effects of parameters on the solutions.
Mathematical analysis of the Navier-Stokes equations with non standard boundary conditions
Tidriri, M. D.
1995-01-01
One of the major applications of the domain decomposition time marching algorithm is the coupling of the Navier-Stokes systems with Boltzmann equations in order to compute transitional flows. Another important application is the coupling of a global Navier-Stokes problem with a local one in order to use different modelizations and/or discretizations. Both of these applications involve a global Navier-Stokes system with nonstandard boundary conditions. The purpose of this work is to prove, using the classical Leray-Schauder theory, that these boundary conditions are admissible and lead to a well posed problem.
Piecewise oblique boundary treatment for the elastic-plastic wave equation on a cartesian grid
Giese, Guido
2009-11-01
Numerical schemes for hyperbolic conservation laws in 2-D on a Cartesian grid usually have the advantage of being easy to implement and showing good computational performances, without allowing the simulation of “real-world” problems on arbitrarily shaped domains. In this paper a numerical treatment of boundary conditions for the elastic-plastic wave equation is developed, which allows the simulation of problems on an arbitrarily shaped physical domain surrounded by a piece-wise smooth boundary curve, but using a PDE solver on a rectangular Cartesian grid with the afore-mentioned advantages.
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.
Backward stochastic Volterra integral equations- a brief survey
Institute of Scientific and Technical Information of China (English)
YONG Jiong-min
2013-01-01
In this paper, we present a brief survey on the updated theory of backward stochas-tic Volterra integral equations (BSVIEs, for short). BSVIEs are a natural generalization of backward stochastic diff erential equations (BSDEs, for short). Some interesting motivations of studying BSVIEs are recalled. With proper solution concepts, it is possible to establish the corresponding well-posedness for BSVIEs. We also survey various comparison theorems for solutions to BSVIEs.
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.
A diffusive Fisher-KPP equation with free boundaries and time-periodic advections
Sun, Ningkui; Lou, Bendong; Zhou, Maolin
2016-01-01
We consider a reaction-diffusion-advection equation of the form: $u_t=u_{xx}-\\beta(t)u_x+f(t,u)$ for $x\\in (g(t),h(t))$, where $\\beta(t)$ is a $T$-periodic function representing the intensity of the advection, $f(t,u)$ is a Fisher-KPP type of nonlinearity, $T$-periodic in $t$, $g(t)$ and $h(t)$ are two free boundaries satisfying Stefan conditions. This equation can be used to describe the population dynamics in time-periodic environment with advection. Its homogeneous version (that is, both $...
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. PMID:26543723
Li, Fang; Liang, Xing; Shen, Wenxian
2015-01-01
In this series of papers, we investigate the spreading and vanishing dynamics of time almost periodic diffusive KPP equations with free boundaries. Such equations are used to characterize the spreading of a new species in time almost periodic environments with free boundaries representing the spreading fronts. In the first part of the series, we showed that a spreading-vanishing dichotomy occurs for such free boundary problems (see [16]). In this second part of the series, we investigate the ...
An integral equation solution for multistage turbomachinery design calculations
Mcfarland, Eric R.
1993-01-01
A method was developed to calculate flows in multistage turbomachinery. The method is an extension of quasi-three-dimensional blade-to-blade solution methods. Governing equations for steady compressible inviscid flow are linearized by introducing approximations. The linearized flow equations are solved using integral equation techniques. The flows through both stationary and rotating blade rows are determined in a single calculation. Multiple bodies can be modelled for each blade row, so that arbitrary blade counts can be analyzed. The method's benefits are its speed and versatility.
Recursive integral equations with positive kernel for lattice calculations
International Nuclear Information System (INIS)
A Kirkwood-Salzburg integral equation, with positive defined kernel, for the states of lattice models of statistical mechanics and quantum field theory is derived. The equation is defined in the thermodynamic limit, and its iterative solution is convergent. Moreover, positivity leads to an exact a priori bound on the iteration. The equation's relevance as a reliable algorithm for lattice calculations is therefore suggested, and it is illustrated with a simple application. It should provide a viable alternative to Monte Carlo methods for models of statistical mechanics and lattice gauge theories. 10 refs
Improved outer boundary conditions for Einstein's field equations
Energy Technology Data Exchange (ETDEWEB)
Buchman, Luisa T [Center for Relativity, University of Texas at Austin, 1 University Station C1606, Austin, TX 78712-1081 (United States); Sarbach, Olivier C A [Instituto de Fisica y Matematicas, Universidad Michoacana de San Nicolas de Hidalgo, Edificio C-3, Cd. Universitaria, C P 58040 Morelia, Michoacan (Mexico)
2007-06-21
In a recent article, we constructed a hierarchy B{sub L} of outer boundary conditions for Einstein's field equations with the property that, for a spherical outer boundary, it is perfectly absorbing for linearized gravitational radiation up to a given angular momentum number L. In this paper, we generalize B{sub L} so that it can be applied to fairly general foliations of spacetime by space-like hypersurfaces and general outer boundary shapes and further, we improve B{sub L} in two steps: (i) we give a local boundary condition C{sub L}which is perfectly absorbing including first-order contributions in 2M/R of curvature corrections for quadrupolar waves (where M is the mass of the spacetime and R is a typical radius of the outer boundary) and which significantly reduces spurious reflections due to backscatter, and (ii) we give a non-local boundary condition D{sub L} which is exact when first-order corrections in 2M/R for both curvature and backscatter are considered, for quadrupolar radiation.
The Navier-Stokes Equations Under a Unilateral Boundary Condition of Signorini's Type
Zhou, Guanyu; Saito, Norikazu
2016-09-01
We propose a new outflow boundary condition, a unilateral condition of Signorini's type, for the incompressible Navier-Stokes equations. The condition is a generalization of the standard free-traction condition. Its variational formulation is given by a variational inequality. We also consider a penalty approximation, a kind of the Robin condition, to deduce a suitable formulation for numerical computations. Under those conditions, we can obtain energy inequalities that are key features for numerical computations. The main contribution of this paper is to establish the well-posedness of the Navier-Stokes equations under those boundary conditions. Particularly, we prove the unique existence of strong solutions of Ladyzhenskaya's class using the standard Galerkin's method. For the proof of the existence of pressures, however, we offer a new method of analysis.
Mixed Initial-Boundary Value Problem for the Capillary Wave Equation
Directory of Open Access Journals (Sweden)
B. Juarez Campos
2016-01-01
Full Text Available We study the mixed initial-boundary value problem for the capillary wave equation: iut+u2u=∂x3/2u, t>0, x>0; u(x,0=u0(x, x>0; u(0,t+βux(0,t=h(t, t>0, where ∂x3/2u=(1/2π∫0∞signx-y/x-yuyy(y dy. We prove the global in-time existence of solutions of IBV problem for nonlinear capillary equation with inhomogeneous Robin boundary conditions. Also we are interested in the study of the asymptotic behavior of solutions.
Normalized RBF networks: application to a system of integral equations
Energy Technology Data Exchange (ETDEWEB)
Golbabai, A; Seifollahi, S; Javidi, M [Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16844 (Iran, Islamic Republic of)], E-mail: golbabai@iust.ac.ir, E-mail: seif@iust.ac.ir, E-mail: mojavidi@yahoo.com
2008-07-15
Linear integral and integro-differential equations of Fredholm and Volterra types have been successfully treated using radial basis function (RBF) networks in previous works. This paper deals with the case of a system of integral equations of Fredholm and Volterra types with a normalized radial basis function (NRBF) network. A novel learning algorithm is developed for the training of NRBF networks in which the BFGS backpropagation (BFGS-BP) least-squares optimization method as a recursive model is used. In the approach presented here, a trial solution is given by an NRBF network of incremental architecture with a set of unknown parameters. Detailed learning algorithms and concrete examples are also included.
Dhage Iteration Method for Generalized Quadratic Functional Integral Equations
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Bapurao C. Dhage
2015-01-01
Full Text Available In this paper we prove the existence as well as approximations of the solutions for a certain nonlinear generalized quadratic functional integral equation. An algorithm for the solutions is developed and it is shown that the sequence of successive approximations starting at a lower or upper solution converges monotonically to the solutions of related quadratic functional integral equation under some suitable mixed hybrid conditions. We rely our main result on Dhage iteration method embodied in a recent hybrid fixed point theorem of Dhage (2014 in partially ordered normed linear spaces. An example is also provided to illustrate the abstract theory developed in the paper.
Approximating positive solutions of quadratic functional integral equations
Directory of Open Access Journals (Sweden)
Dnyaneshwar V. Mule
2016-07-01
Full Text Available In this paper we prove the existence as well as approximations of the positive solutions for a nonlinear quadratic functional integral equation. An algorithm for the solutions is developed and it is shown that the sequence of successive approximations converges monotonically to the positive solution of related quadratic functional integral equation under some suitable mixed hybrid conditions. We rely our results on Dhage iteration method embodied in a recent hybrid fixed point theorem of Dhage (2014 in partially ordered normed linear spaces. An example is also provided to illustrate the abstract theory developed in the paper.
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.
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.
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.
Directory of Open Access Journals (Sweden)
Rabah Haoua
2015-04-01
Full Text Available In this article we give some new results on abstract second-order differential equations of elliptic type with variable operator coefficients and general Robin boundary conditions, in the framework of Holder spaces. We assume that the family of variable coefficients verify the well known Labbas-Terreni assumption used in the sum theory. We use Dunford calculus, interpolation spaces and the semigroup theory to obtain existence, uniqueness and maximal regularity results for the solution of the problem.
Jawerth, Bjoern; Sweldens, Wim
1993-01-01
We present ideas on how to use wavelets in the solution of boundary value ordinary differential equations. Rather than using classical wavelets, we adapt their construction so that they become (bi)orthogonal with respect to the inner product defined by the operator. The stiffness matrix in a Galerkin method then becomes diagonal and can thus be trivially inverted. We show how one can construct an O(N) algorithm for various constant and variable coefficient operators.
Oberman, Adam M.; Zwiers, Ian
2014-01-01
Monotone finite difference methods provide stable convergent discretizations of a class of degenerate elliptic and parabolic Partial Differential Equations (PDEs). These methods are best suited to regular rectangular grids, which leads to low accuracy near curved boundaries or singularities of solutions. In this article we combine monotone finite difference methods with an adaptive grid refinement technique to produce a PDE discretization and solver which is applied to a broad class of equati...
Solutions to fourth-order random differential equations with periodic boundary conditions
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Xiaoling Han
2012-12-01
Full Text Available Existence of solutions and of extremal random solutions are proved for periodic boundary-value problems of fourth-order ordinary random differential equations. Our investigation is done in the space of continuous real-valued functions defined on closed and bounded intervals. Also we study the applications of the random version of a nonlinear alternative of Leray-Schauder type and an algebraic random fixed point theorem by Dhage.
Scheuer, Jacob; Malomed, Boris A.
2001-01-01
We study, analytically and numerically, the dynamical behavior of the solutions of the complex Ginzburg-Landau equation with diffraction but without diffusion, which governs the spatial evolution of the field in an active nonlinear laser cavity. Accordingly, the solutions are subject to periodic boundary conditions. The analysis reveals regions of stable stationary solutions in the model’s parameter space, and a wide range of oscillatory and chaotic behaviors. Close to the first bifurca...
Institute of Scientific and Technical Information of China (English)
Chen Guowang; Xue Hongxia
2008-01-01
In this article, the existence, uniqueness and regularities of the global gener-alized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equation utt -uxx-auxxtt+bux4 - duxxt= f(u)xx are proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.
The second boundary value problem for equations of viscoelastic diffusion in polymers
Vorotnikov, Dmitry A.
2009-01-01
The classical approach to diffusion processes is based on Fick's law that the flux is proportional to the concentration gradient. Various phenomena occurring during propagation of penetrating liquids in polymers show that this type of diffusion exhibits anomalous behavior and contradicts the just mentioned law. However, they can be explained in the framework of non-Fickian diffusion theories based on viscoelasticity of polymers. Initial-boundary value problems for viscoelastic diffusion equat...
International Nuclear Information System (INIS)
In this paper, the accuracy of the Frensley inflow boundary condition of the Wigner equation is analyzed in computing the I-V characteristics of a resonant tunneling diode (RTD). It is found that the Frensley inflow boundary condition for incoming electrons holds only exactly infinite away from the active device region and its accuracy depends on the length of contacts included in the simulation. For this study, the non-equilibrium Green's function (NEGF) with a Dirichlet to Neumann mapping boundary condition is used for comparison. The I-V characteristics of the RTD are found to agree between self-consistent NEGF and Wigner methods at low bias potentials with sufficiently large GaAs contact lengths. Finally, the relation between the negative differential conductance (NDC) of the RTD and the sizes of contact and buffer in the RTD is investigated using both methods.
A Cartesian grid embedded boundary method for Poisson`s equation on irregular domains
Energy Technology Data Exchange (ETDEWEB)
Johansen, H. [Univ. of California, Berkeley, CA (United States). Dept. of Mechanical Engineering; Colella, P. [Lawrence Berkeley National Lab., CA (United States). Center for Computational Sciences and Engineering
1997-01-31
The authors 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. They 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 conservation 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 them to use multi-grid iterations with a simple point relaxation strategy. They have combined this with an adaptive mesh refinement (AMR) procedure. They provide evidence that the algorithm is second-order accurate on various exact solutions, and compare the adaptive and non-adaptive calculations.
Biswas, Debabrata; Kumar, Raghwendra
2015-01-01
Numerical solution of the Poisson equation in metallic enclosures, open at one or more ends, is important in many practical situations such as High Power Microwave (HPM) or photo-cathode devices. It requires imposition of a suitable boundary condition at the open end. In this paper, methods for solving the Poisson equation are investigated for various charge densities and aspect ratios of the open ends. It is found that a mixture of second order and third order local asymptotic boundary condition (ABC) is best suited for large aspect ratios while a proposed non-local matching method, based on the solution of the Laplace equation, scores well when the aspect ratio is near unity for all charge density variations, including ones where the centre of charge is close to an open end or the charge density is non-localized. The two methods complement each other and can be used in electrostatic calculations where the computational domain needs to be terminated at the open boundaries of the metallic enclosure.
Energy Technology Data Exchange (ETDEWEB)
Biswas, Debabrata; Singh, Gaurav; Kumar, Raghwendra [Bhabha Atomic Research Centre, Mumbai 400085 (India)
2015-09-15
Numerical solution of the Poisson equation in metallic enclosures, open at one or more ends, is important in many practical situations, such as high power microwave or photo-cathode devices. It requires imposition of a suitable boundary condition at the open end. In this paper, methods for solving the Poisson equation are investigated for various charge densities and aspect ratios of the open ends. It is found that a mixture of second order and third order local asymptotic boundary conditions is best suited for large aspect ratios, while a proposed non-local matching method, based on the solution of the Laplace equation, scores well when the aspect ratio is near unity for all charge density variations, including ones where the centre of charge is close to an open end or the charge density is non-localized. The two methods complement each other and can be used in electrostatic calculations where the computational domain needs to be terminated at the open boundaries of the metallic enclosure.
Biswas, Debabrata; Singh, Gaurav; Kumar, Raghwendra
2015-09-01
Numerical solution of the Poisson equation in metallic enclosures, open at one or more ends, is important in many practical situations, such as high power microwave or photo-cathode devices. It requires imposition of a suitable boundary condition at the open end. In this paper, methods for solving the Poisson equation are investigated for various charge densities and aspect ratios of the open ends. It is found that a mixture of second order and third order local asymptotic boundary conditions is best suited for large aspect ratios, while a proposed non-local matching method, based on the solution of the Laplace equation, scores well when the aspect ratio is near unity for all charge density variations, including ones where the centre of charge is close to an open end or the charge density is non-localized. The two methods complement each other and can be used in electrostatic calculations where the computational domain needs to be terminated at the open boundaries of the metallic enclosure.
Numerical solution for Laplace equation with mixed boundary condition for ship problem in the sea
Silalahi, Fitriani Tupa R.; Budhi, Wono Setya; Adytia, Didit; van Groesen, E.
2015-09-01
One interesting phenomena is investigating the movement of ships at the sea. To start with the investigation in modelling of this problem, we will assume that the ship is only a one-dimensional object that is floating on the sea surface. Similarly, we assume that the water flow is uniform in parallel directions to the ship. Therefore, we simply use the two-dimensional Laplace equation in this problem. In the section that describes the surface of sea, Neumann boundary condition is imposed in part related to the ship and the Dirichlet boundary condition for others. Then on the other three boundaries, we imposed the Neumann boundary condition by assuming that the water does not flow on the bottom, and both end. The model is solved by numerical solution using the finite element method. Velocity potential solution on the whole domain is demonstrated as a result of the implementation of the finite element method. In this paper, we initiate an investigation with assuming that the ship is on the water so that the domain of the Laplace equation is rectangular. Then we assume the drift ship. Furthermore, we also study the dependence of width and depth of the domain to the velocity potential.
Stability analysis of Boundary Layer in Poiseuille Flow Through A Modified Orr-Sommerfeld Equation
Monwanou, A V; Orou, J B Chabi; 10.5539/apr.v4n4p138
2013-01-01
For applications regarding transition prediction, wing design and control of boundary layers, the fundamental understanding of disturbance growth in the flat-plate boundary layer is an important issue. In the present work we investigate the stability of boundary layer in Poiseuille flow. We normalize pressure and time by inertial and viscous effects. The disturbances are taken to be periodic in the spanwise direction and time. We present a set of linear governing equations for the parabolic evolution of wavelike disturbances. Then, we derive modified Orr-Sommerfeld equations that can be applied in the layer. Contrary to what one might think, we find that Squire's theorem is not applicable for the boundary layer. We find also that normalization by inertial or viscous effects leads to the same order of stability or instability. For the 2D disturbances flow ($\\theta=0$), we found the same critical Reynolds number for our two normalizations. This value coincides with the one we know for neutral stability of the k...
Integrated care in the daily work: coordination beyond organisational boundaries
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Alexandra Petrakou
2009-07-01
Full Text Available Objectives: In this paper, integrated care in an inter-organisational cooperative setting of in-home elderly care is studied. The aim is to explore how home care workers coordinate their daily work, identify coordination issues in situ and discuss possible actions for supporting seamless and integrated elderly care at home. Method: The empirical findings are drawn from an ethnographic workplace study of the cooperation and coordination taking place between home care workers in a Swedish county. Data were collected through observational studies, interviews and group discussions. Findings: The paper identifies a need to support two core issues. Firstly, it must be made clear how the care interventions that are currently defined as ‘self-treatment’ by the home health care should be divided. Secondly, the distributed and asynchronous coordination between all care workers involved, regardless of organisational belonging must be better supported. Conclusion: Integrated care needs to be developed between organisations as well as within each organisation. As a matter of fact, integrated care needs to be built up beyond organisational boundaries. Organisational boundaries affect the planning of the division of care interventions, but not the coordination during the home care process. During the home care process, the main challenge is the coordination difficulties that arise from the fact that workers are distributed in time and/or space, regardless of organisational belonging. A core subject for future practice and research is to develop IT tools that reach beyond formal organisational boundaries and processes while remaining adaptable in view of future structure changes.
Extension to the integral transport equation of an iterative method
Energy Technology Data Exchange (ETDEWEB)
Jehouani, A. [EPRA, Department of Physics, Faculty of Sciences, Semlalia, PO Box 2390, 40000 Marrakech (Morocco)]. E-mail: jehouani@ucam.ac.ma; Elmorabiti, A. [Centre d' Etudes Nucleaires de Maamoura, CNESTEN (Morocco); Ghassoun, J. [EPRA, Department of Physics, Faculty of Sciences, Semlalia, PO Box 2390, 40000 Marrakech (Morocco)
2006-09-15
In this paper we describe an extension to the neutron integral transport equation of an iterative method. Indeed an iterative scheme is used for both energy and space including external iteration for the multiplication factor and internal iteration for flux calculations. The Monte Carlo method is used to evaluate the spatial transfer integrals. The results were compared with those obtained by using the APOLLO2 code for a cylindrical cell.
Extension to the integral transport equation of an iterative method
International Nuclear Information System (INIS)
In this paper we describe an extension to the neutron integral transport equation of an iterative method. Indeed an iterative scheme is used for both energy and space including external iteration for the multiplication factor and internal iteration for flux calculations. The Monte Carlo method is used to evaluate the spatial transfer integrals. The results were compared with those obtained by using the APOLLO2 code for a cylindrical cell
Energy Technology Data Exchange (ETDEWEB)
Corcelli, S.A.; Kress, J.D.; Pratt, L.R.
1995-08-07
This paper develops and characterizes mixed direct-iterative methods for boundary integral formulations of continuum dielectric solvation models. We give an example, the Ca{sup ++}{hor_ellipsis}Cl{sup {minus}} pair potential of mean force in aqueous solution, for which a direct solution at thermal accuracy is difficult and, thus for which mixed direct-iterative methods seem necessary to obtain the required high resolution. For the simplest such formulations, Gauss-Seidel iteration diverges in rare cases. This difficulty is analyzed by obtaining the eigenvalues and the spectral radius of the non-symmetric iteration matrix. This establishes that those divergences are due to inaccuracies of the asymptotic approximations used in evaluation of the matrix elements corresponding to accidental close encounters of boundary elements on different atomic spheres. The spectral radii are then greater than one for those diverging cases. This problem is cured by checking for boundary element pairs closer than the typical spatial extent of the boundary elements and for those cases performing an ``in-line`` Monte Carlo integration to evaluate the required matrix elements. These difficulties are not expected and have not been observed for the thoroughly coarsened equations obtained when only a direct solution is sought. Finally, we give an example application of hybrid quantum-classical methods to deprotonation of orthosilicic acid in water.
Dault, Daniel Lawrence
The moment method is the predominant approach for the solution of electromagnetic boundary integral equations. Traditional moment method discretizations rely on the projection of solution currents onto basis sets that must satisfy strict continuity properties to model physical currents. The choice of basis sets is further restricted by the tight coupling of traditional functional descriptions to the underlying geometrical approximation of the scattering or radiating body. As a result, the choice of approximation function spaces and geometry discretizations for a given boundary integral equation is significantly limited. A quasi-meshless partition of unity based method called the Generalized Method of Moments (GMM) was recently introduced to overcome some of these limitations. The GMM partition of unity scheme affords automatic continuity of solution currents, and therefore permits the use of a much wider range of basis functions than traditional moment methods. However, prior to the work in this thesis, GMM was limited in practical applicability because it was only formulated for a few geometry types, could not be accurately applied to arbitrary scatterers, e.g. those with mixtures of geometrical features, and was not amenable to traditional acceleration methodologies that would permit its application to electrically large problems. The primary contribution of this thesis is to introduce several new GMM formulations that significantly expand the capabilities of the method to make it a practical, broadly applicable approach for solving boundary integral equations and overcoming the limitations inherent in traditional moment method discretizations. Additionally, several of the topics covered address continuing open problems in electromagnetic boundary integral equations with applicability beyond GMM. The work comprises five broad contributions. The first is a new GMM formulation capable of mixing both GMM-type basis sets and traditional basis sets in the same
Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations
Durrett, Rick
2009-01-01
We consider a branching-selection system in $\\rr$ with $N$ particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as $N\\to\\infty$, the empirical measure process associated to the system converges in distribution to a deterministic measure-valued process whose densities solve a free boundary integro-differential equation. We also show that this equation has a unique traveling wave solution traveling at speed $c$ or no such solution depending on whether $c>a$ or $c\\leq a$, where $a$ is the asymptotic speed of the branching random walk obtained by ignoring the removal of the leftmost particles in our process. The traveling wave solutions correspond to solutions of Wiener-Hopf equations.
Difference Scheme for Hyperbolic Heat Conduction Equation with Pulsed Heating Boundary
Institute of Scientific and Technical Information of China (English)
LiJi; ZhangZhengfang; 等
2000-01-01
In this paper,the authors have analyzed the second-order hyperbolic differential equation with pulsed heating boundary,and tried to solve this kind of equation with numerical method.After analyzing the error of the difference scheme and comparing the numerical results with the theoretical results.The authors present some examples to show how the thermal wave propagates in materials.By analyzing the calculation results,the conditions to observe the thermal wave phenomena in the laboratory are conferred.Finally the heat transfter in a complex combined structure is calculated with this method.The result is quite different from the calculated result from the aprabolic heat conduction equation.
On energy boundary layer equations in power law non-Newtonian fluids
Institute of Scientific and Technical Information of China (English)
郑连存; 张欣欣
2008-01-01
The hear transfer mechanism and the constitutive models for energy boundary layer in power law fluids were investigated.Two energy transfer constitutive equations models were proposed based on the assumption of similarity of velocity field momentum diffusion and temperature field heat transfer.The governing systems of partial different equations were transformed into ordinary differential equations respectively by using the similarity transformation group.One model was assumed that Prandtl number is a constant,and the other model was assumed that viscosity diffusion is analogous to thermal diffusion.The solutions were presented analytically and numerically by using the Runge-Kutta formulas and shooting technique and the associated transfer characteristics were discussed.
ADAPTIVE INTERVAL WAVELET PRECISE INTEGRATION METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
MEI Shu-li; LU Qi-shao; ZHANG Sen-wen; JIN Li
2005-01-01
The quasi-Shannon interval wavelet is constructed based on the interpolation wavelet theory, and an adaptive precise integration method, which is based on extrapolation method is presented for nonlinear ordinary differential equations (ODEs). And then, an adaptive interval wavelet precise integration method (AIWPIM) for nonlinear partial differential equations(PDEs) is proposed. The numerical results show that the computational precision of AIWPIM is higher than that of the method constructed by combining the wavelet and the 4th Runge-Kutta method, and the computational amounts of these two methods are almost equal. For convenience, the Burgers equation is taken as an example in introducing this method, which is also valid for more general cases.
Integrated vehicle dynamics control using State Dependent Riccati Equations
Bonsen, B.; Mansvelders, R.; Vermeer, E.
2010-01-01
In this paper we discuss a State Dependent Riccati Equations (SDRE) solution for Integrated Vehicle Dynamics Control (IVDC). The SDRE approach is a nonlinear variant of the well known Linear Quadratic Regulator (LQR) and implements a quadratic cost function optimization. A modified version of this m
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.
Directory of Open Access Journals (Sweden)
Timurkhan S. Aleroev
2013-12-01
Full Text Available We consider a linear heat equation involving a fractional derivative in time, with a nonlocal boundary condition. We determine a source term independent of the space variable, and the temperature distribution for a problem with an over-determining condition of integral type. We prove the existence and uniqueness of the solution, and its continuous dependence on the data.
Amirkhanov, I V; Sarker, N R; Sarhadov, I
2004-01-01
In this work the solutions of different boundary-value problems are retrieved analytically and numerically for the equation of high order with small parameter at a higher derivative. The analysis of these solutions is given. It is found that for some variants of symmetric boundary conditions the solutions of a boundary-value problem for the equations of the 4th, 6th, $\\ldots$ orders transfer into the solution of a Schrödinger equation at $\\varepsilon \\to 0$ ($\\varepsilon $ is small parameter). The retrieved solutions with different knots are orthogonal among themselves. The results of numerical calculations are given.
Leise, Tanya L.
2009-08-19
We consider the problem of the dynamic, transient propagation of a semi-infinite, mode I crack in an infinite elastic body with a nonlinear, viscoelastic cohesize zone. Our problem formulation includes boundary conditions that preclude crack face interpenetration, in contrast to the usual mode I boundary conditions that assume all unloaded crack faces are stress-free. The nonlinear viscoelastic cohesive zone behavior is motivated by dynamic fracture in brittle polymers in which crack propagation is preceeded by significant crazing in a thin region surrounding the crack tip. We present a combined analytical/numerical solution method that involves reducing the problem to a Dirichlet-to-Neumann map along the crack face plane, resulting in a differo-integral equation relating the displacement and stress along the crack faces and within the cohesive zone. © 2009 Springer Science+Business Media B.V.
Wang, K.; Liu, F. C.; Xue, P.; Wang, D.; Xiao, B. L.; Ma, Z. Y.
2016-01-01
Fifteen Al-Mg-Sc samples with subgrain/grain sizes in the range of 1.8 to 4.9 μm were prepared through the processing methods of friction stir processing (FSP), equal-channel-angular pressing (ECAP), rolling, annealing, and combinations of the above. The percentages of high-angle grain boundaries (HAGBs) of these fine-grained alloys were distributed from 39 to 97 pct. The samples processed through FSP had a higher percentage of HAGBs compared to other samples. Superplasticity was achieved in all fifteen samples, but the FSP samples exhibited better superplasticity than other samples because their fine equiaxed grains, which were mostly surrounded by HAGBs, were conducive to the occurrence of grain boundary sliding (GBS) during superplastic deformation. The dominant deformation mechanism was the same for all fifteen samples, i.e., GBS controlled by grain boundary diffusion. However, the subgrains were the GBS units for the rolled or ECAP samples, which contained high percentages of unrecrystallized grains, whereas the fine grains were the GBS units for the FSP samples. Superplastic data analysis revealed that the dimensionless A in the classical constitutive equation for superplasticity of fine-grained Al alloys was not a constant, but increased with an increase in the percentage of HAGBs, demonstrating that the enhanced superplastic deformation kinetics can be ascribed to the high percentage of HAGBs. A modified superplastic constitutive equation with the percentage of HAGBs as a new microstructural parameter was established.
Arbitrary Difference Precise Integration Method for Solving the Seismic Wave Equation
Institute of Scientific and Technical Information of China (English)
Jia Xiaofeng; Wang Runqiu; Hu Tianyue
2004-01-01
Wave equation migration is often applied to solve seismic imaging problems. Usually, the finite difference method is used to obtain the numerical solution of the wave equation. In this paper,the arbitrary difference precise integration (ADPI) method is discussed and applied in seismic migration. The ADPI method has its own distinctive idea. When dispersing coordinates in the space domain, it employs a relatively unrestrained form instead of the one used by the conventional finite difference method. Moreover, in the time domain it adopts the sub-domain precise integration method. As a result, it not only takes the merits of high precision and narrow bandwidth, but also can process various boundary conditions and describe the feature of an inhomogeneous medium better. Numerical results show the benefit of the presented algorithm using the ADPI method.
Rational first integrals of geodesic equations and generalised hidden symmetries
Aoki, Arata; Tomoda, Kentaro
2016-01-01
We discuss novel generalisations of Killing tensors, which are introduced by considering rational first integrals of geodesic equations. We introduce the notion of inconstructible generalised Killing tensors, which cannot be constructed from ordinary Killing tensors. Moreover, we introduce inconstructible rational first integrals, which are constructed from inconstructible generalised Killing tensors, and provide a method for checking the inconstructibility of a rational first integral. Using the method, we show that the rational first integral of the Collinson-O'Donnell solution is not inconstructible. We also provide several examples of metrics admitting an inconstructible rational first integral in two and four dimensions, by using the Maciejewski-Przybylska system. Furthermore, we attempt to generalise other hidden symmetries such as Killing-Yano tensors.
Institute of Scientific and Technical Information of China (English)
Qi-yuan Chen; Tao Tang; Zhen-huan Teng
2004-01-01
The aim of this paper is to develop a fast numerical method for two-dimensional boundary integral equations of the first kind with logarithm kernels when the boundary of the domain is smooth and closed. In this case, the use of the conventional boundary element methods gives linear systems with dense matrix. In this paper, we demonstrate that the dense matrix can be replaced by a sparse one if appropriate graded meshes are used in the quadrature rules. It will be demonstrated that this technique can increase the numerical efficiency significantly.
Institute of Scientific and Technical Information of China (English)
Qingfeng ZHU; Yufeng SHI
2012-01-01
Backward doubly stochastic differential equations driven by Brownian motions and Poisson process (BDSDEP) with non-Lipschitz coefficients on random time interval are studied.The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations (SPDIEs) is treated with BDSDEP.Under non-Lipschitz conditions,the existence and uniqueness results for measurable solutions to BDSDEP are established via the smoothing technique.Then,the continuous dependence for solutions to BDSDEP is derived.Finally,the probabilistic interpretation for the solutions to a class of quasilinear SPDIEs is given.
Lagrangian structures, integrability and chaos for 3D dynamical equations
International Nuclear Information System (INIS)
In this paper, we consider the general setting for constructing action principles for three-dimensional first-order autonomous equations. We present the results for some integrable and non-integrable cases of the Lotka-Volterra equation, and show Lagrangian descriptions which are valid for systems satisfying Shil'nikov criteria on the existence of strange attractors, though chaotic behaviour has not been verified up to now. The Euler-Lagrange equations we get for these systems usually present 'time reparametrization' invariance, though other kinds of invariance may be found according to the kernel of the associated symplectic 2-form. The formulation of a Hamiltonian structure (Poisson brackets and Hamiltonians) for these systems from the Lagrangian viewpoint leads to a method of finding new constants of the motion starting from known ones, which is applied to some systems found in the literature known to possess a constant of the motion, to find the other and thus showing their integrability. In particular, we show that the so-called ABC system is completely integrable if it possesses one constant of the motion
Coarse projective kMC integration: forward/reverse initial and boundary value problems
International Nuclear Information System (INIS)
In 'equation-free' multiscale computation a dynamic model is given at a fine, microscopic level; yet we believe that its coarse-grained, macroscopic dynamics can be described by closed equations involving only coarse variables. These variables are typically various low-order moments of the distributions evolved through the microscopic model. We consider the problem of integrating these unavailable equations by acting directly on kinetic Monte Carlo microscopic simulators, thus circumventing their derivation in closed form. In particular, we use projective multi-step integration to solve the coarse initial value problem forward in time as well as backward in time (under certain conditions). Macroscopic trajectories are thus traced back to unstable, source-type, and even sometimes saddle-like stationary points, even though the microscopic simulator only evolves forward in time. We also demonstrate the use of such projective integrators in a shooting boundary value problem formulation for the computation of 'coarse limit cycles' of the macroscopic behavior, and the approximation of their stability through estimates of the leading 'coarse Floquet multipliers'
Yang, Zhiguo; Rong, Zhijian; Wang, Bo; Zhang, Baile
2015-01-01
In this paper, we present an efficient spectral-element method (SEM) for solving general two-dimensional Helmholtz equations in anisotropic media, with particular applications in accurate simulation of polygonal invisibility cloaks, concentrators and circular rotators arisen from the field of transformation electromagnetics (TE). In practice, we adopt a transparent boundary condition (TBC) characterized by the Dirichlet-to-Neumann (DtN) map to reduce wave propagation in an unbounded domain to a bounded domain. We then introduce a semi-analytic technique to integrate the global TBC with local curvilinear elements seamlessly, which is accomplished by using a novel elemental mapping and analytic formulas for evaluating global Fourier coefficients on spectral-element grids exactly. From the perspective of TE, an invisibility cloak is devised by a singular coordinate transformation of Maxwell's equations that leads to anisotropic materials coating the cloaked region to render any object inside invisible to observe...
Boundary behavior and estimates for solutions of equations containing the $p$-laplacian
Directory of Open Access Journals (Sweden)
Jacqueline Fleckinger
1999-09-01
Full Text Available We use ``Hardy-type'' inequalities to derive $L^q$ estimates for solutions of equations containing the $p$-Laplacian with $p>1$. We begin by deriving some inequalities using elementary ideas from an early article [B3] which has been largely overlooked. Then we derive $L^q$ estimates of the boundary behavior of test functions of finite energy, and consequently of principal (positive eigenfunctions of functionals containing the $p$-Laplacian. The estimates contain exponents known to be sharp when $p=2$. These lead to estimates of the effect of boundary perturbation on the fundamental eigenvalue. Finally, we present global $L^q$ estimates of solutions of the Cauchy problem for some initial-value problems containing the $p$-Laplacian.
Numerical study of a parametric parabolic equation and a related inverse boundary value problem
Mustonen, Lauri
2016-10-01
We consider a time-dependent linear diffusion equation together with a related inverse boundary value problem. The aim of the inverse problem is to determine, based on observations on the boundary, the nonhomogeneous diffusion coefficient in the interior of an object. The method in this paper relies on solving the forward problem for a whole family of diffusivities by using a spectral Galerkin method in the high-dimensional parameter domain. The evaluation of the parametric solution and its derivatives is then completely independent of spatial and temporal discretizations. In the case of a quadratic approximation for the parameter dependence and a direct solver for linear least squares problems, we show that the evaluation of the parametric solution does not increase the complexity of any linearized subproblem arising from a Gauss-Newtonian method that is used to minimize a Tikhonov functional. The feasibility of the proposed algorithm is demonstrated by diffusivity reconstructions in two and three spatial dimensions.
Blow-up in p-Laplacian heat equations with nonlinear boundary conditions
Ding, Juntang; Shen, Xuhui
2016-10-01
In this paper, we investigate the blow-up of solutions to the following p-Laplacian heat equations with nonlinear boundary conditions: {l@{quad}l}(h(u))_t =nabla\\cdot(|nabla u|pnabla u)+k(t)f(u) &{in } Ω×(0,t^{*}), |nabla u|ppartial u/partial n=g(u) &on partialΩ×(0,t^{*}), u(x,0)=u0(x) ≥ 0 & {in } overline{Ω},. where {p ≥ 0} and {Ω} is a bounded convex domain in {RN}, {N ≥ 2} with smooth boundary {partialΩ}. By constructing suitable auxiliary functions and using a first-order differential inequality technique, we establish the conditions on the nonlinearities and data to ensure that the solution u( x, t) blows up at some finite time. Moreover, the upper and lower bounds for the blow-up time, when blow-up does occur, are obtained.
Advances in the study of boundary value problems for nonlinear integrable PDEs
Pelloni, Beatrice
2015-02-01
In this review I summarize some of the most significant advances of the last decade in the analysis and solution of boundary value problems for integrable partial differential equations (PDEs) in two independent variables. These equations arise widely in mathematical physics, and in order to model realistic applications, it is essential to consider bounded domain and inhomogeneous boundary conditions. I focus specifically on a general and widely applicable approach, usually referred to as the unified transform or Fokas transform, that provides a substantial generalization of the classical inverse scattering transform. This approach preserves the conceptual efficiency and aesthetic appeal of the more classical transform approaches, but presents a distinctive and important difference. While the inverse scattering transform follows the ‘separation of variables’ philosophy, albeit in a nonlinear setting, the unified transform is based on the idea of synthesis, rather than separation, of variables. I will outline the main ideas in the case of linear evolution equations, and then illustrate their generalization to certain nonlinear cases of particular significance.
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
This paper is concerned with the existence of positive solutions of two-point Dirichlet singular and nonsingular boundary problems for second-order quasi-linear differential equations with changing sign nonlinearities.
Solutions and Multiple Solutions for p(x)-Laplacian Equations with Nonlinear Boundary Condition
Institute of Scientific and Technical Information of China (English)
Zifei SHEN; Chenyin QIAN
2009-01-01
The authors study the p(x)-Laplacian equations with nonlinear boundary condition.By using the variational method,under appropriate assumptions on the perturbation terms f1(x,u),f2(x,u) and h1(x),h2(x),such that the associated functional satisfies the "mountain pass lemma" and "fountain theorem" respectively,the existence and multiplicity of solutions are obtained.The discussion is based on the theory of variable exponent Lebesgue and Sobolev spaces.
Kuiper, Logan K
2016-01-01
An approximate solution to the two dimensional Navier Stokes equation with periodic boundary conditions is obtained by representing the x any y components of fluid velocity with complex Fourier basis vectors. The chosen space of basis vectors is finite to allow for numerical calculations, but of variable size. Comparisons of the resulting approximate solutions as they vary with the size of the chosen vector space allow for extrapolation to an infinite basis vector space. Results suggest that such a solution, with the full basis vector space and which would give the exact solution, would fail for certain initial velocity configurations when initial velocity and time t exceed certain limits.
Hamiltonian Formalism of mKdV Equation with Non-vanishing Boundary Values
Institute of Scientific and Technical Information of China (English)
HE Jing-Song; CHEN Shi-Rong
2005-01-01
Hamiltonian formalism of the mKdV equation with non-vanishing boundary valueis re-examined by a revised form of the standard procedure. It is known that the previous papers did not give the final results and involved some questionable points [T.C. Au Yeung and P.C.W. Fung, J. Phys. A 21 (1988) 3575]. In this note, simple results are obtained in terms of an affine parameter and a Galileo transformation is introduced to ensure the results compatible with those derived from the inverse scattering transform.
Burlutskaya, M. Sh.
2014-01-01
The Fourier method is used to find a classical solution of the mixed problem for a first-order differential equation with involution and periodic boundary conditions. The application of the Fourier method is substantiated using refined asymptotic formulas obtained for the eigenvalues and eigenfunctions of the corresponding spectral problem. The Fourier series representing the formal solution is transformed using certain techniques, and the possibility of its term-by-term differentiation is proved. Minimal requirements are imposed on the initial data of the problem.
Solution of the time-dependent Schr(o)dinger equation with absorbing boundary conditions
Institute of Scientific and Technical Information of China (English)
Chen Zhidong; Zhang Jinyu; Yu Zhiping
2009-01-01
The performances of absorbing boundary conditions (ABCs) in four widely used finite difference time domain (FDTD) methods, I.e. Explicit, implicit, explicit staggered-time, and Chebyshev methods, for solving the time-dependent Schr(o)dinger equation are assessed and compared. The computation efficiency for each approach is also evaluated. A typical evolution problem of a single Gaussian wave packet is chosen to demonstrate the perfor-mances of the four methods combined with ABCs. It is found that ABCs perfectly eliminate reflection in implicit and explicit staggered-time methods. However, small reflection still exists in explicit and Chebyshev methods even though ABCs are applied.
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
Stability and Bifurcation in a Delayed Reaction-Diffusion Equation with Dirichlet Boundary Condition
Guo, Shangjiang; Ma, Li
2016-04-01
In this paper, we study the dynamics of a diffusive equation with time delay subject to Dirichlet boundary condition in a bounded domain. The existence of spatially nonhomogeneous steady-state solution is investigated by applying Lyapunov-Schmidt reduction. The existence of Hopf bifurcation at the spatially nonhomogeneous steady-state solution is derived by analyzing the distribution of the eigenvalues. The direction of Hopf bifurcation and stability of the bifurcating periodic solution are also investigated by means of normal form theory and center manifold reduction. Moreover, we illustrate our general results by applications to the Nicholson's blowflies models with one- dimensional spatial domain.
Institute of Scientific and Technical Information of China (English)
Sergio VESSELLA
2005-01-01
Let Г be a portion of a C1,α boundary of an n-dimensional domain D. Let u be a solution to a second order parabolic equation in D × (-T,T) and assume that u = 0 on Г× (-T,T), 0 ∈Г. We result we prove that if u (x,t) = O (|x|k) for every t ∈ (-T,T) and every k ∈ N, then u is identically equal to zero.
Integrability + Supersymmetry + Boundary: Life on the edge is not so dull after all!
Nepomechie, R I
2004-01-01
After a brief review of integrability, first in the absence and then in the presence of a boundary, I outline the construction of actions for the N=1 and N=2 boundary sine-Gordon models. The key point is to introduce Fermionic boundary degrees of freedom in the boundary actions.
Directory of Open Access Journals (Sweden)
A. A. Dosiyev
2010-01-01
Full Text Available The block-grid method (see Dosiyev, 2004 for the solution of the Dirichlet problem on polygons, when a boundary function on each side of the boundary is given from C2,λ, 0<λ<1, is analized. In the integral represetations around each singular vertex, which are combined with the uniform grids on "nonsingular" part the boundary conditions are taken into account with the help of integrals of Poisson type for a half-plane. It is proved that the final uniform error is of order O(h2+ε, where ε is the error of the approximation of the mentioned integrals, h is the mesh step. For the p-order derivatives (p=0,1,… of the difference between the approximate and the exact solution in each "singular" part O((h2+εrj1/αj-p order is obtained, here rj is the distance from the current point to the vertex in question, αjπ is the value of the interior angle of the jth vertex. Finally, the method is illustrated by solving the problem in L-shaped polygon, and a high accurate approximation for the stress intensity factor is given.
Directory of Open Access Journals (Sweden)
Dosiyev AA
2010-01-01
Full Text Available Abstract The block-grid method (see Dosiyev, 2004 for the solution of the Dirichlet problem on polygons, when a boundary function on each side of the boundary is given from , , is analized. In the integral represetations around each singular vertex, which are combined with the uniform grids on "nonsingular" part the boundary conditions are taken into account with the help of integrals of Poisson type for a half-plane. It is proved that the final uniform error is of order , where is the error of the approximation of the mentioned integrals, is the mesh step. For the -order derivatives ( of the difference between the approximate and the exact solution in each "singular" part order is obtained, here is the distance from the current point to the vertex in question, is the value of the interior angle of the th vertex. Finally, the method is illustrated by solving the problem in L-shaped polygon, and a high accurate approximation for the stress intensity factor is given.
Daniel Sevcovic
2008-01-01
The purpose of this survey chapter is to present a transformation technique that can be used in analysis and numerical computation of the early exercise boundary for an American style of vanilla options that can be modelled by class of generalized Black-Scholes equations. We analyze qualitatively and quantitatively the early exercise boundary for a linear as well as a class of nonlinear Black-Scholes equations with a volatility coefficient which can be a nonlinear function of the second deriv...
Directory of Open Access Journals (Sweden)
Zhenlai Han
2012-11-01
Full Text Available In this article, we consider the following boundary-value problem of nonlinear fractional differential equation with $p$-Laplacian operator $$displaylines{ D_{0+}^eta(phi_p(D_{0+}^alpha u(t+a(tf(u=0, quad 0
Li, Fang; Liang, Xing; Shen, Wenxian
2016-08-01
In this series of papers, we investigate the spreading and vanishing dynamics of time almost periodic diffusive KPP equations with free boundaries. Such equations are used to characterize the spreading of a new species in time almost periodic environments with free boundaries representing the spreading fronts. In the first part of the series, we showed that a spreading-vanishing dichotomy occurs for such free boundary problems (see [16]). In this second part of the series, we investigate the spreading speeds of such free boundary problems in the case that the spreading occurs. We first prove the existence of a unique time almost periodic semi-wave solution associated to such a free boundary problem. Using the semi-wave solution, we then prove that the free boundary problem has a unique spreading speed.
Introduction to stochastic analysis integrals and differential equations
Mackevicius, Vigirdas
2013-01-01
This is an introduction to stochastic integration and stochastic differential equations written in an understandable way for a wide audience, from students of mathematics to practitioners in biology, chemistry, physics, and finances. The presentation is based on the naïve stochastic integration, rather than on abstract theories of measure and stochastic processes. The proofs are rather simple for practitioners and, at the same time, rather rigorous for mathematicians. Detailed application examples in natural sciences and finance are presented. Much attention is paid to simulation diffusion pro
Directory of Open Access Journals (Sweden)
Jianxun Su
2015-01-01
Full Text Available A space-domain integral equation method accelerated with adaptive cross approximation (ACA is presented for the fast and accurate analysis of electromagnetic (EM scattering from multilayered metallic photonic crystal (MPC. The method directly solves for the electric field in order to easily enable the periodic boundary condition (PBC in the spatial domain. The ACA is a purely algebraic method allowing the compression of fully populated matrices; hence, its formulation and implementation are independent of integral equation kernel (Green’s function. Therefore, the ACA is very well suited for accelerating integral equation analysis of periodic structure with the integral kernel of the periodic Green’s function (PGF. The computation of the spatial-domain periodic Green’s function (PGF is accelerated by the modified Ewald transformation, such that the multilayered periodic structure can be analyzed efficiently and accurately. An effective interpolation method is also proposed to fast compute the periodic Green’s function, which can greatly reduce the time of matrix filling. Numerical examples show that the proposed method can greatly save the frequency sweep time for multilayered periodic structure.
WAVEFORM RELAXATION METHODS OF NONLINEAR INTEGRAL-DIFFERENTIAL-ALGEBRAIC EQUATIONS
Institute of Scientific and Technical Information of China (English)
Yao-lin Jiang
2005-01-01
In this paper we consider continuous-time and discrete-time waveform relaxation methods for general nonlinear integral-differential-algebraic equations. For the continuous-time case we derive the convergence condition of the iterative methods by invoking the spectral theory on the resulting iterative operators. By use of the implicit difference forms,namely the backward-differentiation formulae, we also yield the convergence condition of the discrete waveforms. Numerical experiments are provided to illustrate the theoretical work reported here.