parabolic equation with integral boundary condition
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M. Denche
2001-01-01
Full Text Available In this paper we study the problem of control by the initial conditions of the heat equation with an integral boundary condition. This problem is ill-posed. Perturbing the final condition we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates.
APPLICATION OF BOUNDARY INTEGRAL EQUATION METHOD FOR THERMOELASTICITY PROBLEMS
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Vorona Yu.V.
2015-12-01
Full Text Available Boundary Integral Equation Method is used for solving analytically the problems of coupled thermoelastic spherical wave propagation. The resulting mathematical expressions coincide with the solutions obtained in a conventional manner.
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Nieto JuanJ
2009-01-01
Full Text Available This paper deals with some existence results for a boundary value problem involving a nonlinear integrodifferential equation of fractional order with integral boundary conditions. Our results are based on contraction mapping principle and Krasnosel'skiĭ's fixed point theorem.
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Yumei Zou
2017-01-01
Full Text Available This paper deals with the integral boundary value problems of fractional differential equations at resonance. By Mawhin’s coincidence degree theory, we present some new results on the existence of solutions for a class of differential equations of fractional order with integral boundary conditions at resonance. An example is also included to illustrate the main results.
DEFF Research Database (Denmark)
Jha, Abhinav K.; Zhu, Yansong; Kang, Jin U.
2016-01-01
An integral Neumann-series implementation of the Radiative Transfer Equation that accounts for boundary conditions is proposed to simulate photon transport through tissue for transcranial optical imaging.......An integral Neumann-series implementation of the Radiative Transfer Equation that accounts for boundary conditions is proposed to simulate photon transport through tissue for transcranial optical imaging....
Singularity Preserving Numerical Methods for Boundary Integral Equations
Kaneko, Hideaki (Principal Investigator)
1996-01-01
In the past twelve months (May 8, 1995 - May 8, 1996), under the cooperative agreement with Division of Multidisciplinary Optimization at NASA Langley, we have accomplished the following five projects: a note on the finite element method with singular basis functions; numerical quadrature for weakly singular integrals; superconvergence of degenerate kernel method; superconvergence of the iterated collocation method for Hammersteion equations; and singularity preserving Galerkin method for Hammerstein equations with logarithmic kernel. This final report consists of five papers describing these projects. Each project is preceeded by a brief abstract.
Costabel, M.; Ervin, V. J.; Stephan, E. P.
1990-07-01
Previously Costabel and Stephan proved the convergence of the collocation method for boundary integral equations on polygonal domains for piecewise linear trial functions which are constant on subintervals next to corners. The convergence and associated error estimates were given in suitable Sobolev spaces with appropriately weighted norms. In this paper we present, for Laplace's equation, the implementation of their method and a slightly modified version. In the latter we use piecewise linear trial functions which are discontinuous at the corners. Of particular note is that the computed experimental convergence rates are in complete agreement with the predicted theoretical rates. In particular, our numerical results underline clearly how the order of convergence depends on the graded mesh.
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
The collocation method for first-kind boundary integral equations on polygonal regions
Yan, Yi
1990-01-01
In this paper the collocation method for first-kind boundary integral equations, by using piecewise constant trial functions with uniform mesh, is shown to be equivalent to a projection method for second-kind Fredholm equations. In a certain sense this projection is an interpolation projection. By introducing this technique of analysis, we particularly consider the case of polygonal boundaries. We give asymptotic error estimates in {L_2} norm on the boundaries, and some superconvergence results for the single layer potential.
Hochstadt, Harry
2011-01-01
This classic work is now available in an unabridged paperback edition. Hochstatdt's concise treatment of integral equations represents the best compromise between the detailed classical approach and the faster functional analytic approach, while developing the most desirable features of each. The seven chapters present an introduction to integral equations, elementary techniques, the theory of compact operators, applications to boundary value problems in more than dimension, a complete treatment of numerous transform techniques, a development of the classical Fredholm technique, and applicatio
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.
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Shayma Adil Murad
2011-01-01
Full Text Available We study the existence and uniqueness of the solutions of mixed Volterra-Fredholm type integral equations with integral boundary condition in Banach space. Our analysis is based on an application of the Krasnosel'skii fixed-point theorem.
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Feng Hanying
2011-01-01
Full Text Available The expression and properties of Green's function for a class of nonlinear fractional differential equations with integral boundary conditions are studied and employed to obtain some results on the existence of positive solutions by using fixed point theorem in cones. The proofs are based on the reduction of the problem considered to the equivalent Fredholm integral equation of the second kind. The results significantly extend and improve many known results even for integer-order cases.
Analyzing diffraction gratings by a boundary integral equation Neumann-to-Dirichlet map method.
Wu, Yumao; Lu, Ya Yan
2009-11-01
For analyzing diffraction gratings, a new method is developed based on dividing one period of the grating into homogeneous subdomains and computing the Neumann-to-Dirichlet (NtD) maps for these subdomains by boundary integral equations. For a subdomain, the NtD operator maps the normal derivative of the wave field to the wave field on its boundary. The integral operators used in this method are simple to approximate, since they involve only the standard Green's function of the Helmholtz equation in homogeneous media. The method retains the advantages of existing boundary integral equation methods for diffraction gratings but avoids the quasi-periodic Green's functions that are expensive to evaluate.
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...
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Yang Xiao-Jun
2015-01-01
Full Text Available In the present paper we investigate the fractal boundary value problems for the Fredholm\\Volterra integral equations, heat conduction and wave equations by using the local fractional decomposition method. The operator is described by the local fractional operators. The four illustrative examples are given to elaborate the accuracy and reliability of the obtained results. [Projekat Ministarstva nauke Republike Srbije, br. OI 174001, III41006 i br. TI 35006
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Jiqiang Jiang
2012-01-01
Full Text Available We consider the existence of positive solutions for a class of nonlinear integral boundary value problems for fractional differential equations. By using some fixed point theorems, the existence and multiplicity results of positive solutions are obtained. The results obtained in this paper improve and generalize some well-known results.
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.
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
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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.
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
A Boundary Value Problem with Multivariables Integral Type Condition for Parabolic Equations
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A. L. Marhoune
2009-01-01
Full Text Available We study a boundary value problem with multivariables integral type condition for a class of parabolic equations. We prove the existence, uniqueness, and continuous dependence of the solution upon the data in the functional wieghted Sobolev spaces. Results are obtained by using a functional analysis method based on two-sided a priori estimates and on the density of the range of the linear operator generated by the considered problem.
Implicit Boundary Integral Methods for the Helmholtz Equation in Exterior Domains
2016-06-01
solutions computed by EIBIM and IBIM using different mesh sizes . The scattering surface is the bean shape shown in Figure (3.3). k = 1, 0 = √ ∆x. Evaluated...methods (IBIMs) We now describe the implicit boundary integral formulation for integral equation of the form ∫ Γ β(y)K(x, y)ds(y) + λβ(x) = f(x), for x...1)sc,0.1) 142.9 141.2 140.9 Cond(w(2)∞,0.1) 1666.5 1419.2 1413.01 Table 2: Condition numbers of matrices formed by using different kernels. The
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.
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 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 Value Problems with Integral Gluing Conditions for Fractional-Order Mixed-Type Equation
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A. S. Berdyshev
2011-01-01
Full Text Available Analogs of the Tricomi and the Gellerstedt problems with integral gluing conditions for mixed parabolic-hyperbolic equation with parameter have been considered. The considered mixed-type equation consists of fractional diffusion and telegraph equation. The Tricomi problem is equivalently reduced to the second-kind Volterra integral equation, which is uniquely solvable. The uniqueness of the Gellerstedt problem is proven by energy integrals' method and the existence by reducing it to the ordinary differential equations. The method of Green functions and properties of integral-differential operators have been used.
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.
Feng, Justin C.; Matzner, Richard A.
2017-11-01
We reexamine the relationship between the path integral and canonical formulation of quantum general relativity. In particular, we present a formal derivation of the Wheeler-DeWitt equation from the path integral for quantum general relativity by way of boundary variations. One feature of this approach is that it does not require an explicit 3 +1 splitting of spacetime in the bulk. For spacetimes with a spatial boundary, we show that the dependence of the transition amplitudes on spatial boundary conditions is determined by a Wheeler-DeWitt equation for the spatial boundary surface. We find that variations in the induced metric at the spatial boundary can be used to describe time evolution—time evolution in quantum general relativity is therefore governed by boundary conditions on the gravitational field at the spatial boundary. We then briefly describe a formalism for computing the dependence of transition amplitudes on spatial boundary conditions. Finally, we argue that for nonsmooth boundaries meaningful transition amplitudes must depend on boundary conditions at the joint surfaces.
Fu, Liwei; Frenner, Karsten; Osten, Wolfgang
2014-07-15
The scattering of electromagnetic waves from rough surfaces has been actively studied for more than a century now because of its involvement in vast application areas. In the past two decades, great advances have been made by incorporating multiple scattering effects into analytical approaches. However, no model can yet be applied to surfaces with arbitrary roughness. It is also very difficult to study the cross-polarization, shadowing, or multiple scattering effects. In order to study more fundamentally the interaction of polarized light with more general rough surfaces of general media, we have developed a rigorous numerical simulator to calculate the resulting speckle fields. The full Maxwell equations were solved using surface integral equations combined with a boundary element method. The rough surface was discretized by higher order quadrilateral edge elements. The effective tangential electric and magnetic fields in each element in terms of 10 edges were first solved. The scattered electric and magnetic fields everywhere in space were then calculated correspondingly. One of the great advantages of such a simulator is that both the near and far fields can be calculated directly. Preliminary results of different kinds of metallic structures are presented, by which the advantages of the method are demonstrated.
Mardanov, M. J.; Mahmudov, N. I.; Sharifov, Y. A.
2014-01-01
We study a boundary value problem for the system of nonlinear impulsive fractional differential equations of order α (0 < α ≤ 1) involving the two-point and integral boundary conditions. Some new results on existence and uniqueness of a solution are established by using fixed point theorems. Some illustrative examples are also presented. We extend previous results even in the integer case α = 1. PMID:24782675
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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.
Tricomi, Francesco Giacomo
1957-01-01
This classic text on integral equations by the late Professor F. G. Tricomi, of the Mathematics Faculty of the University of Turin, Italy, presents an authoritative, well-written treatment of the subject at the graduate or advanced undergraduate level. To render the book accessible to as wide an audience as possible, the author has kept the mathematical knowledge required on the part of the reader to a minimum; a solid foundation in differential and integral calculus, together with some knowledge of the theory of functions is sufficient. The book is divided into four chapters, with two useful
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A. Anguraj
2014-02-01
Full Text Available We study in this paper,the existence of solutions for fractional integro differential equations with impulsive and integral conditions by using fixed point method. We establish the Sufficient conditions and unique solution for given problem. An Example is also explained to the main results.
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.
Klaseboer, Evert; Sepehrirahnama, Shahrokh; Chan, Derek Y C
2017-08-01
The general space-time evolution of the scattering of an incident acoustic plane wave pulse by an arbitrary configuration of targets is treated by employing a recently developed non-singular boundary integral method to solve the Helmholtz equation in the frequency domain from which the space-time solution of the wave equation is obtained using the fast Fourier transform. The non-singular boundary integral solution can enforce the radiation boundary condition at infinity exactly and can account for multiple scattering effects at all spacings between scatterers without adverse effects on the numerical precision. More generally, the absence of singular kernels in the non-singular integral equation confers high numerical stability and precision for smaller numbers of degrees of freedom. The use of fast Fourier transform to obtain the time dependence is not constrained to discrete time steps and is particularly efficient for studying the response to different incident pulses by the same configuration of scatterers. The precision that can be attained using a smaller number of Fourier components is also quantified.
To the boundary value problem of ordinary differential equations
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Serikbai Aisagaliev
2015-09-01
Full Text Available Method for solving of a boundary value problem for ordinary differential equations with boundary conditions at phase and integral constraints is proposed. The base of the method is an immersion principle based on the general solution of the first order Fredholm integral equation which allows to reduce the original boundary value problem to the special problem of the optimal equation.
Hu, Fang; Pizzo, Michelle E.; Nark, Douglas M.
2017-01-01
It has been well-known that under the assumption of a constant uniform mean flow, the acoustic wave propagation equation can be formulated as a boundary integral equation, in both the time domain and the frequency domain. Compared with solving partial differential equations, numerical methods based on the boundary integral equation have the advantage of a reduced spatial dimension and, hence, requiring only a surface mesh. However, the constant uniform mean flow assumption, while convenient for formulating the integral equation, does not satisfy the solid wall boundary condition wherever the body surface is not aligned with the uniform mean flow. In this paper, we argue that the proper boundary condition for the acoustic wave should not have its normal velocity be zero everywhere on the solid surfaces, as has been applied in the literature. A careful study of the acoustic energy conservation equation is presented that shows such a boundary condition in fact leads to erroneous source or sink points on solid surfaces not aligned with the mean flow. A new solid wall boundary condition is proposed that conserves the acoustic energy and a new time domain boundary integral equation is derived. In addition to conserving the acoustic energy, another significant advantage of the new equation is that it is considerably simpler than previous formulations. In particular, tangential derivatives of the solution on the solid surfaces are no longer needed in the new formulation, which greatly simplifies numerical implementation. Furthermore, stabilization of the new integral equation by Burton-Miller type reformulation is presented. The stability of the new formulation is studied theoretically as well as numerically by an eigenvalue analysis. Numerical solutions are also presented that demonstrate the stability of the new formulation.
Stokes equations with penalised slip boundary conditions
Dione, Ibrahima; Tibirna, Cristian; Urquiza, José
2013-07-01
We consider the finite-element approximation of Stokes equations with slip boundary conditions imposed with the penalty method. In the case of a smooth curved boundary, our numerical results suggest that curved finite elements, regularised normal vectors or reduced integration techniques can be used to avoid a Babuska's-type paradox and ensure the convergence of finite-element approximations to the exact solution. Convergence orders with these remedies are also compared.
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Erkinjon T. Karimov
2014-01-01
Full Text Available In the present work we investigate the Tricomi problem with an integral gluing condition for a parabolic-hyperbolic equation involving the Caputo fractional differential operator. Using the method of energy integrals, we prove the uniqueness of the solution for the considered problem. The existence of the solution have been proved applying methods of ordinary differential equations and Fredholm integral equations. The solution is represented in an explicit form.
Feischl, Michael; Gantner, Gregor; Haberl, Alexander; Praetorius, Dirk
2017-01-01
In a recent work (Feischl et al. in Eng Anal Bound Elem 62:141-153, 2016), we analyzed a weighted-residual error estimator for isogeometric boundary element methods in 2D and proposed an adaptive algorithm which steers the local mesh-refinement of the underlying partition as well as the multiplicity of the knots. In the present work, we give a mathematical proof that this algorithm leads to convergence even with optimal algebraic rates. Technical contributions include a novel mesh-size function which also monitors the knot multiplicity as well as inverse estimates for NURBS in fractional-order Sobolev norms.
Mendelson, A.
1977-01-01
Two advances in the numerical techniques of utilizing the BIE method are presented. The boundary unknowns are represented by parabolas over each interval which are integrated in closed form. These integrals are listed for easy use. For problems involving crack tip singularities, these singularities are included in the boundary integrals so that the stress intensity factor becomes just one more unknown in the set of boundary unknowns thus avoiding the uncertainties of plotting and extrapolating techniques. The method is applied to the problems of a notched beam in tension and bending, with excellent results.
Thirard, Christophe; Parot, Jean-Marc
2017-11-01
Solving acoustic equations in the time domain, possibly coupled with the description of flexible structure dynamics, remains attractive as compared to solving the same in the frequency domain: this allows for better consideration of local non-linearities (acoustics/structure), and the boundary integral formulation (also known as BEM) offers an exact description of the infinite acoustic field based on a simple surface mesh (no need for 3D-volume discretization). Some issues remain however: the required memory space and computation time continue to grow rapidly when the number of elements of the surface mesh increases. In the case of a structure with a regular non-slender shape, the computational cost, measured in terms of required memory space, varies by Helmholtz number to the power of 4. This paper illustrates how the accelerating method called NGTD helps overcome this difficulty. This paper shows the applicability of 2 level NGTD to acoustic and vibroacoustic problems described solely by the hypersingular formulation for surfaces. It goes into more detail on some important aspects of the interpolation process and on the memory saving obtained. Implementation within the MOT (;March-On-Time;) ASTRYD code shows the benefits of this method. The memory requirement shows an estimated trend lower than power 1.35 of the number of surface elements.
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...
Yan, Zhenya
2017-05-01
We extend the idea of the Fokas unified transform to investigate the initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4 × 4 Lax pair on the half-line. The solution of this system can be expressed in terms of the solution of a 4 × 4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. The relevant jump matrices of the RH problem can be explicitly found using the two spectral functions s(k) and S(k), which can be defined by the initial data, the Dirichlet-Neumann boundary data at x = 0. The global relation is established between the two dependent spectral functions. The general mappings between Dirichlet and Neumann boundary values are analyzed in terms of the global relation. These results may be of the potential significance in both spinor Bose-Einstein condensates and the theory of multi-component integrable systems.
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A. Becker
2003-01-01
Full Text Available In this paper a hybrid method combining the FDTD/FIT with a Time Domain Boundary-Integral Marching-on-in-Time Algorithm (TD-BIM is presented. Inhomogeneous regions are modelled with the FIT-method, an alternative formulation of the FDTD. Homogeneous regions (which is in the presented numerical example the open space are modelled using a TD-BIM with equivalent electric and magnetic currents flowing on the boundary between the inhomogeneous and the homogeneous regions. The regions are coupled by the tangential magnetic fields just outside the inhomogeneous regions. These fields are calculated by making use of a Mixed Potential Integral Formulation for the magnetic field. The latter consists of equivalent electric and magnetic currents on the boundary plane between the homogeneous and the inhomogeneous region. The magnetic currents result directly from the electric fields of the Yee lattice. Electric currents in the same plane are calculated by making use of the TD-BIM and using the electric field of the Yee lattice as boundary condition. The presented hybrid method only needs the interpolations inherent in FIT and no additional interpolation. A numerical result is compared to a calculation that models both regions with FDTD.
Nonlinear streak computation using boundary region equations
Energy Technology Data Exchange (ETDEWEB)
Martin, J A; Martel, C, E-mail: juanangel.martin@upm.es, E-mail: carlos.martel@upm.es [Depto. de Fundamentos Matematicos, E.T.S.I Aeronauticos, Universidad Politecnica de Madrid, Plaza Cardenal Cisneros 3, 28040 Madrid (Spain)
2012-08-01
The boundary region equations (BREs) are applied for the simulation of the nonlinear evolution of a spanwise periodic array of streaks in a flat plate boundary layer. The well-known BRE formulation is obtained from the complete Navier-Stokes equations in the high Reynolds number limit, and provides the correct asymptotic description of three-dimensional boundary layer streaks. In this paper, a fast and robust streamwise marching scheme is introduced to perform their numerical integration. Typical streak computations present in the literature correspond to linear streaks or to small-amplitude nonlinear streaks computed using direct numerical simulation (DNS) or the nonlinear parabolized stability equations (PSEs). We use the BREs to numerically compute high-amplitude streaks, a method which requires much lower computational effort than DNS and does not have the consistency and convergence problems of the PSE. It is found that the flow configuration changes substantially as the amplitude of the streaks grows and the nonlinear effects come into play. The transversal motion (in the wall normal-streamwise plane) becomes more important and strongly distorts the streamwise velocity profiles, which end up being quite different from those of the linear case. We analyze in detail the resulting flow patterns for the nonlinearly saturated streaks and compare them with available experimental results. (paper)
Reaction diffusion equations with boundary degeneracy
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Huashui Zhan
2016-03-01
Full Text Available In this article, we consider the reaction diffusion equation $$ \\frac{\\partial u}{\\partial t} = \\Delta A(u,\\quad (x,t\\in \\Omega \\times (0,T, $$ with the homogeneous boundary condition. Inspired by the Fichera-Oleinik theory, if the equation is not only strongly degenerate in the interior of $\\Omega$, but also degenerate on the boundary, we show that the solution of the equation is free from any limitation of the boundary condition.
The laminar boundary layer equations
Curle, N
2017-01-01
Thorough introduction to boundary layer problems offers an ordered, logical presentation accessible to undergraduates. The text's careful expositions of the limitations and accuracy of various methods will also benefit professionals. 1962 edition.
Nonlocal boundary value problem for telegraph equations
Ashyralyev, Allaberen; Modanli, Mahmut
2015-09-01
In this work, the nonlocal boundary value problem for a telegraph equation in a Hilbert space is conceived. Stability estimates for the solution of this problem are obtained. The first and second order of accuracy difference schemes for the approximate solution of this problem are constructed. Stability estimates for the solution of these difference schemes are established. In implementations, two mixed problems for telegraph partial differential equations are investigated. The methods are showed by numerical experiments.
Optimal Wentzell Boundary Control of Parabolic Equations
Energy Technology Data Exchange (ETDEWEB)
Luo, Yousong, E-mail: yousong.luo@rmit.edu.au [RMIT University, School of Mathematical and Geospatial Sciences (Australia)
2017-04-15
This paper deals with a class of optimal control problems governed by an initial-boundary value problem of a parabolic equation. The case of semi-linear boundary control is studied where the control is applied to the system via the Wentzell boundary condition. The differentiability of the state variable with respect to the control is established and hence a necessary condition is derived for the optimal solution in the case of both unconstrained and constrained problems. The condition is also sufficient for the unconstrained convex problems. A second order condition is also derived.
Boundary value problems 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
Path Integrals on Manifolds with Boundary
Ludewig, Matthias
2017-09-01
We give time-slicing path integral formulas for solutions to the heat equation corresponding to a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold with boundary. More specifically, we show that such a solution can be approximated by integrals over finite-dimensional path spaces of piecewise geodesics subordinated to increasingly fine partitions of the time interval. We consider a subclass of mixed boundary conditions which includes standard Dirichlet and Neumann boundary conditions.
Existence of solutions to boundary value problem of fractional differential equations with impulsive
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Weihua JIANG
2016-12-01
Full Text Available In order to solve the boundary value problem of fractional impulsive differential equations with countable impulses and integral boundary conditions on the half line, the existence of solutions to the boundary problem is specifically studied. By defining suitable Banach spaces, norms and operators, using the properties of fractional calculus and applying the contraction mapping principle and Krasnoselskii's fixed point theorem, the existence of solutions for the boundary value problem of fractional impulsive differential equations with countable impulses and integral boundary conditions on the half line is proved, and examples are given to illustrate the existence of solutions to this kind of equation boundary value problems.
Multidimensional singular integrals and integral equations
Mikhlin, Solomon Grigorievich; Stark, M; Ulam, S
1965-01-01
Multidimensional Singular Integrals and Integral Equations presents the results of the theory of multidimensional singular integrals and of equations containing such integrals. Emphasis is on singular integrals taken over Euclidean space or in the closed manifold of Liapounov and equations containing such integrals. This volume is comprised of eight chapters and begins with an overview of some theorems on linear equations in Banach spaces, followed by a discussion on the simplest properties of multidimensional singular integrals. Subsequent chapters deal with compounding of singular integrals
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...
A Coordinate Transformation for Unsteady Boundary Layer Equations
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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.
Integral equations and their applications
Rahman, M
2007-01-01
For many years, the subject of functional equations has held a prominent place in the attention of mathematicians. In more recent years this attention has been directed to a particular kind of functional equation, an integral equation, wherein the unknown function occurs under the integral sign. The study of this kind of equation is sometimes referred to as the inversion of a definite integral. While scientists and engineers can already choose from a number of books on integral equations, this new book encompasses recent developments including some preliminary backgrounds of formulations of integral equations governing the physical situation of the problems. It also contains elegant analytical and numerical methods, and an important topic of the variational principles. Primarily intended for senior undergraduate students and first year postgraduate students of engineering and science courses, students of mathematical and physical sciences will also find many sections of direct relevance. The book contains eig...
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.
On the nonlinear Schrodinger equation with nonzero boundary conditions
Fagerstrom, Emily
integral, provided the initial condition satisfies further conditions. Modulational instability (focusing NLS with symmetric nonzero boundary conditions at infinity.) The focusing NLS equation is considered with potentials that are "box-like" piecewise constant functions. Several results are obtained. In particular, it is shown that there are conditions on the parameters of the potential for which there are no discrete eigenvalues. Thus there is a class of potentials for which the corresponding solutions of the NLS equation have no solitons. Hence, solitons cannot be the medium for the modulational instability. This contradicts a recent conjecture by Zakharov. On the other hand, it is shown for a different class of potentials the scattering problem always has a discrete eigenvalue along the imaginary axis. Thus, there exist arbitrarily small perturbations of the constant potential for which solitons exist, so no area theorem is possible. The existence, number and location of discrete eigenvalues in other situations are studied numerically. Finally, the small-deviation limit of the IST is computed and compared with the direct linearization of the NLS equation around a constant background. From this it is shown that there is an interval of the continuous spectrum on which the eigenvalue is imaginary and the scattering parameter is imaginary. The Jost eigenfunctions corresponding to this interval are the nonlinear analogue of the unstable Fourier modes. Defocusing NLS equation with asymmetric boundary conditions at infinity. The defocusing NLS equation with asymmetric boundary conditions is considered. To do so, first the case of symmetric boundary conditions is revisited. While the IST for this case has been formulated in the literature, it is usually done through the use of a uniformization variable. This was done because the eigenvalues of the scattering problem have branching; the uniformization variable allows one to move from a 2-sheeted Riemann surface to the complex
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Bozor Islomov
2015-08-01
Full Text Available We prove the unique solvability of a boundary-value problems for a third-order loaded integro-differential equation with variable coefficients, by reducing the equation to a Volterra integral equation.
Sonoda, Hidenori
2005-01-01
An application of the exact renormalization group equations to the scalar field theory in three dimensional euclidean space is discussed. We show how to modify the original formulation by J. Polchinski in order to find the Wilson-Fisher fixed point using perturbation theory.
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
Yusop, Nur Syaza Mohd; Mohamed, Nurul Akmal
2017-05-01
Boundary Element Method (BEM) is a numerical way to approximate the solutions of a Boundary Value Problem (BVP). The potential problem which involves the Laplace's equation on the square shape domain will be considered where the boundary is divided into four sets of linear boundary elements. We study the derivation system of equation for mixed BVP with one Dirichlet Boundary Condition (BC) is prescribed on one element of the boundary and Neumann BC on the other three elements. The mixed BVP will be reduced to a Boundary Integral Equation (BIE) by using a direct method which involves Green's second identity representation formula. Then, linear interpolation is used where the boundary will be discretized into some linear elements. As the result, we then obtain the system of linear equations. In conclusion, the specific element in the mixed BVP will have the specific prescribe value depends on the type of boundary condition. For Dirichlet BC, it has only one value at each node but for the Neumann BC, there will be different values at the corner nodes due to outward normal. Therefore, the assembly process for the system of equations related to the mixed BVP may not be as straight forward as Dirichlet BVP and Neumann BVP. For the future research, we will consider the different shape domains for mixed BVP with different prescribed boundary conditions.
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.
Feynman integrals and difference equations
Energy Technology Data Exchange (ETDEWEB)
Moch, S. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany); Schneider, C. [Johannes Kepler Univ., Linz (Austria). Research Inst. for Symbolic Computation
2007-09-15
We report on the calculation of multi-loop Feynman integrals for single-scale problems by means of difference equations in Mellin space. The solution to these difference equations in terms of harmonic sums can be constructed algorithmically over difference fields, the so-called {pi}{sigma}{sup *}-fields. We test the implementation of the Mathematica package Sigma on examples from recent higher order perturbative calculations in Quantum Chromodynamics. (orig.)
Stationary solutions and Neumann boundary conditions in the Sivashinsky equation.
Denet, Bruno
2006-09-01
New stationary solutions of the (Michelson) Sivashinsky equation of premixed flames are obtained numerically in this paper. Some of these solutions, of the bicoalescent type recently described by Guidi and Marchetti, are stable with Neumann boundary conditions. With these boundary conditions, the time evolution of the Sivashinsky equation in the presence of a moderate white noise is controlled by jumps between stationary solutions.
Integration rules for scattering equations
Energy Technology Data Exchange (ETDEWEB)
Baadsgaard, Christian; Bjerrum-Bohr, N.E.J.; Bourjaily, Jacob L.; Damgaard, Poul H. [Niels Bohr International Academy and Discovery Center,Niels Bohr Institute, University of Copenhagen,Blegdamsvej 17, DK-2100 Copenhagen Ø (Denmark)
2015-09-21
As described by Cachazo, He and Yuan, scattering amplitudes in many quantum field theories can be represented as integrals that are fully localized on solutions to the so-called scattering equations. Because the number of solutions to the scattering equations grows quite rapidly, the contour of integration involves contributions from many isolated components. In this paper, we provide a simple, combinatorial rule that immediately provides the result of integration against the scattering equation constraints for any Möbius-invariant integrand involving only simple poles. These rules have a simple diagrammatic interpretation that makes the evaluation of any such integrand immediate. Finally, we explain how these rules are related to the computation of amplitudes in the field theory limit of string theory.
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.
A Boundary Control Problem for the Viscous Cahn–Hilliard Equation with Dynamic Boundary Conditions
Energy Technology Data Exchange (ETDEWEB)
Colli, Pierluigi, E-mail: pierluigi.colli@unipv.it; Gilardi, Gianni, E-mail: gianni.gilardi@unipv.it [Universitá di Pavia and Research Associate at the IMATI – C.N.R. PAVIA, Dipartimento di Matematica “F. Casorati” (Italy); Sprekels, Jürgen, E-mail: juergen.sprekels@wias-berlin.de [Weierstrass Institute (Germany)
2016-04-15
A boundary control problem for the viscous Cahn–Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved.
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Bakhtiyor Kadirkulov
2014-02-01
Full Text Available In this article, we study a boundary value problem for a parabolic-hyperbolic equation with Caputo fractional derivative. Under certain conditions, we prove its unique solvability using methods of integral equations and Green's functions.
On the strongly damped wave equation and the heat equation with mixed boundary conditions
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Aloisio F. Neves
2000-01-01
Full Text Available We study two one-dimensional equations: the strongly damped wave equation and the heat equation, both with mixed boundary conditions. We prove the existence of global strong solutions and the existence of compact global attractors for these equations in two different spaces.
A New Algorithm for System of Integral Equations
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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.
Boundary value problemfor multidimensional fractional advection-dispersion equation
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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
Initial-boundary value problems for the wave equation
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Tynysbek Sh. Kalmenov
2014-02-01
Full Text Available In this work we consider an initial-boundary value problem for the one-dimensional wave equation. We prove the uniqueness of the solution and show that the solution coincides with the wave potential.
Boundary integral formulation for cracks at imperfect interfaces
Mishuris, G.; Piccolroaz, A.; Vellender, A.
2013-01-01
We consider an infinite bi-material plane containing a semi-infinite crack situated on a soft imperfect interface. The crack is loaded by a general asymmetrical system of forces distributed along the crack faces. On the basis of the weight function approach and the fundamental reciprocal identity, we derive the corresponding boundary integral formulation, relating physical quantities. The boundary integral equations derived in this paper in the imperfect interface setting show a weak singular...
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.
Comparison of artificial absorbing boundaries for acoustic wave equation modelling
Gao, Yingjie; Song, Hanjie; Zhang, Jinhai; Yao, Zhenxing
2017-12-01
Absorbing boundary conditions are necessary in numerical simulation for reducing the artificial reflections from model boundaries. In this paper, we overview the most important and typical absorbing boundary conditions developed throughout history. We first derive the wave equations of similar methods in unified forms; then, we compare their absorbing performance via theoretical analyses and numerical experiments. The Higdon boundary condition is shown to be the best one among the three main absorbing boundary conditions that are based on a one-way wave equation. The Clayton and Engquist boundary is a special case of the Higdon boundary but has difficulty in dealing with the corner points in implementaion. The Reynolds boundary does not have this problem but its absorbing performance is the poorest among these three methods. The sponge boundary has difficulties in determining the optimal parameters in advance and too many layers are required to achieve a good enough absorbing performance. The hybrid absorbing boundary condition (hybrid ABC) has a better absorbing performance than the Higdon boundary does; however, it is still less efficient for absorbing nearly grazing waves since it is based on the one-way wave equation. In contrast, the perfectly matched layer (PML) can perform much better using a few layers. For example, the 10-layer PML would perform well for absorbing most reflected waves except the nearly grazing incident waves. The 20-layer PML is suggested for most practical applications. For nearly grazing incident waves, convolutional PML shows superiority over the PML when the source is close to the boundary for large-scale models. The Higdon boundary and hybrid ABC are preferred when the computational cost is high and high-level absorbing performance is not required, such as migration and migration velocity analyses, since they are not as sensitive to the amplitude errors as the full waveform inversion.
On a stochastic Burgers equation with Dirichlet boundary conditions
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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.
Partial differential equations and boundary-value problems with applications
Pinsky, Mark A
2011-01-01
Building on the basic techniques of separation of variables and Fourier series, the book presents the solution of boundary-value problems for basic partial differential equations: the heat equation, wave equation, and Laplace equation, considered in various standard coordinate systems-rectangular, cylindrical, and spherical. Each of the equations is derived in the three-dimensional context; the solutions are organized according to the geometry of the coordinate system, which makes the mathematics especially transparent. Bessel and Legendre functions are studied and used whenever appropriate th
Boundary conditions for hyperbolic systems of partial differentials equations.
Guaily, Amr G; Epstein, Marcelo
2013-07-01
An easy-to-apply algorithm is proposed to determine the correct set(s) of boundary conditions for hyperbolic systems of partial differential equations. The proposed approach is based on the idea of the incoming/outgoing characteristics and is validated by considering two problems. The first one is the well-known Euler system of equations in gas dynamics and it proved to yield set(s) of boundary conditions consistent with the literature. The second test case corresponds to the system of equations governing the flow of viscoelastic liquids.
Memory boundary feedback stabilization for Schrodinger equations with variable coefficients
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Abdesselam Nawel
2017-05-01
Full Text Available First we consider the boundary stabilization of Schrodinger equations with constant coefficient memory feedback. This is done by using Riemannian geometry methods and the multipliers technique. Then we explore the stabilization limits of Schrodinger equations whose elliptical part has a variable coefficient. We established the exponential decay of solutions using the multipliers techniques. The introduction of dissipative boundary conditions of memory type allowed us to obtain an accurate estimate on the uniform rate of decay of the energy for Schrodinger equations.
Boundary properties of solutions of equations of minimal surface kind
Miklyukov, V. M.
2001-10-01
Generalized solutions of equations of minimal-surface type are studied. It is shown that a solution makes at most countably many jumps at the boundary. In particular, a solution defined in the exterior of a disc extends by continuity to the boundary circle everywhere outside a countable point set. An estimate of the sum of certain non-local characteristics of the jumps of a solution at the boundary is presented. A result similar to Fatou's theorem on angular boundary values is proved.
PREFACE: Symmetries and Integrability of Difference Equations
Doliwa, Adam; Korhonen, Risto; Lafortune, Stéphane
2007-10-01
M Sergeev on quantization of three-wave equations. Random matrix theory. This section contains a paper by A V Kitaev on the boundary conditions for scaled random matrix ensembles in the bulk of the spectrum. Symmetries and conservation laws. In this section we have five articles. H Gegen, X-B Hu, D Levi and S Tsujimoto consider a difference-analogue of Davey-Stewartson system giving its discrete Gram-type determinant solution and Lax pair. The paper by D Levi, M Petrera, and C Scimiterna is about the lattice Schwarzian KDV equation and its symmetries, while O G Rasin and P E Hydon study the conservation laws for integrable difference equations. S Saito and N Saitoh discuss recurrence equations associated with invariant varieties of periodic points, and P H van der Kamp presents closed-form expressions for integrals of MKDV and sine-Gordon maps. Ultra-discrete systems. This final category contains an article by C Ormerod on connection matrices for ultradiscrete linear problems. We would like to express our sincerest thanks to all contributors, and to everyone involved in compiling this special issue.
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...
Function Substitution in Partial Differential Equations: Nonhomogeneous Boundary Conditions
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T. V. Oblakova
2017-01-01
Full Text Available The paper considers a mixed initial-boundary value problem for a parabolic equation with nonhomogeneous boundary conditions. The classical approach to search for analytical solution of such problems in the first phase involves variable substitution, leading to a problem with homogeneous boundary conditions. Reference materials [1] give, as a rule, the simplest types of variable substitutions where new and old unknown functions differ by a term, linear in the spatial variable. The form of this additive term depends on the type of the boundary conditions, but is in no way related to the equation under consideration. Moreover, in the case of the second boundary-value problem, it is necessary to use a quadratic additive, since a linear substitution for this type of conditions may be unavailable. The courseware [2] - [4], usually, ends only with the first boundary-value problem generally formulated.The paper considers a substitution that takes into account, in principle, the form of a linear differential operator. Namely, as an additive term, it is proposed to use the parametrically time-dependent solution of the boundary value problem for an ordinary differential equation obtained from the original partial differential equation by the method of separation of the Fourier variables.The existence of the proposed substitution for boundary conditions of any type is proved by the example of a non-stationary heat-transfer equation with the heat exchange available with the surrounding medium. In this case, the additive term is a linear combination of hyperbolic functions. It is shown that, in addition to the "insensitivity" to the type of boundary conditions, the advantages of a new substitution in comparison with the traditional linear (or quadratic one include a much simpler structure of the solution obtained. Just the described approach allows us to obtain a solution with a clearly distinguished stationary component, in case a stationarity occurs, for
1986-07-01
tool for the stress analysis of three-dimensional cracked components in linea elastic Isotropic bodies. Ibs fundamntals of the Boundary Zntegral...linear, algebraic equatioma* which are solved mubj eat to the bousdery cown- diticas (eithei displacement or stress) of the problem. Diplacts u* tioas of
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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.
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...... it is demonstrated how a singular integral equation with infinite support can be solved by use of the preceding formulae....
A boundary value problem for the wave equation
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Nezam Iraniparast
1999-01-01
Full Text Available Traditionally, boundary value problems have been studied for elliptic differential equations. The mathematical systems described in these cases turn out to be “well posed”. However, it is also important, both mathematically and physically, to investigate the question of boundary value problems for hyperbolic partial differential equations. In this regard, prescribing data along characteristics as formulated by Kalmenov [5] is of special interest. The most recent works in this area have resulted in a number of interesting discoveries [3, 4, 5, 7, 8]. Our aim here is to extend some of these results to a more general domain which includes the characteristics of the underlying wave equation as a part of its boundary.
On a Nonlinear Degenerate Evolution Equation with Nonlinear Boundary Damping
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A. T. Lourêdo
2015-01-01
Full Text Available This paper deals essentially with a nonlinear degenerate evolution equation of the form Ku″-Δu+∑j=1nbj∂u′/∂xj+uσu=0 supplemented with nonlinear boundary conditions of Neumann type given by ∂u/∂ν+h·, u′=0. Under suitable conditions the existence and uniqueness of solutions are shown and that the boundary damping produces a uniform global stability of the corresponding solutions.
Surface integrals approach to solution of some free boundary problems
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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.
Method for solving moving boundary value problems for linear evolution equations
Fokas; Pelloni
2000-05-22
We introduce a method of solving initial boundary value problems for linear evolution equations in a time-dependent domain, and we apply it to an equation with dispersion relation omega(k), in the domain l(t)integral representation in the complex k plane, involving either an integral of exp[ikx-iomega(k)t]rho(k) along a time-dependent contour, or an integral of exp[ikx-iomega(k)t]rho(k, &kmacr;) over a fixed two-dimensional domain. The functions rho(k) and rho(k,&kmacr;) can be computed through the solution of a system of Volterra linear integral equations. This method can be generalized to nonlinear integrable partial differential equations.
Existence theory for nonlinear volterra integral and differential equations
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Sikorska Aneta
2001-01-01
Full Text Available In this paper we prove the existence theorems for the integrodifferential equation where in first part are functions with values in a Banach space and the integral is taken in the sense of Bochner. In second part are weakly–weakly sequentially continuous functions and the integral is the Pettis integral. Additionaly, the functions and satisfy some boundary conditions and conditions expressed in terms of measure of noncompactness or measure of weak noncompactness.
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.
Positive Solutions for Some Beam Equation Boundary Value Problems
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Xu Weiya
2009-01-01
Full Text Available A new fixed point theorem in a cone is applied to obtain the existence of positive solutions of some fourth-order beam equation boundary value problems with dependence on the first-order derivative where is continuous.
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.
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...
High order integral equation method for diffraction gratings.
Lu, Wangtao; Lu, Ya Yan
2012-05-01
Conventional integral equation methods for diffraction gratings require lattice sum techniques to evaluate quasi-periodic Green's functions. The boundary integral equation Neumann-to-Dirichlet map (BIE-NtD) method in Wu and Lu [J. Opt. Soc. Am. A 26, 2444 (2009)], [J. Opt. Soc. Am. A 28, 1191 (2011)] is a recently developed integral equation method that avoids the quasi-periodic Green's functions and is relatively easy to implement. In this paper, we present a number of improvements for this method, including a revised formulation that is more stable numerically, and more accurate methods for computing tangential derivatives along material interfaces and for matching boundary conditions with the homogeneous top and bottom regions. Numerical examples indicate that the improved BIE-NtD map method achieves a high order of accuracy for in-plane and conical diffractions of dielectric gratings. © 2012 Optical Society of America
Integral equations with contrasting kernels
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Theodore Burton
2008-01-01
Full Text Available In this paper we study integral equations of the form $x(t=a(t-\\int^t_0 C(t,sx(sds$ with sharply contrasting kernels typified by $C^*(t,s=\\ln (e+(t-s$ and $D^*(t,s=[1+(t-s]^{-1}$. The kernel assigns a weight to $x(s$ and these kernels have exactly opposite effects of weighting. Each type is well represented in the literature. Our first project is to show that for $a\\in L^2[0,\\infty$, then solutions are largely indistinguishable regardless of which kernel is used. This is a surprise and it leads us to study the essential differences. In fact, those differences become large as the magnitude of $a(t$ increases. The form of the kernel alone projects necessary conditions concerning the magnitude of $a(t$ which could result in bounded solutions. Thus, the next project is to determine how close we can come to proving that the necessary conditions are also sufficient. The third project is to show that solutions will be bounded for given conditions on $C$ regardless of whether $a$ is chosen large or small; this is important in real-world problems since we would like to have $a(t$ as the sum of a bounded, but badly behaved function, and a large well behaved function.
Boundary-value problems for wave equations with data on the whole boundary
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Makhmud A. Sadybekov
2016-10-01
Full Text Available In this article 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. 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.
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.
Hypersingular integral equations and their applications
Lifanov, IK; Vainikko, MGM
2003-01-01
A number of new methods for solving singular and hypersingular integral equations have emerged in recent years. This volume presents some of these new methods along with classical exact, approximate, and numerical methods. The authors explore the analysis of hypersingular integral equations based on the theory of pseudodifferential operators and consider one-, two- and multi-dimensional integral equations. The text also presents the discrete closed vortex frame method and some other numerical methods for solving hypersingular integral equations. The treatment includes applications to problems in areas such as aerodynamics, elasticity, diffraction, and heat and mass transfer.
Why are some boundary currents blocked by the equator?
Nof, Doron
1990-05-01
Existing nonlinear theories for the effect of the equator on northward-flowing boundary jets adjacent to a finite depth upper layer suggest that such inertial currents can flow from the southern to the northern hemisphere with the equator playing no special role ( ANDERSON and MOORE, 1979, Deep-Sea Research, 26, 1-22). It is demonstrated in this paper that, in contrast to these flows, nonlinear currents bounded by a front on their open ocean side (i.e. no adjacent upper layer) experience traumatic changes in their structure as they approach the equator. Within a deformation radius away from the equator, they separate from the coast and turn eastward to form a swift narrow jet. The equator blocks their advancement into the northern hemisphere. The above behavior is demonstrated by examining two nonlinear analytical models. First, we consider a simple highly idealized 1 1/2 layer model of a (southern hemisphere) wedge-like boundary current flowing northward with a wall on its left and a front on its right (looking downstream). It is almost a trivial matter to demonstrate analytically that such a current (which is just a special case of the Anderson and Moore flow) cannot penetrate into the northern hemisphere. Instead, it must separate from the coast in the vicinity of the equator. Secondly, a more realistic and more complicated 2 1/2 layer model is considered. In this nonlinear analytical model the current consists of two active layers. As in the previous model, the core of the current is a wedge-like layer (i.e. it is bounded by a front on the oceanic side) but the remaining surrounding field (i.e. the second moving layer) extends to infinity. It is shown that the flow in the second layer car cross the equator without any traumatic changes (as suggested by ANDERSON and MOORE, 1979), but the core must again separate from the coast in the vicinity of the equator. As in the previous model, the current associated with the core is blocked by the equator. The general
Exact solution for the fractional cable equation with nonlocal boundary conditions
Bazhlekova, Emilia; Dimovski, Ivan
2013-10-01
The fractional cable equation is studied on a bounded space domain. One of the prescribed boundary conditions is of Dirichlet type, the other is of a general form, which includes the case of nonlocal boundary conditions. In real problems nonlocal boundary conditions are prescribed when the data on the boundary can not be measured directly. We apply spectral projection operators to convert the problem to a system of integral equations in any generalized eigenspace. In this way we prove uniqueness of the solution and give an algorithm for constructing the solution in the form of an expansion in terms of the generalized eigenfunctions and three-parameter Mittag-Leffler functions. Explicit representation of the solution is given for the case of double eigenvalues. We consider some examples and as a particular case we recover a recent result. The asymptotic behavior of the solution is also studied.
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.
Problems in Hydrodynamics and Partial Differential Equations.
CALCULUS OF VARIATIONS, RESEARCH MANAGEMENT ), NONLINEAR DIFFERENTIAL EQUATIONS, PARTIAL DIFFERENTIAL EQUATIONS, BOUNDARY VALUE PROBLEMS, EQUATIONS, INEQUALITIES, MEASURE THEORY , INTEGRALS, ABSTRACTS
Asymptotically periodic solutions of Volterra integral equations
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Muhammad N. Islam
2016-03-01
Full Text Available We study the existence of asymptotically periodic solutions of a nonlinear Volterra integral equation. In the process, we obtain the existence of periodic solutions of an associated nonlinear integral equation with infinite delay. Schauder's fixed point theorem is used in the analysis.
Boundary-value problems for fractional heat equation involving Caputo-Fabrizio derivative
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Erkinjon Karimov
2016-10-01
Full Text Available In this work, we consider a number of boundary-value problems for time-fractional heat equation with the recently introduced Caputo-Fabrizio derivative. Using the method of separation of variables, we prove a unique solvability of the stated problems. Moreover, we have found an explicit solution to certain initial value problem for Caputo-Fabrizio fractional order differential equation by reducing the problem to a Volterra integral equation. Different forms of solution were presented depending on the values of the parameter appeared in the problem.
Energy Technology Data Exchange (ETDEWEB)
Novikov, I.A.
1979-01-01
A mathematical solution is worked out for solving direct and inverse linear boundary problems on thermal conductivity where the heat transfer process is assumed to occur at a finite velocity. Three-dimensional potentials for a simple and binary layers and volume are introduced for the telegrapher's equation. Within the limiting cases of a parabolic and wave equation, they transfer to the corresponding heat and wave potentials. The introduced potentials make it possible to reduce the internal boundary problems of thermal conductivity to second order integral-differential equations. In addition, the potentials of the telegrapher's equation are used to reduce a linear three-dimensional boundary inverse problem on thermal conductivity to a system composed of integral-differential equations of the second order and a certain integral condition for two coordinate and time functions. 9 references.
Algebraic Bethe Ansatz for O(2N) sigma models with integrable diagonal boundaries
Energy Technology Data Exchange (ETDEWEB)
Gombor, Tamás [MTA Lendület Holographic QFT Group, Wigner Research Centre,H-1525 Budapest 114, P.O.B. 49 (Hungary); Institute for Theoretical Physics, Roland Eötvös University,1117 Budapest, Pázmány s. 1/A (Hungary); Palla, László [Institute for Theoretical Physics, Roland Eötvös University,1117 Budapest, Pázmány s. 1/A (Hungary)
2016-02-24
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.
Isogeometric Analysis of Boundary Integral Equations
2015-04-21
grants from the Office of Naval Research (N00014-08-1-0992), the National ScienceFoundation ( CMMI -01101007), and SINTEF (UTA10-000374) with the... CMMI -01101007). We thank these organizations for their generous support. We are also grateful to Professor Michael Scott (Brigham Young University) for
Solvability of boundary-value problems for Poisson equations with Hadamard type boundary operator
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Batirkhan Turmetov
2016-06-01
Full Text Available In this article we study properties of some integro-differential operators of fractional order. As an application of the properties of these operators for Poisson equation we examine questions on solvability of a fractional analogue of Neumann problem and analogues of periodic boundary-value problems for circular domains. The exact conditions for solvability of these problems are found.
New integral equations for the study of electromagnetic radiation within cavities
Darling, B. T.; Imbeau, J. A.
1980-09-01
For stationary solutions of Maxwell's equations in the interior of a closed surface, two new vectorial integral equations are derived. They consist of integrals over the surface that the values of the fields on the surface must satisfy. These integral equations have two advantages over integral equations with singular kernels obtained by taking the limiting form of the equations of Stratton and Chu as the point of observation approaches the surface. In the present integral equations the kernel is regular, and the point of observation may be taken anywhere inside, outside, or on the surface. The new method of solution of Maxwell's equations for boundary value problems independent of time which is proposed is analogous to that which was proposed previously for the boundary value problem of Helmholtz's equation.
Optimal control problems for impulsive systems with integral boundary conditions
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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.
The boundary-domain integral method for elliptic systems
Pomp, Andreas
1998-01-01
This monograph gives a description of all algorithmic steps and a mathematical foundation for a special numerical method, namely the boundary-domain integral method (BDIM). This method is a generalization of the well-known boundary element method, but it is also applicable to linear elliptic systems with variable coefficients, especially to shell equations. The text should be understandable at the beginning graduate-level. It is addressed to researchers in the fields of numerical analysis and computational mechanics, and will be of interest to everyone looking at serious alternatives to the well-established finite element methods.
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Paul Eloe
2002-01-01
Full Text Available The method of quasilinearization for nonlinear impulsive differential equations with linear boundary conditions is studied. The boundary conditions include periodic boundary conditions. It is proved the convergence is quadratic.
Random integral equations on time scales
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Vasile Lupulescu
2013-01-01
Full Text Available In this paper, we present the existence and uniqueness of random solution of a random integral equation of Volterra type on time scales. We also study the asymptotic properties of the unique random solution.
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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.
Radiation Boundary Conditions for the Two-Dimensional Wave Equation from a Variational Principle
Broeze, J.; Broeze, Jan; van Daalen, Edwin F.G.; van Daalen, E.F.G.
1992-01-01
A variational principle is used to derive a new radiation boundary condition for the two-dimensional wave equation. This boundary condition is obtained from an expression for the local energy flux velocity on the boundary in normal direction. The wellposedness of the wave equation with this boundary
Periodic boundary value problems of second order random differential equations
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Bapurao Dhage
2009-04-01
Full Text Available In this paper, an existence and the existence of extremal random solutions are proved for a periodic boundary value problem of second order ordinary random differential equations. Our investigations have been placed in the space of real-valued functions defined and continuous on closed and bounded intervals of real line together with the applications of the random version of a nonlinear alternative of Leray-Schauder type and an algebraic random fixed point theorem of Dhage. An example is also indicated for demonstrating the realizations of the abstract theory developed in this paper.
Boundary controllability of Maxwell's equations in a spherical region
Kime, Katherine A.
1987-02-01
This paper examines the question of control of electromagnetic fields in a three dimensional spherical region by means of control currents on the boundary of that region. Results of this type are significant in connection with the theory of waveguides, electromagnetic pulse devices, and stealth technology, as well as being of independent mathematical interest. The necessary theory of divergence-free solutions of the vector wave equation is developed. By use of eigenfunctions of the vector Laplacian in appropriate divergence-free domains and moment problem techniques, sufficient conditions for controllability are set forth.
Initial-boundary value problem with a nonlocal condition for a viscosity equation
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Abdelfatah Bouziani
2002-01-01
Full Text Available This paper deals with the proof of the existence, uniqueness, and continuous dependence of a strong solution upon the data, for an initial-boundary value problem which combine Neumann and integral conditions for a viscosity equation. The proof is based on an energy inequality and on the density of the range of the linear operator corresponding to the abstract formulation of the studied problem.
On a third order parabolic equation with a nonlocal boundary condition
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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.
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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.
Functional differential inclusions with integral boundary conditions
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Mouffak Benchohra
2007-07-01
Full Text Available In this paper, we investigate the existence of solutions for a class of second order functional differential inclusions with integral boundary conditions. By using suitable fixed point theorems, we study the case when the right hand side has convex as well as nonconvex values.
Panafricanism, African Boundaries and Regional Integration ...
African Journals Online (AJOL)
It is indicated that this division has led to the emergence of small states with small market economics competing rather than completing each other's economy. It is argued that regional economic integration cannot take place with boundaries as obstacles. The Pan African idea of closer unity is examined. Regional economic ...
On a Volterra Stieltjes integral equation
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P. T. Vaz
1990-01-01
Full Text Available The paper deals with a study of linear Volterra integral equations involving Lebesgue-Stieltjes integrals in two independent variables. The authors prove an existence theorem using the Banach fixed-point principle. An explicit example is also considered.
Numerical integration of damped Maxwell equations
Botchev, M.A.; Verwer, J.G.
2009-01-01
We study the numerical time integration of Maxwell's equations from electromagnetism. Following the method of lines approach we start from a general semidiscrete Maxwell system for which a number of time-integration methods are considered. These methods have in common an explicit treatment of the
Numerical integration of damped Maxwell equations
Bochev, Mikhail A.; Verwer, J.G.
We study the numerical time integration of Maxwell's equations from electromagnetism. Following the method of lines approach we start from a general semi-discrete Maxwell system for which a number of time-integration methods are considered. These methods have in common an explicit treatment of the
Numerical integration of damped Maxwell equations
Bochev, Mikhail A.; Verwer, J.G.
We study the numerical time integration of Maxwell's equations from electromagnetism. Following the method of lines approach we start from a general semidiscrete Maxwell system for which a number of time-integration methods are considered. These methods have in common an explicit treatment of the
The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential
Liu, Shuangqian; Yang, Xiongfeng
2017-01-01
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.
Numerical methods for solution of singular integral equations
Boykov, I. V.
2016-01-01
This paper is devoted to overview of the authors works for numerical solution of singular integral equations (SIE), polysingular integral equations and multi-dimensional singular integral equations of the second kind. The authors investigated onsidered iterative - projective methods and parallel methods for solution of singular integral equations, polysingular integral equations and multi-dimensional singular integral equations. The paper is the second part of overview of the authors works de...
Oscillatory integrals generated by the Ostrovsky equation
Varlamov, Vladimir
2005-11-01
The Ostrovsky equation governs the propagation of long nonlinear surface waves in the presence of rotation. It is related to the Korteweg-de Vries (KdV) and the Kadomtsev-Petviashvili models. KdV can be obtained from the equation in question when the rotation parameter γ equals zero. A fundamental solution of the Cauchy problem for the linear Ostrovsky equation is presented in the form of an oscillatory Fourier integral. Another integral representation involving Airy and Bessel functions is derived for it. It is shown that its asymptotic expansion as γ → 0 contains the KdV fundamental solution as the zero term. The Airy transform is used to establish some of its properties. Higher-order asymptotics for γ → 0 on a bounded time interval are obtained for both the fundamental solution and the solution of the linear Cauchy problem for the Ostrovsky equation.
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
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Juan Carlos Ceballos V.
2005-10-01
Full Text Available The exact boundary controllability of the higher order nonlinear Schrodinger equation with constant coefficients on a bounded domain with various boundary conditions is studied. We derive the exact boundary controllability for this equation for sufficiently small initial and final states.
Spheroidal Integral Equations for Geodetic Inversion of Geopotential Gradients
Novák, Pavel; Šprlák, Michal
2017-12-01
The static Earth's gravitational field has traditionally been described in geodesy and geophysics by the gravitational potential (geopotential for short), a scalar function of 3-D position. Although not directly observable, geopotential functionals such as its first- and second-order gradients are routinely measured by ground, airborne and/or satellite sensors. In geodesy, these observables are often used for recovery of the static geopotential at some simple reference surface approximating the actual Earth's surface. A generalized mathematical model is represented by a surface integral equation which originates in solving Dirichlet's boundary-value problem of the potential theory defined for the harmonic geopotential, spheroidal boundary and globally distributed gradient data. The mathematical model can be used for combining various geopotential gradients without necessity of their re-sampling or prior continuation in space. The model extends the apparatus of integral equations which results from solving boundary-value problems of the potential theory to all geopotential gradients observed by current ground, airborne and satellite sensors. Differences between spherical and spheroidal formulations of integral kernel functions of Green's kind are investigated. Estimated differences reach relative values at the level of 3% which demonstrates the significance of spheroidal approximation for flattened bodies such as the Earth. The observation model can be used for combined inversion of currently available geopotential gradients while exploring their spectral and stochastic characteristics. The model would be even more relevant to gravitational field modelling of other bodies in space with more pronounced spheroidal geometry than that of the Earth.
Abel integral equations analysis and applications
Gorenflo, Rudolf
1991-01-01
In many fields of application of mathematics, progress is crucially dependent on the good flow of information between (i) theoretical mathematicians looking for applications, (ii) mathematicians working in applications in need of theory, and (iii) scientists and engineers applying mathematical models and methods. The intention of this book is to stimulate this flow of information. In the first three chapters (accessible to third year students of mathematics and physics and to mathematically interested engineers) applications of Abel integral equations are surveyed broadly including determination of potentials, stereology, seismic travel times, spectroscopy, optical fibres. In subsequent chapters (requiring some background in functional analysis) mapping properties of Abel integral operators and their relation to other integral transforms in various function spaces are investi- gated, questions of existence and uniqueness of solutions of linear and nonlinear Abel integral equations are treated, and for equatio...
On integrable boundaries in the 2 dimensional O(N) σ-models
Aniceto, Inês; Bajnok, Zoltán; Gombor, Tamás; Kim, Minkyoo; Palla, László
2017-09-01
We make an attempt to map the integrable boundary conditions for 2 dimensional non-linear O(N) σ-models. We do it at various levels: classically, by demanding the existence of infinitely many conserved local charges and also by constructing the double row transfer matrix from the Lax connection, which leads to the spectral curve formulation of the problem; at the quantum level, we describe the solutions of the boundary Yang-Baxter equation and derive the Bethe-Yang equations. We then show how to connect the thermodynamic limit of the boundary Bethe-Yang equations to the spectral curve.
Integrable boundary interaction in 3D target space: The “pillow-brane” model
Energy Technology Data Exchange (ETDEWEB)
Lukyanov, Sergei L., E-mail: sergei@physics.rutgers.edu [NHETC, Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849 (United States); L.D. Landau Institute for Theoretical Physics, Chernogolovka, 142432 (Russian Federation); Zamolodchikov, Alexander B. [NHETC, Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849 (United States); Institute for Information Transmission Problems, Moscow (Russian Federation)
2013-08-21
We propose a model of boundary interaction, with three-dimensional target space, and the boundary values of the field X∈R{sup 3} constrained to lay on a two-dimensional surface of the “pillow” shape. We argue that the model is integrable, and suggest that its exact solution is described in terms of certain linear ordinary differential equation.
Integral analysis of boundary layer flows with pressure gradient
Wei, Tie; Maciel, Yvan; Klewicki, Joseph
2017-09-01
This Rapid Communication investigates boundary layer flows with a pressure gradient using a similarity/integral analysis of the continuity equation and momentum equation in the streamwise direction. The analysis yields useful analytical relations for Ve, the mean wall-normal velocity at the edge of the boundary layer, and for the skin friction coefficient Cf in terms of the boundary layer parameters and in particular βRC, the Rotta-Clauser pressure gradient parameter. The analytical results are compared with experimental and numerical data and are found to be valid. One of the main findings is that for large positive βRC (an important effect of an adverse pressure gradient), the friction coefficient is closely related to βRC as Cf∝1 /βRC , because δ /δ1,δ1/δ2=H , and d δ /d x become approximately constant. Here, δ is the boundary layer thickness, δ1 is the displacement thickness, δ2 is the momentum thickness, and H is the shape factor. Another finding is that the mean wall-normal velocity at the edge of the boundary layer is related to other flow variables as UeVe/uτ2=H +(1 +δ /δ1+H ) βRC , where Ue is the streamwise velocity at the edge of the boundary layer. At zero pressure gradient, this relation reduces to U∞V∞/uτ2=H , as recently derived by Wei and Klewicki [Phys. Rev. Fluids 1, 082401 (2016), 10.1103/PhysRevFluids.1.082401].
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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.
Fractional boundary value problems with multiple orders of fractional derivatives and integrals
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Sotiris K. Ntouyas
2017-04-01
Full Text Available In this article we study a new class of boundary value problems for fractional differential equations and inclusions with multiple orders of fractional derivatives and integrals, in both fractional differential equation and boundary conditions. The Sadovski's fixed point theorem is applied in the single-valued case while, in multi-valued case, the nonlinear alternative for contractive maps is used. Some illustrative examples are also included.
Nonlocal singular problem with integral condition for a second-order parabolic equation
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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.
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Hytham. A. Alkresheh
2016-02-01
Full Text Available In this paper, an algorithm based on a new modification, developed by Duan and Rach, for the Adomian decomposition method (ADM is generalized to find positive solutions for boundary value problems involving nonlinear fractional ordinary differential equations. In the proposed algorithm the boundary conditions are used to convert the nonlinear fractional differential equations to an equivalent integral equation and then a recursion scheme is used to obtain the analytical solution components without the use of undetermined coefficients. Hence, there is no requirement to solve a nonlinear equation or a system of nonlinear equations of undetermined coefficients at each stage of approximation solution as per in the standard ADM. The fractional derivative is described in the Caputo sense. Numerical examples are provided to demonstrate the feasibility of the proposed algorithm.
Integral equations for free-molecule ow in MEMS: recent advancements
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Fedeli Patrick
2017-03-01
Full Text Available We address a Boundary Integral Equation (BIE approach for the analysis of gas dissipation in near-vacuum for Micro Electro Mechanical Systems (MEMS. Inspired by an analogy with the radiosity equation in computer graphics, we discuss an efficient way to compute the visible domain of integration. Moreover, we tackle the issue of near singular integrals by developing a set of analytical formulas for planar polyhedral domains. Finally a validation with experimental results taken from the literature is presented.
Complementary equations: a fractional differential equation and a Volterra integral equation
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Leigh Becker
2015-03-01
\\] on an interval $(0, T]$ if and only if it satisfies the Volterra integral equation \\[ x(t = x^{0}t^{q-1}+\\frac{1}{\\Gamma (q}\\int_{0}^{t}(t-s^{q-1}f(s, x(s\\,ds \\] on this same interval. In contradistinction to established existence theorems for these equations, no Lipschitz condition is imposed on $f(t,x$. Examples with closed-form solutions illustrate this result.
Some Antiperiodic Boundary Value Problem for Nonlinear Fractional Impulsive Differential Equations
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Xianghu Liu
2014-01-01
Full Text Available This paper is concerned with the sufficient conditions for the existence of solutions for a class of generalized antiperiodic boundary value problem for nonlinear fractional impulsive differential equations involving the Riemann-Liouville fractional derivative. Firstly, we introduce the fractional calculus and give the generalized R-L fractional integral formula of R-L fractional derivative involving impulsive. Secondly, the sufficient condition for the existence and uniqueness of solutions is presented. Finally, we give some examples to illustrate our main results.
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Aqlan Mohammed H.
2016-01-01
Full Text Available We develop the existence theory for sequential fractional differential equations involving Liouville-Caputo fractional derivative equipped with anti-periodic type (non-separated and nonlocal integral boundary conditions. Several existence criteria depending on the nonlinearity involved in the problems are presented by means of a variety of tools of the fixed point theory. The applicability of the results is shown with the aid of examples. Our results are not only new in the given configuration but also yield some new special cases for specific choices of parameters involved in the problems.
Ene, Remus-Daniel; Marinca, Vasile; Marinca, Bogdan
2016-01-01
Analytic approximate solutions using Optimal Homotopy Perturbation Method (OHPM) are given for steady boundary layer flow over a nonlinearly stretching wall in presence of partial slip at the boundary. The governing equations are reduced to nonlinear ordinary differential equation by means of similarity transformations. Some examples are considered and the effects of different parameters are shown. OHPM is a very efficient procedure, ensuring a very rapid convergence of the solutions after only two iterations.
Integral equation models for endemic infectious diseases.
Hethcote, H W; Tudor, D W
1980-03-01
Endemic infectious diseases for which infection confers permanent immunity are described by a system of nonlinear Volterra integral equations of convolution type. These constant-parameter models include vital dynamics (birth and deaths), immunization and distributed infectious period. The models are shown to be well posed, the threshold criteria are determined and the asymptotic behavior is analysed. It is concluded that distributed delays do not change the thresholds and the asymptotic behaviors of the models.
Directory of Open Access Journals (Sweden)
Emran Tohidi
2016-01-01
Full Text Available This article contributes a matrix approach by using Taylor approximation to obtain the numerical solution of one-dimensional time-dependent parabolic partial differential equations (PDEs subject to nonlocal boundary integral conditions. We first impose the initial and boundary conditions to the main problems and then reach to the associated integro-PDEs. By using operational matrices and also the completeness of the monomials basis, the obtained integro-PDEs will be reduced to the generalized Sylvester equations. For solving these algebraic systems, we apply a famous technique in Krylov subspace iterative methods. A numerical example is considered to show the efficiency of the proposed idea.
Nonlinear integral equations for the sausage model
Ahn, Changrim; Balog, Janos; Ravanini, Francesco
2017-08-01
The sausage model, first proposed by Fateev, Onofri, and Zamolodchikov, is a deformation of the O(3) sigma model preserving integrability. The target space is deformed from the sphere to ‘sausage’ shape by a deformation parameter ν. This model is defined by a factorizable S-matrix which is obtained by deforming that of the O(3) sigma model by a parameter λ. Clues for the deformed sigma model are provided by various UV and IR information through the thermodynamic Bethe ansatz (TBA) analysis based on the S-matrix. Application of TBA to the sausage model is, however, limited to the case of 1/λ integer where the coupled integral equations can be truncated to a finite number. In this paper, we propose a finite set of nonlinear integral equations (NLIEs), which are applicable to generic value of λ. Our derivation is based on T-Q relations extracted from the truncated TBA equations. For a consistency check, we compute next-leading order corrections of the vacuum energy and extract the S-matrix information in the IR limit. We also solved the NLIE both analytically and numerically in the UV limit to get the effective central charge and compared with that of the zero-mode dynamics to obtain exact relation between ν and λ. Dedicated to the memory of Petr Petrovich Kulish.
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.
A Tensor-Train accelerated solver for integral equations in complex geometries
Corona, Eduardo; Rahimian, Abtin; Zorin, Denis
2017-04-01
We present a framework using the Quantized Tensor Train (QTT) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the QTT decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations. For a broad range of problems, computational and storage costs of the inversion scheme are extremely modest O (log N) and once the inverse is computed, it can be applied in O (Nlog N) . We analyze the QTT ranks for hierarchically low rank matrices and discuss its relationship to commonly used hierarchical compression techniques such as FMM and HSS. We prove that the QTT ranks are bounded for translation-invariant systems and argue that this behavior extends to non-translation invariant volume and boundary integrals. For volume integrals, the QTT decomposition provides an efficient direct solver requiring significantly less memory compared to other fast direct solvers. We present results demonstrating the remarkable performance of the QTT-based solver when applied to both translation and non-translation invariant volume integrals in 3D. For boundary integral equations, we demonstrate that using a QTT decomposition to construct preconditioners for a Krylov subspace method leads to an efficient and robust solver with a small memory footprint. We test the QTT preconditioners in the iterative solution of an exterior elliptic boundary value problem (Laplace) formulated as a boundary integral equation in complex, multiply connected geometries.
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.
Solvability of some Neumann-type boundary value problems for biharmonic equations
Directory of Open Access Journals (Sweden)
Valery Karachik
2017-09-01
Full Text Available We study some boundary-value problems for inhomogeneous biharmonic equation with periodic boundary conditions. These problems are generalization to periodic data of the Neumann-type boundary-value problems considered before by the authors. We obtain existence and uniqueness of solutions for the problems under consideration.
Boundary value problems for third order differential equations by solution matching
Directory of Open Access Journals (Sweden)
Johnny Henderson
2009-10-01
Full Text Available For the ordinary differential equation, $y''' = f(x,y,y',$ $y'',$ solutions of 3-point boundary value problems on $[a,b]$ are matched with solutions of 3-point boundary value problems on $[b,c]$ to obtain solutions satisfying 5-point boundary conditions on $[a,c]$.
Solution to the one-dimensional telegrapher's equation subject to a backreaction boundary condition
Prüstel, Thorsten; Meier-Schellersheim, Martin
2013-01-01
We discuss solutions of the one-dimensional telegrapher's equation in the presence of boundary conditions. We revisit the case of a radiation boundary condition and obtain an alternative expression for the already known Green's function. Furthermore, we formulate a backreaction boundary condition, which has been widely used in the context of diffusion-controlled reversible reactions, for a one-dimensional telegrapher's equation and derive the corresponding Green's function.
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A. Sakabekov
2016-01-01
Full Text Available We prove existence and uniqueness of the solution of the problem with initial and Maxwell-Auzhan boundary conditions for nonstationary nonlinear one-dimensional Boltzmann’s six-moment system equations in space of functions continuous in time and summable in square by a spatial variable. In order to obtain a priori estimation of the initial and boundary value problem for nonstationary nonlinear one-dimensional Boltzmann’s six-moment system equations we get the integral equality and then use the spherical representation of vector. Then we obtain the initial value problem for Riccati equation. We have managed to obtain a particular solution of this equation in an explicit form.
Solution of Moving Boundary Space-Time Fractional Burger’s Equation
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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.
Nonlinear impulsive Volterra integral equations in Banach spaces and applications
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Dajun Guo
1993-01-01
Full Text Available In this paper, we first extend results on the existence of maximal solutions for nonlinear Volterra integral equations in Banach spaces to impulsive Volterra integral equations. Then, we give some applications to initial value problems for first order impulsive differential equations in Banach spaces. The results are demonstrated by means of an example of an infinite system for impulsive differential equations.
Invariant imbedding and a matrix integral equation of neuronal networks.
Kalaba, R.; Ruspini, E. H.
1971-01-01
A matrix Fredholm integral equation of neuronal networks is transformed into a Cauchy system suited for numerical and analytical studies. A special case is discussed, and a connection with the classical renewal integral equation of stochastic point processes is presented.
Sarna, Neeraj; Torrilhon, Manuel
2017-11-01
We define certain criteria, using the characteristic decomposition of the boundary conditions and energy estimates, which a set of stable boundary conditions for a linear initial boundary value problem, involving a symmetric hyperbolic system, must satisfy. We first use these stability criteria to show the instability of the Maxwell boundary conditions proposed by Grad (Commun Pure Appl Math 2(4):331-407, 1949). We then recognise a special block structure of the moment equations which arises due to the recursion relations and the orthogonality of the Hermite polynomials; the block structure will help us in formulating stable boundary conditions for an arbitrary order Hermite discretization of the Boltzmann equation. The formulation of stable boundary conditions relies upon an Onsager matrix which will be constructed such that the newly proposed boundary conditions stay close to the Maxwell boundary conditions at least in the lower order moments.
Integral solutions of fractional evolution equations with nondense domain
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Haibo Gu
2017-06-01
Full Text Available In this article, we study the existence of integral solutions for two classes of fractional order evolution equations with nondensely defined linear operators. First, we consider the nonhomogeneous fractional order evolution equation and obtain its integral solution by Laplace transform and probability density function. Subsequently, based on the form of integral solution for nonhomogeneous fractional order evolution equation, we investigate the existence of integral solution for nonlinear fractional order evolution equation by noncompact measure method.
Boundary Layer Equations and Lie Group Analysis of a Sisko Fluid
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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.
Adomian Method for Solving Fuzzy Fredholm-Volterra Integral Equations
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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 local fractional Volterra integral equations in fractal heat transfer
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Wu Zhong-Hua
2016-01-01
Full Text Available In the article, the fractal heat-transfer models are described by the local fractional integral equations. The local fractional linear and nonlinear Volterra integral equations are employed to present the heat transfer problems in fractal media. The local fractional integral equations are derived from the Fourier law in fractal media.
Integral equations formulation of plasmonic transmission lines.
Sallam, Mai O; Vandenbosch, Guy A E; Gielen, Georges; Soliman, Ezzeldin A
2014-09-22
In this paper, a comprehensive integral equation formulation of plasmonic transmission lines is presented for the first time. Such lines are made up of a number of metallic strips with arbitrary shapes and dimensions immersed within a stack of planar dielectric or metallic layers. These lines support a number of propagating modes. Each mode has its own phase constant, attenuation constant, and field distribution. The presented integral equation formulation is solved using the Method of Moments (MoM). It provides all the propagation characteristics of the modes. The new formulation is applied to a number of plasmonic transmission lines, such as: single rectangular strip, horizontally coupled strips, vertically coupled strips, triangular strip, and circular strip. The numerical study is performed in the frequency (wavelength) range of 150-450 THz (0.66-2.0 μm). The results of the proposed technique are compared with those obtained using Lumerical mode solution, and CST. Very good agreement has been observed. The main advantage of the MoM is its intrinsic speed for this type of problem compared to general purpose solvers.
Right focal boundary value problems for difference equations
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Johnny Henderson
2010-01-01
Full Text Available An application is made of a new Avery et al. fixed point theorem of compression and expansion functional type in the spirit of the original fixed point work of Leggett and Williams, to obtain positive solutions of the second order right focal discrete boundary value problem. In the application of the fixed point theorem, neither the entire lower nor entire upper boundary is required to be mapped inward or outward. A nontrivial example is also provided.
Large time behavior of solutions to parabolic equations with Neumann boundary conditions
da Lio, Francesca
2008-03-01
In this paper we are interested in the large time behavior as t-->+[infinity] of the viscosity solutions of parabolic equations with nonlinear Neumann type boundary conditions in connection with ergodic boundary problems which have been recently studied by Barles and the author in [G. Barles, F. Da Lio, On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linèaire 22 (5) (2005) 521-541].
Zuo, Chao; Chen, Qian; Asundi, Anand
2014-04-21
The transport of intensity equation (TIE) is a two-dimensional second order elliptic partial differential equation that must be solved under appropriate boundary conditions. However, the boundary conditions are difficult to obtain in practice. The fast Fourier transform (FFT) based TIE solutions are widely adopted for its speed and simplicity. However, it implies periodic boundary conditions, which lead to significant boundary artifacts when the imposed assumption is violated. In this work, TIE phase retrieval is considered as an inhomogeneous Neumann boundary value problem with the boundary values experimentally measurable around a hard-edged aperture, without any assumption or prior knowledge about the test object and the setup. The analytic integral solution via Green's function is given, as well as a fast numerical implementation for a rectangular region using the discrete cosine transform. This approach is applicable for the case of non-uniform intensity distribution with no extra effort to extract the boundary values from the intensity derivative signals. Its efficiency and robustness have been verified by several numerical simulations even when the objects are complex and the intensity measurements are noisy. This method promises to be an effective fast TIE solver for quantitative phase imaging applications.
Homogenization of the stochastic Navier–Stokes equation with a stochastic slip boundary condition
Bessaih, Hakima
2015-11-02
The two-dimensional Navier–Stokes equation in a perforated domain with a dynamical slip boundary condition is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the holes. We consider a scaling (ᵋ for the viscosity and 1 for the density) that will lead to a time-dependent limit problem. However, the noncritical scaling (ᵋ, β > 1) is considered in front of the nonlinear term. The homogenized system in the limit is obtained as a Darcy’s law with memory with two permeabilities and an extra term that is due to the stochastic perturbation on the boundary of the holes. The nonhomogeneity on the boundary contains a stochastic part that yields in the limit an additional term in the Darcy’s law. We use the two-scale convergence method after extending the solution with 0 inside the holes to pass to the limit. By Itô stochastic calculus, we get uniform estimates on the solution in appropriate spaces. Due to the stochastic integral, the pressure that appears in the variational formulation does not have enough regularity in time. This fact made us rely only on the variational formulation for the passage to the limit on the solution. We obtain a variational formulation for the limit that is solution of a Stokes system with two pressures. This two-scale limit gives rise to three cell problems, two of them give the permeabilities while the third one gives an extra term in the Darcy’s law due to the stochastic perturbation on the boundary of the holes.
Integral transformation of Heun's equation and some applications
竹村, 剛一
2011-01-01
It is known that the Fuchsian differential equation which produces the sixth Painlev\\'e equation corresponds to the Fuchsian differential equation with different parameters via Euler's integral transformation, and Heun's equation also corresponds to Heun's equation with different parameters, again via Euler's integral transformation. In this paper we study the correspondences in detail. After investigating correspondences with respect to monodromy, it is demonstrated that the existence of pol...
Integrable and continuous solutions of a nonlinear quadratic integral equation
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Ahmed El-Sayed
2008-08-01
Full Text Available We are concerned here with a nonlinear quadratic integral equation of Volterra type. The existence of at least one $L_1-$ positive solution will be proved under the Carath\\`{e}odory condition. Secondly we will make a link between Peano condition and Carath\\`{e}odory condition to prove the existence of at least one positive continuous solution. Finally the existence of the maximal and minimal solutions will be proved.
High-Order Accurate Solutions to the Helmholtz Equation in the Presence of Boundary Singularities
2015-03-31
elliptic equations . In Proceedings of the Soviet-American Conference on Partial Differential Equations , pages 303–304, Novosibirsk, Moscow, Russia, 1963...193, 2012. [19] L. Fox and R. Sankar. Boundary singularities in linear elliptic differential equations . J. Inst. Math. Appl., 5:340–350, 1969. [20] D.S...57] R.S. Lehman. Developments at an analytic corner of solutions of elliptic partial differential equations . J. Math. Mech., 8:727–760, 1959. [58
Analysis of Blasius Equation for Flat-Plate Flow with Infinite Boundary Value
DEFF Research Database (Denmark)
Miansari, M. O.; Miansari, M. E.; Barari, Amin
2010-01-01
and write the nonlinear differential equation in the state space format, and then solve the initial value problem instead of boundary value problem. The significance of linear part is a key factor in convergence. A first seen linear part may lead to an unstable solution, therefore an extra term is added......This paper applies the homotopy perturbation method (HPM) to determine the well-known Blasius equation with infinite boundary value for Flat-plate Flow. We study here the possibility of reducing the momentum and continuity equations to ordinary differential equations by a similarity transformation...
On singular solutions of a magnetohydrodynamic nonlinear boundary layer equation
Mohammed Guedda; Abdelilah Gmira; Mohammed Benlahsen
2007-01-01
This paper concerns the singular solutions of the equation $$ f''' +kappa ff''-eta {f'}^2 = 0, $$ where $eta < 0$ and $kappa = 0$ or 1. This equation arises when modelling heat transfer past a vertical flat plate embedded in a saturated porous medium with an applied magnetic field. After suitable normalization, $f'$ represents the velocity parallel to the surface or the non-dimensional fluid temperature. Our interest is in solutions which develop a singularity at some point (t...
Existence of arbitrarily smooth solutions of the LLG equation in 3D with natural boundary conditions
Feischl, Michael; Tran, Thanh
2016-01-01
We prove that the Landau-Lifshitz-Gilbert equation in three space dimensions with homogeneous Neumann boundary conditions admits arbitrarily smooth solutions, given that the initial data is sufficiently close to a constant function.
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.
Asymptotic behavior of solutions to nonlinear parabolic equation with nonlinear boundary conditions
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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.
A simple and efficient outflow boundary condition for the incompressible Navier–Stokes equations
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Yibao Li
2017-01-01
Full Text Available Many researchers have proposed special treatments for outlet boundary conditions owing to lack of information at the outlet. Among them, the simplest method requires a large enough computational domain to prevent or reduce numerical errors at the boundaries. However, an efficient method generally requires special treatment to overcome the problems raised by the outlet boundary condition used. For example, mass flux is not conserved and the fluid field is not divergence-free at the outlet boundary. Overcoming these problems requires additional computational cost. In this paper, we present a simple and efficient outflow boundary condition for the incompressible Navier–Stokes equations, aiming to reduce the computational domain for simulating flow inside a long channel in the streamwise direction. The proposed outflow boundary condition is based on the transparent equation, where a weak formulation is used. The pressure boundary condition is derived by using the Navier–Stokes equations and the outlet flow boundary condition. In the numerical algorithm, a staggered marker-and-cell grid is used and temporal discretization is based on a projection method. The intermediate velocity boundary condition is consistently adopted to handle the velocity–pressure coupling. Characteristic numerical experiments are presented to demonstrate the robustness and accuracy of the proposed numerical scheme. Furthermore, the agreement of computational results from small and large domains suggests that our proposed outflow boundary condition can significantly reduce computational domain sizes.
A memory type boundary stabilization of a mildly damped wave equation
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Mokhtar Kirane
1999-01-01
Full Text Available We consider the wave equation with a mild internal dissipation. It is proved that any small dissipation inside the domain is sufficient to uniformly stabilize the solution of this equation by means of a nonlinear feedback of memory type acting on a part of the boundary. This is established without any restriction on the space dimension and without geometrical conditions on the domain or its boundary.
Integrable coupling system of fractional soliton equation hierarchy
Energy Technology Data Exchange (ETDEWEB)
Yu Fajun, E-mail: yfajun@163.co [College of Maths and Systematic Science, Shenyang Normal University, Shenyang 110034 (China)
2009-10-05
In this Letter, we consider the derivatives and integrals of fractional order and present a class of the integrable coupling system of the fractional order soliton equations. The fractional order coupled Boussinesq and KdV equations are the special cases of this class. Furthermore, the fractional AKNS soliton equation hierarchy is obtained.
Integrability of two coupled Kadomtsev–Petviashvili equations
Indian Academy of Sciences (India)
2011-08-02
Aug 2, 2011 ... of Hirota's bilinear method is used to examine the integrability of each coupled equation. Multiple- ... It is a natural generalization of the KdV equation and is completely integrable by the inverse ... Several distinct approaches have been employed to establish coupled KdV and coupled KP equations. For.
A Cartesian grid embedded boundary method for the heat equation on irregular domains
Energy Technology Data Exchange (ETDEWEB)
McCorquodale, Peter; Colella, Phillip; Johansen, Hans
2001-03-14
We present an algorithm for solving the heat equation on irregular time-dependent domains. It is based on the Cartesian grid embedded boundary algorithm of Johansen and Colella (J. Comput. Phys. 147(2):60--85) for discretizing Poisson's equation, combined with a second-order accurate discretization of the time derivative. This leads to a method that is second-order accurate in space and time. For the case where the boundary is moving, we convert the moving-boundary problem to a sequence of fixed-boundary problems, combined with an extrapolation procedure to initialize values that are uncovered as the boundary moves. We find that, in the moving boundary case, the use of Crank--Nicolson time discretization is unstable, requiring us to use the L{sub 0}-stable implicit Runge--Kutta method of Twizell, Gumel, and Arigu.
On Coupled p-Laplacian Fractional Differential Equations with Nonlinear Boundary Conditions
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Aziz Khan
2017-01-01
Full Text Available This paper is related to the existence and uniqueness of solutions to a coupled system of fractional differential equations (FDEs with nonlinear p-Laplacian operator by using fractional integral boundary conditions with nonlinear term and also to checking the Hyers-Ulam stability for the proposed problem. The functions involved in the proposed coupled system are continuous and satisfy certain growth conditions. By using topological degree theory some conditions are established which ensure the existence and uniqueness of solution to the proposed problem. Further, certain conditions are developed corresponding to Hyers-Ulam type stability for the positive solution of the considered coupled system of FDEs. Also, from applications point of view, we give an example.
Brito, Irene; Mena, Filipe C
2017-08-01
We prove that, for a given spherically symmetric fluid distribution with tangential pressure on an initial space-like hypersurface with a time-like boundary, there exists a unique, local in time solution to the Einstein equations in a neighbourhood of the boundary. As an application, we consider a particular elastic fluid interior matched to a vacuum exterior.
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Liu Yuji
2008-01-01
Full Text Available Abstract This paper deals with the existence of solutions of the periodic boundary value problem of the impulsive Duffing equations: . Sufficient conditions are established for the existence of at least one solution of above-mentioned boundary value problem. Our method is based upon Schaeffer's fixed-point theorem. Examples are presented to illustrate the efficiency of the obtained results.
Brito, Irene; Mena, Filipe C.
2017-08-01
We prove that, for a given spherically symmetric fluid distribution with tangential pressure on an initial space-like hypersurface with a time-like boundary, there exists a unique, local in time solution to the Einstein equations in a neighbourhood of the boundary. As an application, we consider a particular elastic fluid interior matched to a vacuum exterior.
On singular solutions of a magnetohydrodynamic nonlinear boundary layer equation
Directory of Open Access Journals (Sweden)
Mohammed Guedda
2007-05-01
Full Text Available This paper concerns the singular solutions of the equation $$ f''' +kappa ff''-eta {f'}^2 = 0, $$ where $eta < 0$ and $kappa = 0$ or 1. This equation arises when modelling heat transfer past a vertical flat plate embedded in a saturated porous medium with an applied magnetic field. After suitable normalization, $f'$ represents the velocity parallel to the surface or the non-dimensional fluid temperature. Our interest is in solutions which develop a singularity at some point (the blow-up point. In particular, we shall examine in detail the behavior of $f$ near the blow-up point.
Angelani, Luca
2015-12-01
Absorption problems of run-and-tumble particles, described by the telegrapher's equation, are analyzed in one space dimension considering partially reflecting boundaries. Exact expressions for the probability distribution function in the Laplace domain and for the mean time to absorption are given, discussing some interesting limits (Brownian and wave limit, large volume limit) and different case studies (semi-infinite segment, equal and symmetric boundaries, totally/partially reflecting boundaries).
Boundary value problems for a nonlinear elliptic equation
Egorov, Yu. V.
2017-06-01
It is proved that the Dirichlet and Neumann problems for a nonlinear second-order elliptic equation have infinitely many solutions. The spectrum of these problems is studied and the weak convergence of the normed eigenfunctions to zero is established. Bibliography: 10 titles.
Eigenvalues of boundary value problems for higher order differential equations
Wong, Patricia J. Y.; Agarwal, Ravi P.
1996-01-01
We shall consider the boundary value problem y ( n ) + λ Q ( t , y , y 1 , ⋅ ⋅ ⋅ , y ( n − 2 ) ) = λ P ( t , y , y 1 , ⋅ ⋅ ⋅ , y ( n − 1 ) ) , n ≥ 2 , t ∈ ( 0 , 1 ) , y ( i ) ( 0 ) = 0 , 0 ≤ i ≤ n − 3 , α y ( n − 2 ) ( 0 ) − β y ( n − 1 ) ( 0 ) = 0 , γ y ( n − 2 ) ( 1 ) + δ y ( n...
Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions
Grunau, H.-Ch.; Sweers, G.
1996-01-01
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).
Three-Point Boundary Value Problems for Conformable Fractional Differential Equations
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H. Batarfi
2015-01-01
Full Text Available We study a fractional differential equation using a recent novel concept of fractional derivative with initial and three-point boundary conditions. We first obtain Green's function for the linear problem and then we study the nonlinear differential equation.
On The Stabilization of the Linear Kawahara Equation with Periodic Boundary Conditions
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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.
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...
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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.
Multicomponent integrable wave equations. I. Darboux-dressing transformation
Degasperis, A.; Lombardo, S.
2007-01-01
The Darboux-dressing transformations are applied to the Lax pair associated with systems of coupled nonlinear wave equations in the case of boundary values which are appropriate to both bŕight' and dárk' soliton solutions. The general formalism is set up and the relevant equations are explicitly
Existence of solutions for mixed Volterra-Fredholm integral equations
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Asadollah Aghajani
2012-08-01
Full Text Available In this article, we give some results concerning the continuity of the nonlinear Volterra and Fredholm integral operators on the space $L^{1}[0,infty$. Then by using the concept of measure of weak noncompactness, we prove an existence result for a functional integral equation which includes several classes of nonlinear integral equations. Our results extend some previous works.
Solving Abel integral equations of first kind via fractional calculus
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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.
Regular boundary value problems for the heat equation with scalar parameters
Kalmenov, Tynysbek Sh.; Besbaev, Gani; Medetbekova, Ryskul
2017-09-01
This paper belongs to the general theory of well-posed initial-boundary value problems for parabolic equations. The classical construction of a boundary value problem is as follows: an equation and a boundary condition are given. It is necessary to investigate the solvability of this problem and properties of the solution if it exists (in the sense of belonging to some space). Beginning with the papers of J. von Neumann and M.I. Vishik (1951), there exists another more general approach: an equation and a space are given, right-hand parts of the equation and boundary conditions, and a solution must belong to this space. It is necessary to describe all the boundary conditions, for which the problem is correctly solvable in this space. Further development of this theory was given by M. Otelbaev, who constructed a complete theory for ordinary differential operators and for symmetric semibounded operators in a Banach space. In this paper we find regular solution of the regular boundary problem for the heat equation with scalar parameter.
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Suphawat Asawasamrit
2014-01-01
Full Text Available We study a class of fractional q-integrodifference equations with nonlocal fractional q-integral boundary conditions which have different quantum numbers. By applying the Banach contraction principle, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative, the existence and uniqueness of solutions are obtained. In addition, some examples to illustrate our results are given.
Exponential integrators for the incompressible Navier-Stokes equations.
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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
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
ICM: an Integrated Compartment Method for numerically solving partial differential equations
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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.
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.
Rahmouni, Lyes; Mitharwal, Rajendra; Andriulli, Francesco P.
2017-11-01
This work presents two new volume integral equations for the Electroencephalography (EEG) forward problem which, differently from the standard integral approaches in the domain, can handle heterogeneities and anisotropies of the head/brain conductivity profiles. The new formulations translate to the quasi-static regime some volume integral equation strategies that have been successfully applied to high frequency electromagnetic scattering problems. This has been obtained by extending, to the volume case, the two classical surface integral formulations used in EEG imaging and by introducing an extra surface equation, in addition to the volume ones, to properly handle boundary conditions. Numerical results corroborate theoretical treatments, showing the competitiveness of our new schemes over existing techniques and qualifying them as a valid alternative to differential equation based methods.
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Hussein A. H. Salem
2013-01-01
Full Text Available The object of this paper is to investigate the existence of a class of solutions for some boundary value problems of fractional order with integral boundary conditions. The considered problems are very interesting and important from an application point of view. They include two, three, multipoint, and nonlocal boundary value problems as special cases. We stress on single and multivalued problems for which the nonlinear term is assumed only to be Pettis integrable and depends on the fractional derivative of an unknown function. Some investigations on fractional Pettis integrability for functions and multifunctions are also presented. An example illustrating the main result is given.
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Adriana C. Briozzo
2006-02-01
Full Text Available We prove the existence and uniqueness, local in time, of a solution for a one-phase Stefan problem of a non-classical heat equation for a semi-infinite material with temperature boundary condition at the fixed face. We use the Friedman-Rubinstein integral representation method and the Banach contraction theorem in order to solve an equivalent system of two Volterra integral equations.
Effective nonlinear Neumann boundary conditions for 1D nonconvex Hamilton-Jacobi equations
Guerand, Jessica
2017-09-01
We study Hamilton-Jacobi equations in [ 0 , + ∞) of evolution type with nonlinear Neumann boundary conditions in the case where the Hamiltonian is not necessarily convex with respect to the gradient variable. In this paper, we give two main results. First, we prove for a nonconvex and coercive Hamiltonian that general boundary conditions in a relaxed sense are equivalent to effective ones in a strong sense. Here, we exhibit the effective boundary conditions while for a quasi-convex Hamiltonian, we already know them (Imbert and Monneau, 2016). Second, we give a comparison principle for a nonconvex and nonnecessarily coercive Hamiltonian where the boundary condition can have constant parts.
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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.
Acoustic 3D modeling by the method of integral equations
Malovichko, M.; Khokhlov, N.; Yavich, N.; Zhdanov, M.
2018-02-01
This paper presents a parallel algorithm for frequency-domain acoustic modeling by the method of integral equations (IE). The algorithm is applied to seismic simulation. The IE method reduces the size of the problem but leads to a dense system matrix. A tolerable memory consumption and numerical complexity were achieved by applying an iterative solver, accompanied by an effective matrix-vector multiplication operation, based on the fast Fourier transform (FFT). We demonstrate that, the IE system matrix is better conditioned than that of the finite-difference (FD) method, and discuss its relation to a specially preconditioned FD matrix. We considered several methods of matrix-vector multiplication for the free-space and layered host models. The developed algorithm and computer code were benchmarked against the FD time-domain solution. It was demonstrated that, the method could accurately calculate the seismic field for the models with sharp material boundaries and a point source and receiver located close to the free surface. We used OpenMP to speed up the matrix-vector multiplication, while MPI was used to speed up the solution of the system equations, and also for parallelizing across multiple sources. The practical examples and efficiency tests are presented as well.
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.
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Leigh Becker
2016-09-01
\\end{equation} where $D^{q}$ denotes the Riemann-Liouville fractional differential operator. Finally, the resolvent integral function $\\int_0^t R(s\\,ds$ is shown to be the unique continuous solution of an integral equation closely related to (R$_\\lambda$. Closed-form expressions for it and $R(t$ are derived.
An integrable semi-discrete Degasperis-Procesi equation
Feng, Bao-Feng; Maruno, Ken-ichi; Ohta, Yasuhiro
2017-06-01
Based on our previous work on the Degasperis-Procesi equation (Feng et al J. Phys. A: Math. Theor. 46 045205) and the integrable semi-discrete analogue of its short wave limit (Feng et al J. Phys. A: Math. Theor. 48 135203), we derive an integrable semi-discrete Degasperis-Procesi equation by Hirota’s bilinear method. Furthermore, N-soliton solution to the semi-discrete Degasperis-Procesi equation is constructed. It is shown that both the proposed semi-discrete Degasperis-Procesi equation, and its N-soliton solution converge to ones of the original Degasperis-Procesi equation in the continuum limit.
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M. W. Dunlop
1999-12-01
Full Text Available Magnetic field measurements, taken by the magnetometer experiment (MAM on board the German Equator-S spacecraft, have been used to identify and categorise 131 crossings of the dawn-side magnetopause at low latitude, providing unusual, long duration coverage of the adjacent magnetospheric regions and near magnetosheath. The crossings occurred on 31 orbits, providing unbiased coverage over the full range of local magnetic shear from 06:00 to 10:40 LT. Apogee extent places the spacecraft in conditions associated with intermediate, rather than low, solar wind dynamic pressure, as it processes into the flank region. The apogee of the spacecraft remains close to the magnetopause for mean solar wind pressure. The occurrence of the magnetopause encounters are summarised and are found to compare well with predicted boundary location, where solar wind conditions are known. Most scale with solar wind pressure. Magnetopause shape is also documented and we find that the magnetopause orientation is consistently sunward of a model boundary and is not accounted for by IMF or local magnetic shear conditions. A number of well-established crossings, particularly those at high magnetic shear, or exhibiting unusually high-pressure states, were observed and have been analysed for their boundary characteristics and some details of their boundary and near magnetosheath properties are discussed. Of particular note are the occurrence of mirror-like signatures in the adjacent magnetosheath during a significant fraction of the encounters and a high number of multiple crossings over a long time period. The latter is facilitated by the spacecraft orbit which is designed to remain in the near magnetosheath for average solar wind pressure. For most encounters, a well-ordered, tangential (draped magnetosheath field is observed and there is little evidence of large deviations in local boundary orientations. Two passes corresponding to close conjunctions of the Geotail spacecraft
Directory of Open Access Journals (Sweden)
M. W. Dunlop
Full Text Available Magnetic field measurements, taken by the magnetometer experiment (MAM on board the German Equator-S spacecraft, have been used to identify and categorise 131 crossings of the dawn-side magnetopause at low latitude, providing unusual, long duration coverage of the adjacent magnetospheric regions and near magnetosheath. The crossings occurred on 31 orbits, providing unbiased coverage over the full range of local magnetic shear from 06:00 to 10:40 LT. Apogee extent places the spacecraft in conditions associated with intermediate, rather than low, solar wind dynamic pressure, as it processes into the flank region. The apogee of the spacecraft remains close to the magnetopause for mean solar wind pressure. The occurrence of the magnetopause encounters are summarised and are found to compare well with predicted boundary location, where solar wind conditions are known. Most scale with solar wind pressure. Magnetopause shape is also documented and we find that the magnetopause orientation is consistently sunward of a model boundary and is not accounted for by IMF or local magnetic shear conditions. A number of well-established crossings, particularly those at high magnetic shear, or exhibiting unusually high-pressure states, were observed and have been analysed for their boundary characteristics and some details of their boundary and near magnetosheath properties are discussed. Of particular note are the occurrence of mirror-like signatures in the adjacent magnetosheath during a significant fraction of the encounters and a high number of multiple crossings over a long time period. The latter is facilitated by the spacecraft orbit which is designed to remain in the near magnetosheath for average solar wind pressure. For most encounters, a well-ordered, tangential (draped magnetosheath field is observed and there is little evidence of large deviations in local boundary orientations. Two passes corresponding to close conjunctions of the Geotail spacecraft
On Some Classes of Linear Volterra Integral Equations
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Anatoly S. Apartsyn
2014-01-01
Full Text Available The sufficient conditions are obtained for the existence and uniqueness of continuous solution to the linear nonclassical Volterra equation that appears in the integral models of developing systems. The Volterra integral equations of the first kind with piecewise smooth kernels are considered. Illustrative examples are presented.
Simplifying Differential Equations for Multiscale Feynman Integrals beyond Multiple Polylogarithms.
Adams, Luise; Chaubey, Ekta; Weinzierl, Stefan
2017-04-07
In this Letter we exploit factorization properties of Picard-Fuchs operators to decouple differential equations for multiscale Feynman integrals. The algorithm reduces the differential equations to blocks of the size of the order of the irreducible factors of the Picard-Fuchs operator. As a side product, our method can be used to easily convert the differential equations for Feynman integrals which evaluate to multiple polylogarithms to an ϵ form.
Studies on the discrete integrable equations over finite fields
Kanki, Masataka
2013-01-01
Discrete dynamical systems over finite fields are investigated and their integrability is discussed. In particular, the discrete Painlev\\'{e} equations and the discrete KdV equation are defined over finite fields and their special solutions are obtained. Their investigation over the finite fields has not been done thoroughly, partly because of the indeterminacies that appear in defining the equations. In this paper we introduce two methods to well-define the equations over the finite fields a...
Linearization properties, first integrals, nonlocal transformation for heat transfer equation
Orhan, Özlem; Özer, Teoman
2016-08-01
We examine first integrals and linearization methods of the second-order ordinary differential equation which is called fin equation in this study. Fin is heat exchange surfaces which are used widely in industry. We analyze symmetry classification with respect to different choices of thermal conductivity and heat transfer coefficient functions of fin equation. Finally, we apply nonlocal transformation to fin equation and examine the results for different functions.
An energy absorbing far-field boundary condition for the elastic wave equation
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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.
Nonlinear Schrodinger equations on the half-line with nonlinear boundary conditions
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Ahmet Batal
2016-08-01
Full Text Available In this article, we study the initial boundary value problem for nonlinear Schrodinger equations on the half-line with nonlinear boundary conditions $$ u_x(0,t+\\lambda|u(0,t|^ru(0,t=0,\\quad \\lambda\\in\\mathbb{R}-\\{0\\},\\; r> 0. $$ We discuss the local well-posedness when the initial data $u_0=u(x,0$ belongs to an $L^2$-based inhomogeneous Sobolev space $H^s(\\mathbb{R}_+$ with $s\\in (\\frac{1}{2},\\frac{7}{2}-\\{\\frac{3}{2}\\}$. We deal with the nonlinear boundary condition by first studying the linear Schrodinger equation with a time-dependent inhomogeneous Neumann boundary condition $u_x(0,t=h(t$ where $h\\in H^{\\frac{2s-1}{4}}(0,T$.
Boundary conditions in conformal and integrable theories
Petkova, V B
2000-01-01
The study of boundary conditions in rational conformal field theories is not only physically important. It also reveals a lot on the structure of the theory ``in the bulk''. The same graphs classify both the torus and the cylinder partition functions and provide data on their hidden ``quantum symmetry''. The Ocneanu triangular cells -- the 3j-symbols of these symmetries, admit various interpretations and make a link between different problems.
Povstenko, Y.
2013-09-01
The axisymmetric time-fractional diffusion-wave equation with the Caputo derivative of the order 0 function and the values of its normal derivative at the boundary. The fundamental solutions to the Cauchy, source, and boundary problems are investigated. The Laplace transform with respect to time and finite Hankel transform with respect to the radial coordinate are used. The solutions are obtained in terms of Mittag-Leffler functions. The numerical results are illustrated graphically.
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.
Angelani, Luca
2016-01-01
Absorption problems of run-and-tumble particles, described by the telegrapher's equation, are analyzed in one space dimension considering partially reflecting boundaries. Exact expressions for the probability distribution function in the Laplace domain and for the mean time to absorption are given, discussing some interesting limits (Brownian and wave limit, large volume limit) and different case studies (semi-infinite segment, equal and symmetric boundaries, totally/partially reflecting boun...
Qualitative properties of nonlinear Volterra integral equations
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Muhammad Islam
2008-03-01
Full Text Available In this article, the contraction mapping principle and Liapunov's method are used to study qualitative properties of nonlinear Volterra equations of the form $x(t = a(t -\\int^{t}_{0}C(t,sg(s,x(s\\;ds,t\\geq0.$ In particular, the existence of bounded solutions and solutions with various $L^p$ properties are studied under suitable conditions on the functions involved with this equation.
Exact controllability for a wave equation with mixed boundary conditions in a non-cylindrical domain
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Lizhi Cui
2014-04-01
Full Text Available In this article we study the exact controllability of a one-dimensional wave equation with mixed boundary conditions in a non-cylindrical domain. The fixed endpoint has a Dirichlet-type boundary condition, while the moving end has a Neumann-type condition. When the speed of the moving endpoint is less than the characteristic speed, the exact controllability of this equation is established by Hilbert Uniqueness Method. Moreover, we shall give the explicit dependence of the controllability time on the speed of the moving endpoint.
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.
Singh, Randhir; Das, Nilima; Kumar, Jitendra
2017-06-01
An effective analytical technique is proposed for the solution of the Lane-Emden equations. The proposed technique is based on the variational iteration method (VIM) and the convergence control parameter h . In order to avoid solving a sequence of nonlinear algebraic or complicated integrals for the derivation of unknown constant, the boundary conditions are used before designing the recursive scheme for solution. The series solutions are found which converges rapidly to the exact solution. Convergence analysis and error bounds are discussed. Accuracy, applicability of the method is examined by solving three singular problems: i) nonlinear Poisson-Boltzmann equation, ii) distribution of heat sources in the human head, iii) second-kind Lane-Emden equation.
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G. K. Parks
1999-12-01
Full Text Available An electrostatic analyser (ESA onboard the Equator-S spacecraft operating in coordination with a potential control device (PCD has obtained the first accurate electron energy spectrum with energies ≈7 eV–100 eV in the vicinity of the magnetopause. On 8 January, 1998, a solar wind pressure increase pushed the magnetopause inward, leaving the Equator-S spacecraft in the magnetosheath. On the return into the magnetosphere approximately 80 min later, the magnetopause was observed by the ESA and the solid state telescopes (the SSTs detected electrons and ions with energies ≈20–300 keV. The high time resolution (3 s data from ESA and SST show the boundary region contains of multiple plasma sources that appear to evolve in space and time. We show that electrons with energies ≈7 eV–100 eV permeate the outer regions of the magnetosphere, from the magnetopause to ≈6Re. Pitch-angle distributions of ≈20–300 keV electrons show the electrons travel in both directions along the magnetic field with a peak at 90° indicating a trapped configuration. The IMF during this interval was dominated by Bx and By components with a small Bz.Key words. Magnetospheric physics (magnetopause · cusp · and boundary layers; magnetospheric configuration and dynamics; solar wind · magnetosphere interactions
Directory of Open Access Journals (Sweden)
G. K. Parks
Full Text Available An electrostatic analyser (ESA onboard the Equator-S spacecraft operating in coordination with a potential control device (PCD has obtained the first accurate electron energy spectrum with energies ≈7 eV–100 eV in the vicinity of the magnetopause. On 8 January, 1998, a solar wind pressure increase pushed the magnetopause inward, leaving the Equator-S spacecraft in the magnetosheath. On the return into the magnetosphere approximately 80 min later, the magnetopause was observed by the ESA and the solid state telescopes (the SSTs detected electrons and ions with energies ≈20–300 keV. The high time resolution (3 s data from ESA and SST show the boundary region contains of multiple plasma sources that appear to evolve in space and time. We show that electrons with energies ≈7 eV–100 eV permeate the outer regions of the magnetosphere, from the magnetopause to ≈6Re. Pitch-angle distributions of ≈20–300 keV electrons show the electrons travel in both directions along the magnetic field with a peak at 90° indicating a trapped configuration. The IMF during this interval was dominated by Bx and By components with a small Bz.
Key words. Magnetospheric physics (magnetopause · cusp · and boundary layers; magnetospheric configuration and dynamics; solar wind · magnetosphere interactions
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Zhang Sheng
2016-01-01
Full Text Available In this paper, variable separation method combined with the properties of Mittag-Leffler function is used to solve a variable-coefficient time fractional advection-dispersion equation with initial and boundary conditions. As a result, a explicit exact solution is obtained. It is shown that the variable separation method can provide a useful mathematical tool for solving the time fractional heat transfer equations.
Non-homogeneous boundary-value problems of higher order differential equations with p-Laplacian
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Yuji Liu
2008-02-01
Full Text Available We establish sufficient conditions for the existence of positive solutions to five multi-point boundary value problems. These problems have a common equation (in different function domains and different boundary conditions. It is interesting note that the methods for solving all these problems and most of the reference are based on the Mawhin's coincidence degree theory. First, we present a survey of multi-point boundary-value problems and the motivation of this paper. Then we present the main results which generalize and improve results in the references. We conclude this article with examples of problems that can not solved by methods known so far.
Seslija, Marko; Perunicic, Branislava; Salihbegovic, A; Supic, H; Velagic, J; Sadzak, A
2009-01-01
This paper considers the application of extrapolation techniques in finding approximate solutions of some optimization problems with constraints defined by the Robin boundary problem for the Laplace equation. When applied extrapolation techniques produce very accurate solutions of the boundary
The Kadomtsev{endash}Petviashvili equation as a source of integrable model equations
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Maccari, A. [Technical Institute ``G. Cardano,`` Piazza della Resistenza 1, 00015 Monterotondo Rome (Italy)
1996-12-01
A new integrable and nonlinear partial differential equation (PDE) in 2+1 dimensions is obtained, by an asymptotically exact reduction method based on Fourier expansion and spatiotemporal rescaling, from the Kadomtsev{endash}Petviashvili equation. The integrability property is explicitly demonstrated, by exhibiting the corresponding Lax pair, that is obtained by applying the reduction technique to the Lax pair of the Kadomtsev{endash}Petviashvili equation. This model equation is likely to be of applicative relevance, because it may be considered a consistent approximation of a large class of nonlinear evolution PDEs. {copyright} {ital 1996 American Institute of Physics.}
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.
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.
Mosayebidorcheh, Sobhan
2013-01-01
The boundary layer equation of the pseudoplastic fluid over a flat plate is considered. This equation is a boundary value problem (BVP) with the high nonlinearity and a boundary condition at infinity. To solve such problems, powerful numerical techniques are usually used. Here, through using differential transform method (DTM), the BVP is replaced by two initial value problems (IVP) and then multi-step differential transform method (MDTM) is applied to solve them. The differential equation an...
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...
Multicomponent integrable wave equations: II. Soliton solutions
Energy Technology Data Exchange (ETDEWEB)
Degasperis, A [Dipartimento di Fisica, Universita di Roma ' La Sapienza' , and Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Rome (Italy); Lombardo, S [School of Mathematics, University of Manchester, Alan Turing Building, Upper Brook Street, Manchester M13 9EP (United Kingdom)], E-mail: antonio.degasperis@roma1.infn.it, E-mail: sara.lombardo@manchester.ac.uk, E-mail: sara@few.vu.nl
2009-09-25
The Darboux-dressing transformations developed in Degasperis and Lombardo (2007 J. Phys. A: Math. Theor. 40 961-77) are here applied to construct soliton solutions for a class of boomeronic-type equations. The vacuum (i.e. vanishing) solution and the generic plane wave solution are both dressed to yield one-soliton solutions. The formulae are specialized to the particularly interesting case of the resonant interaction of three waves, a well-known model which is of boomeronic type. For this equation a novel solution which describes three locked dark pulses (simulton) is introduced.
Iterative Method for Solving the Second Boundary Value Problem for Biharmonic-Type Equation
Directory of Open Access Journals (Sweden)
Dang Quang A.
2012-01-01
Full Text Available Solving boundary value problems (BVPs for the fourth-order differential equations by the reduction of them to BVPs for the second-order equations with the aim to use the achievements for the latter ones attracts attention from many researchers. In this paper, using the technique developed by ourselves in recent works, we construct iterative method for the second BVP for biharmonic-type equation, which describes the deflection of a plate resting on a biparametric elastic foundation. The convergence rate of the method is established. The optimal value of the iterative parameter is found. Several numerical examples confirm the efficiency of the proposed method.
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.
Yousef, Hamood Mohammed; Ismail, Ahmad Izani
2017-11-01
In this paper, Laplace Adomian decomposition method (LADM) was applied to solve Delay differential equations with Boundary Value Problems. The solution is in the form of a convergent series which is easy to compute. This approach is tested on two test problem. The findings obtained exhibit the reliability and efficiency of the proposed method.
On an initial-boundary value problem for the nonlinear Schrödinger equation
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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.
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...
Directory of Open Access Journals (Sweden)
Tengfei Shen
2015-12-01
Full Text Available This paper deals with the multiplicity of solutions for Dirichlet boundary conditions of second-order quasilinear equations with impulsive effects. By using critical point theory, a new result is obtained. An example is given to illustrate the main result.
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Mohamed Jleli
2014-01-01
Full Text Available A class of nonlinear multipoint boundary value problems for singular fractional differential equations is considered. By means of a coupled fixed point theorem on ordered sets, some results on the existence and uniqueness of positive solutions are obtained.
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Valery Romanovski
2008-12-01
Full Text Available We prove existence results for second-order impulsive differential equations with antiperiodic boundary value conditions in the presence of classical fixed point theorems. We also obtain the expression of Green's function of related linear operator in the space of piecewise continuous functions.
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Yuji Liu
2004-01-01
Full Text Available A new fixed point theorem on cones is applied to obtain the existence of at least two positive solutions of a higher-order three-point boundary value problem for the differential equation subject to a class ofboundary value conditions. The associated Green's function is given. Some results obtained recently are generalized.
Positive solutions of multi-point boundary value problem of fractional differential equation
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De-xiang Ma
2015-07-01
Full Text Available By means of two fixed-point theorems on a cone in Banach spaces, some existence and multiplicity results of positive solutions of a nonlinear fractional differential equation boundary value problem are obtained. The proofs are based upon some properties of Green’s function, which are also the key of the paper.
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Romanovski Valery
2008-01-01
Full Text Available We prove existence results for second-order impulsive differential equations with antiperiodic boundary value conditions in the presence of classical fixed point theorems. We also obtain the expression of Green's function of related linear operator in the space of piecewise continuous functions.
BOUNDARY VALUE PROBLEM FOR A LOADED EQUATION ELLIPTIC-HYPERBOLIC TYPE IN A DOUBLY CONNECTED DOMAIN
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O.Kh. Abdullaev
2014-06-01
Full Text Available We study the existence and uniqueness of the solution of one boundary value problem for the loaded elliptic-hyperbolic equation of the second order with two lines of change of type in double-connected domain. Similar results have been received by D.M.Kuryhazov, when investigated domain is one-connected.
Null exact controllability of the parabolic equations with equivalued surface boundary condition
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2006-01-01
Full Text Available This paper is devoted to showing the null exact controllability for a class of parabolic equations with equivalued surface boundary condition. Our method is based on the duality argument and global Carleman-type estimate for a parabolic operator.
L1-Solutions of Boundary Value Problems for Implicit Fractional Order Differential Equations
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Mouffak Benchohra
2015-12-01
Full Text Available The aim of this paper is to present new results on the existence of solutions for a class of boundary value problem for fractional order implicit differential equations involving the Caputo fractional derivative. Our results are based on Schauder's fixed point theorem and the Banach contraction principle fixed point theorem.
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Saowaluk Chasreechai
2015-01-01
Full Text Available We study existence and uniqueness results for Caputo fractional sum-difference equations with fractional sum boundary value conditions, by using the Banach contraction principle and Schaefer’s fixed point theorem. Our problem contains different numbers of order in fractional difference and fractional sums. Finally, we present some examples to show the importance of these results.
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.
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Zhiqiang Zhou
2017-01-01
Full Text Available We study the pricing of the American options with fractal transmission system under two-state regime switching models. This pricing problem can be formulated as a free boundary problem of time-fractional partial differential equation (FPDE system. Firstly, applying Laplace transform to the governing FPDEs with respect to the time variable results in second-order ordinary differential equations (ODEs with two free boundaries. Then, the solutions of ODEs are expressed in an explicit form. Consequently the early exercise boundaries and the values for the American option are recovered using the Gaver-Stehfest formula. Numerical comparisons of the methods with the finite difference methods are carried out to verify the efficiency of the methods.
Attractor of Beam Equation with Structural Damping under Nonlinear Boundary Conditions
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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.
Integrable fourth-order difference equations
Energy Technology Data Exchange (ETDEWEB)
Maheswari, C Uma; Sahadevan, R, E-mail: umagenie2004@hotmail.co, E-mail: ramajayamsaha@yahoo.co.i [Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai-600 005, Tamil Nadu (India)
2010-06-11
In this paper an attempt is made to find four-dimensional analogs of two-dimensional Quispel, Roberts and Thompson mappings and identified four distinct cases have been identified. The obtained mappings are measure preserving. The integrability of the isolated mappings is examined by constructing a sufficient number of integrals and their symplectic structure wherever possible.
Evaluation of outflow boundary conditions for two-phase lattice Boltzmann equation.
Lou, Qin; Guo, Zhaoli; Shi, Baochang
2013-06-01
Outflow boundary condition (OBC) is a critical issue in computational fluid dynamics. As a type of numerical method for fluid flows, the lattice Boltzmann equation (LBE) method has gained much success in a variety of complex flows, and certain OBCs have been suggested for the LBE in simulating simple single-phase flows. However, very few discussions on the OBCs have been made for the two-phase LBE method. In this work, three types of OBCs that are widely used in the LBE for single-phase flows, i.e., the Neumann boundary condition, the convective boundary condition, and the extrapolation boundary condition, are extended to a two-phase LBE method and their performances are investigated. The comprehensive results of several two-phase flows show that these boundary conditions behave quite differently in the simulations of two-phase flows. Specifically, it is found that the Neumann boundary condition and the extrapolation boundary condition give rather poor predictions, while the type of convective boundary conditions work well, although the choice of the convection velocity has some slight influences on the results. We also apply these OBC schemes to some other two-phase models, and similar observations are found.
Kirsch, Andreas
2015-01-01
This book gives a concise introduction to the basic techniques needed for the theoretical analysis of the Maxwell Equations, and filters in an elegant way the essential parts, e.g., concerning the various function spaces needed to rigorously investigate the boundary integral equations and variational equations. The book arose from lectures taught by the authors over many years and can be helpful in designing graduate courses for mathematically orientated students on electromagnetic wave propagation problems. The students should have some knowledge on vector analysis (curves, surfaces, divergence theorem) and functional analysis (normed spaces, Hilbert spaces, linear and bounded operators, dual space). Written in an accessible manner, topics are first approached with simpler scale Helmholtz Equations before turning to Maxwell Equations. There are examples and exercises throughout the book. It will be useful for graduate students and researchers in applied mathematics and engineers working in the theoretical ap...
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Seppo Seikkala
2008-08-01
Full Text Available In this article we derive existence and comparison results for discontinuous non-absolute functional integral equations of Volterra type in an ordered Banach space which has a regular order cone. The obtained results are then applied to first-order impulsive differential equations.
Tisdell, C. C.
2017-01-01
Solution methods to exact differential equations via integrating factors have a rich history dating back to Euler (1740) and the ideas enjoy applications to thermodynamics and electromagnetism. Recently, Azevedo and Valentino presented an analysis of the generalized Bernoulli equation, constructing a general solution by linearizing the problem…
Symmetry Analysis and Exact Solutions of the 2D Unsteady Incompressible Boundary-Layer Equations
Han, Zhong; Chen, Yong
2017-01-01
To find intrinsically different symmetry reductions and inequivalent group invariant solutions of the 2D unsteady incompressible boundary-layer equations, a two-dimensional optimal system is constructed which attributed to the classification of the corresponding Lie subalgebras. The comprehensiveness and inequivalence of the optimal system are shown clearly under different values of invariants. Then by virtue of the optimal system obtained, the boundary-layer equations are directly reduced to a system of ordinary differential equations (ODEs) by only one step. It has been shown that not only do we recover many of the known results but also find some new reductions and explicit solutions, which may be previously unknown. Supported by the Global Change Research Program of China under Grant No. 2015CB953904, National Natural Science Foundation of China under Grant Nos. 11275072, 11435005, 11675054, and Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things under Grant No. ZF1213
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Paul W. Eloe
2016-11-01
Full Text Available We consider families of two-point boundary value problems for fractional differential equations where the fractional derivative is assumed to be the Riemann-Liouville fractional derivative. The problems considered are such that appropriate differential operators commute and the problems can be constructed as nested boundary value problems for lower order fractional differential equations. Green's functions are then constructed as convolutions of lower order Green's functions. Comparison theorems are known for the Green's functions for the lower order problems and so, we obtain analogous comparison theorems for the two families of higher order equations considered here. We also pose a related open question for a family of Green's functions that do not apparently have convolution representations.
POSITIVE SOLUTIONS OF A NONLINEAR THREE-POINT EIGENVALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS
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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.
Adomian solution of a nonlinear quadratic integral equation
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E.A.A. Ziada
2013-04-01
Full Text Available We are concerned here with a nonlinear quadratic integral equation (QIE. The existence of a unique solution will be proved. Convergence analysis of Adomian decomposition method (ADM applied to these type of equations is discussed. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of Adomian’s series solution. Two methods are used to solve these type of equations; ADM and repeated trapezoidal method. The obtained results are compared.
Symmetries, integrals and solutions of ordinary differential equations ...
Indian Academy of Sciences (India)
first observation was of the transformation of one first integral of the nonlinear ordinary differential equation ... The representative second-order ordinary differential equation of maximal point symmetry, videlicet y = 0,. (2.1) ...... Noetherian Symmetries, Advances in Systems, Signals, Control and Computers,. Bajic VB ed ...
Nonlinear partial differential equations: Integrability, geometry and related topics
Krasil'shchik, Joseph; Rubtsov, Volodya
2017-03-01
Geometry and Differential Equations became inextricably entwined during the last one hundred fifty years after S. Lie and F. Klein's fundamental insights. The two subjects go hand in hand and they mutually enrich each other, especially after the "Soliton Revolution" and the glorious streak of Symplectic and Poisson Geometry methods in the context of Integrability and Solvability problems for Non-linear Differential Equations.
Iterative estimate of the solution of nonlinear integral equations by ...
African Journals Online (AJOL)
The paper considered the application of Picard's iteration scheme in the approximation of solutions of operator equations in Banach spaces. Using Lipschitz continuity condition and the prescribed auxiliary scalar function, the location of existence of solution for a nonlinear integral equation of Fredholm type and second kind ...
Computational Methods for Solving Linear Fuzzy Volterra Integral Equation
National Research Council Canada - National Science Library
Jihan Hamaydi; Naji Qatanani
2017-01-01
Two numerical schemes, namely, the Taylor expansion and the variational iteration methods, have been implemented to give an approximate solution of the fuzzy linear Volterra integral equation of the second kind...
On solutions of a Volterra integral equation with deviating arguments
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M. Diana Julie
2009-04-01
Full Text Available In this article, we establish the existence and asymptotic characterization of solutions to a nonlinear Volterra integral equation with deviating arguments. Our proof is based on measure of noncompactness and the Schauder fixed point theorem.
-Stability Approach to Variational Iteration Method for Solving Integral Equations
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Rhoades BE
2009-01-01
Full Text Available We consider -stability definition according to Y. Qing and B. E. Rhoades (2008 and we show that the variational iteration method for solving integral equations is -stable. Finally, we present some text examples to illustrate our result.
Application of radial basis function to approximate functional integral equations
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Reza Firouzdor
2016-06-01
Full Text Available In the present paper, Radial Basis Function (RBF interpolation is applied to approximate the numerical solution of both Fredlholm and Volterra functional integral equations. RBF interpolation is based on linear combinations of terms which include a single univariate function. Applying RBF in functional integral equation, a linear system $ \\Psi C=G $ will be obtain in which by defining coefficient vector $ C $, target function will be approximiated. Finally, validity of the method is illustrated by some examples.
Extending the diffusion approximation to the boundary using an integrated diffusion model
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Chen, Chen; Du, Zhidong; Pan, Liang, E-mail: liangpan@purdue.edu [School of Mechanical Engineering, Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907 (United States)
2015-06-15
The widely used diffusion approximation is inaccurate to describe the transport behaviors near surfaces and interfaces. To solve such stochastic processes, an integro-differential equation, such as the Boltzmann transport equation (BTE), is typically required. In this work, we show that it is possible to keep the simplicity of the diffusion approximation by introducing a nonlocal source term and a spatially varying diffusion coefficient. We apply the proposed integrated diffusion model (IDM) to a benchmark problem of heat conduction across a thin film to demonstrate its feasibility. We also validate the model when boundary reflections and uniform internal heat generation are present.
Extending the diffusion approximation to the boundary using an integrated diffusion model
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Chen Chen
2015-06-01
Full Text Available The widely used diffusion approximation is inaccurate to describe the transport behaviors near surfaces and interfaces. To solve such stochastic processes, an integro-differential equation, such as the Boltzmann transport equation (BTE, is typically required. In this work, we show that it is possible to keep the simplicity of the diffusion approximation by introducing a nonlocal source term and a spatially varying diffusion coefficient. We apply the proposed integrated diffusion model (IDM to a benchmark problem of heat conduction across a thin film to demonstrate its feasibility. We also validate the model when boundary reflections and uniform internal heat generation are present.
Boundary value problem for one-dimensional fractional differential advection-dispersion equation
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Khasambiev Mokhammad Vakhaevich
2014-07-01
Full Text Available An equation commonly used to describe solute transport in aquifers has attracted more attention in recent years. After a formal study of some aspects of the advection-diffusion equation, basically from the mathematical point of view with the solution of a differential equation with fractional derivative, the main interest to this problem shifted onto physical aspects of the dynamical system, such as the total energy and the dynamical response. In this regard it should be pointed out that the interaction with environment is expressed in terms of stochastic arrow of time. This allows one also to reach a progress in one more issue. Formerly the equation of advection-diffusion was not obtained from any physical principles. However, mainly the success concerns linear fractional systems. In fact, there are many cases in which linear treatments are not sufficient. The more general systems described by nonlinear fractional differential equations have not been studied enough. The ordinary calculus brings out clearly that essentially new phenomena occur in nonlinear systems, which generally cannot occur in linear systems. Due to vast range of application of the fractional advection-dispersion equation, a lot of work has been done to find numerical solution and fundamental solution of this equation. The research on the analytical solution of initial-boundary problem for space-fractional advection-dispersion equation is relatively new and is still at an early stage of development. In this paper, we will take use of the method of variable separation to solve space-fractional advection-dispersion equation with initial boundary data.
On the numerical solution of the diffusion equation with a nonlocal boundary condition
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Dehghan Mehdi
2003-01-01
Full Text Available Parabolic partial differential equations with nonlocal boundary specifications feature in the mathematical modeling of many phenomena. In this paper, numerical schemes are developed for obtaining approximate solutions to the initial boundary value problem for one-dimensional diffusion equation with a nonlocal constraint in place of one of the standard boundary conditions. The method of lines (MOL semidiscretization approach is used to transform the model partial differential equation into a system of first-order linear ordinary differential equations (ODEs. The partial derivative with respect to the space variable is approximated by a second-order finite-difference approximation. The solution of the resulting system of first-order ODEs satisfies a recurrence relation which involves a matrix exponential function. Numerical techniques are developed by approximating the exponential matrix function in this recurrence relation. We use a partial fraction expansion to compute the matrix exponential function via Pade approximations, which is particularly useful in parallel processing. The algorithm is tested on a model problem from the literature.
Hosseini, Vahid Reza; Shivanian, Elyas; Chen, Wen
2015-02-01
In this article, a general type of two-dimensional time-fractional telegraph equation explained by the Caputo derivative sense for (1 < α ≤ 2) is considered and analyzed by a method based on the Galerkin weak form and local radial point interpolant (LRPI) approximation subject to given appropriate initial and Dirichlet boundary conditions. In the proposed method, so-called meshless local radial point interpolation (MLRPI) method, a meshless Galerkin weak form is applied to the interior nodes while the meshless collocation method is used for the nodes on the boundary, so the Dirichlet boundary condition is imposed directly. The point interpolation method is proposed to construct shape functions using the radial basis functions. In the MLRPI method, it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares. Two numerical examples are presented and satisfactory agreements are achieved.
Rebelo, Raphaël; Winternitz, Pavel
2017-01-01
This book shows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference equations. Difference equations are playing an increasingly important role in the natural sciences. Indeed, many phenomena are inherently discrete and thus naturally described by difference equations. More fundamentally, in subatomic physics, space-time may actually be discrete. Differential equations would then just be approximations of more basic discrete ones. Moreover, when using differential equations to analyze continuous processes, it is often necessary to resort to numerical methods. This always involves a discretization of the differential equations involved, thus replacing them by difference ones. Each of the nine peer-reviewed chapters in this volume serves as a self-contained treatment of a topic, containing introductory material as well as the latest research results and exercises. Each chapter is presented by one or more early career researchers...
Ge, Liang; Sotiropoulos, Fotis
2007-08-01
A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [1]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus.
Integral representation of a solution of Heun's general equation
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Jomo Batola
2007-06-01
Full Text Available We establish an integral representation for the Frobenius solution with an exponent zero at $z=0$ of the general Heun equation. First we present an extension of Mellin's lemma which provides a powerful method that takes into account differential equations which are not of the form studied by Mellin. That is the case for equations of Heun's type. It is that aspect which makes our work different from Valent's work. The method is powerful because it allows obtaining directly the nucleus equation of the representation. The integral representation formula obtained with this method leads quickly and naturally to already known results in the case of hypergeometric functions. The generalisation of this method gives a type of differential equations which form is a novelty and deserves to be studied further.
Ding, Xiao-Li; Nieto, Juan J.
2017-11-01
In this paper, we consider the analytical solutions of coupling fractional partial differential equations (FPDEs) with Dirichlet boundary conditions on a finite domain. Firstly, the method of successive approximations is used to obtain the analytical solutions of coupling multi-term time fractional ordinary differential equations. Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the coupling FPDEs to the coupling multi-term time fractional ordinary differential equations. By applying the obtained analytical solutions to the resulting multi-term time fractional ordinary differential equations, the desired analytical solutions of the coupling FPDEs are given. Our results are applied to derive the analytical solutions of some special cases to demonstrate their applicability.
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.
Commentary on local and boundary regularity of weak solutions to Navier-Stokes equations
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Zdenek Skalak
2004-01-01
Full Text Available We present results on local and boundary regularity for weak solutions to the Navier-Stokes equations. Beginning with the regularity criterion proved recently in [14] for the Cauchy problem, we show that this criterion holds also locally. This is also the case for some other results: We present two examples concerning the regularity of weak solutions stemming from the regularity of two components of the vorticity ([2] or from the regularity of the pressure ([3]. We conclude by presenting regularity criteria near the boundary based on the results in [10] and [16].
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.
Fringe integral equation method for a truncated grounded dielectric slab
DEFF Research Database (Denmark)
Jørgensen, Erik; Maci, S.; Toccafondi, A.
2001-01-01
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...... 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...
Periodic solutions of Volterra integral equations
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M. N. Islam
1988-01-01
Full Text Available Consider the system of equationsx(t=f(t+∫−∞tk(t,sx(sds, (1andx(t=f(t+∫−∞tk(t,sg(s,x(sds. (2Existence of continuous periodic solutions of (1 is shown using the resolvent function of the kernel k. Some important properties of the resolvent function including its uniqueness are obtained in the process. In obtaining periodic solutions of (1 it is necessary that the resolvent of k is integrable in some sense. For a scalar convolution kernel k some explicit conditions are derived to determine whether or not the resolvent of k is integrable. Finally, the existence and uniqueness of continuous periodic solutions of (1 and (2 are btained using the contraction mapping principle as the basic tool.
The Solutions of Mixed Monotone Fredholm-Type Integral Equations in Banach Spaces
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Hua Su
2013-01-01
Full Text Available By introducing new definitions of ϕ convex and -φ concave quasioperator and v0 quasilower and u0 quasiupper, by means of the monotone iterative techniques without any compactness conditions, we obtain the iterative unique solution of nonlinear mixed monotone Fredholm-type integral equations in Banach spaces. Our results are even new to ϕ convex and -φ concave quasi operator, and then we apply these results to the two-point boundary value problem of second-order nonlinear ordinary differential equations in the ordered Banach spaces.
Kosaka, Atsushi
2012-01-01
We consider a positive solution of the Emden equation with the critical Sobolev exponent on a geodesic ball in S3. In the case of the Dirichlet boundary condition, Bandle and Peletier [2] proved the precise result on the existence of a positive radial solution. We investigate the same equation with the third kind boundary condition and obtain a more general result. Namely we prove that the existence and the nonexistence of solutions depend on the geodesic radius and the boundary condition. Mo...
Collins, J. D.; Volakis, John L.
1992-01-01
A method that combines the finite element and boundary integral techniques for the numerical solution of electromagnetic scattering problems is presented. The finite element method is well known for requiring a low order storage and for its capability to model inhomogeneous structures. Of particular emphasis in this work is the reduction of the storage requirement by terminating the finite element mesh on a boundary in a fashion which renders the boundary integrals in convolutional form. The fast Fourier transform is then used to evaluate these integrals in a conjugate gradient solver, without a need to generate the actual matrix. This method has a marked advantage over traditional integral equation approaches with respect to the storage requirement of highly inhomogeneous structures. Rectangular, circular, and ogival mesh termination boundaries are examined for two-dimensional scattering. In the case of axially symmetric structures, the boundary integral matrix storage is reduced by exploiting matrix symmetries and solving the resulting system via the conjugate gradient method. In each case several results are presented for various scatterers aimed at validating the method and providing an assessment of its capabilities. Important in methods incorporating boundary integral equations is the issue of internal resonance. A method is implemented for their removal, and is shown to be effective in the two-dimensional and three-dimensional applications.
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.
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R. C. Mittal
2014-01-01
Full Text Available We present a technique based on collocation of cubic B-spline basis functions to solve second order one-dimensional hyperbolic telegraph equation with Neumann boundary conditions. The use of cubic B-spline basis functions for spatial variable and its derivatives reduces the problem into system of first order ordinary differential equations. The resulting system subsequently has been solved by SSP-RK54 scheme. The accuracy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and in good agreement with the exact solution.
Solutions of the Helmholtz equation with boundary conditions for force-free magnetic fields
Rasband, S. N.; Turner, L.
1981-01-01
It is shown that the solution, with one ignorable coordinate, for the Taylor minimum energy state (resulting in a force-free magnetic field) in either a straight cylindrical or a toroidal geometry with arbitrary cross section can be reduced to the solution of either an inhomogeneous Helmholtz equation or a Grad-Shafranov equation with simple boundary conditions. Standard Green's function theory is, therefore, applicable. Detailed solutions are presented for the Taylor state in toroidal and cylindrical domains having a rectangular cross section. The focus is on solutions corresponding to the continuous eigenvalue spectra. Singular behavior at 90 deg corners is explored in detail.
Energy Technology Data Exchange (ETDEWEB)
Ackroyd, R.T. (UKAEA Risley Nuclear Power Development Establishment. Process Technology and Safety Directorate)
1983-01-01
A completely boundary-free maximum principle for the first-order Boltzmann equation is derived from the completely boundary-free maximum principle for the mixed-parity Boltzmann equation. When continuity is imposed on the trial function for directions crossing interfaces the completely boundary-free principle for the first-order Boltzmann equation reduces to a maximum principle previously established directly from first principles and indirectly by the Euler-Lagrange method. Present finite element methods for the first-order Boltzmann equation are based on a weighted-residual method which permits the use of discontinuous trial functions. The new principle for the first-order equation can be used as a basis for finite-element methods with the same freedom from boundary conditions as those based on the weighted-residual method. The extremum principle as the parent of the variationally-derived weighted-residual equations ensures their good behaviour.
An algorithm of computing inhomogeneous differential equations for definite integrals
Nakayama, Hiromasa; Nishiyama, Kenta
2010-01-01
We give an algorithm to compute inhomogeneous differential equations for definite integrals with parameters. The algorithm is based on the integration algorithm for $D$-modules by Oaku. Main tool in the algorithm is the Gr\\"obner basis method in the ring of differential operators.
Applied partial differential equations
DuChateau, Paul
2012-01-01
Book focuses mainly on boundary-value and initial-boundary-value problems on spatially bounded and on unbounded domains; integral transforms; uniqueness and continuous dependence on data, first-order equations, and more. Numerous exercises included.
Niu, Jun; Ren, Yi; Liu, Qing Huo
2017-10-02
In this work, we propose a numerical solver combining the spectral element - boundary integral (SEBI) method with the periodic layered medium dyadic Green's function. The periodic layered medium dyadic Green's function is formulated under matrix representation. The surface integral equations (SIEs) are then implemented as the radiation boundary condition to truncate the top and bottom computation domain. After describing the interior computation domain with the vector wave equations, and treating the lateral boundaries with Bloch periodic boundary conditions, the whole computation domains are discretized with mixed-order Gauss- Lobatto-Legendre basis functions in the SEBI method. This method avoids the discretization of the top and bottom layered media, so it can be much more efficient than conventional methods. Numerical results validate the proposed solver with fast convergence throughout the whole computation domain and good performance for typical multiscale nano-optical applications.
Nonzero solutions of nonlinear integral equations modeling infectious disease
Energy Technology Data Exchange (ETDEWEB)
Williams, L.R. (Indiana Univ., South Bend); Leggett, R.W.
1982-01-01
Sufficient conditions to insure the existence of periodic solutions to the nonlinear integral equation, x(t) = ..integral../sup t//sub t-tau/f(s,x(s))ds, are given in terms of simple product and product integral inequalities. The equation can be interpreted as a model for the spread of infectious diseases (e.g., gonorrhea or any of the rhinovirus viruses) if x(t) is the proportion of infectives at time t and f(t,x(t)) is the proportion of new infectives per unit time.
Directory of Open Access Journals (Sweden)
Vrabel Robert
2011-01-01
Full Text Available Abstract This paper deals with the existence and asymptotic behavior of the solutions to the singularly perturbed second-order nonlinear differential equations. For example, feedback control problems, such as the steady states of the thermostats, where the controllers add or remove heat, depending upon the temperature detected by the sensors in other places, can be interpreted with a second-order ordinary differential equation subject to a nonlocal four-point boundary condition. Singular perturbation problems arise in the heat transfer problems with large Peclet numbers. We show that the solutions of mathematical model, in general, start with fast transient which is the so-called boundary layer phenomenon, and after decay of this transient they remain close to the solution of reduced problem with an arising new fast transient at the end of considered interval. Our analysis relies on the method of lower and upper solutions.
Mixed Initial-Boundary Value Problem for the Capillary Wave Equation
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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.
Blow-up analysis for a system of heat equations coupled through a nonlinear boundary condition
DEFF Research Database (Denmark)
Pedersen, M.; Lin, Zhigui
2001-01-01
Consider the system of heat equations uit - Δui = 0 (i = 1 , . . . , k, uk+i := u1) in Ω x (0, T) coupled through nonlinear boundary conditions ∂ui/∂η = up1i+1 on ∂Ω x [0, T). The upper and lower bounds of the blow-up rate is derived. © 2000 Elsevier Science Ltd. All rights reserved.......Consider the system of heat equations uit - Δui = 0 (i = 1 , . . . , k, uk+i := u1) in Ω x (0, T) coupled through nonlinear boundary conditions ∂ui/∂η = up1i+1 on ∂Ω x [0, T). The upper and lower bounds of the blow-up rate is derived. © 2000 Elsevier Science Ltd. All rights reserved....
DEFF Research Database (Denmark)
Mariegaard, Jesper Sandvig
We consider a control problem for the wave equation: Given the initial state, find a specific boundary condition, called a control, that steers the system to a desired final state. The Hilbert uniqueness method (HUM) is a mathematical method for the solution of such control problems. It builds....... As an example, we employ a HUM solution to an inverse source problem for the wave equation: Given boundary measurements for a wave problem with a separable source, find the spatial part of the source term. The reconstruction formula depends on a set of HUM eigenfunction controls; we suggest a discretization...... and show its convergence. We compare results obtained by L-FEM controls and DG-FEM controls. The reconstruction formula is seen to be quite sensitive to control inaccuracies which indeed favors DG-FEM over L-FEM....
Scattering of surface waves modelled by the integral equation method
Lu, Laiyu; Maupin, Valerie; Zeng, Rongsheng; Ding, Zhifeng
2008-09-01
The integral equation method is used to model the propagation of surface waves in 3-D structures. The wavefield is represented by the Fredholm integral equation, and the scattered surface waves are calculated by solving the integral equation numerically. The integration of the Green's function elements is given analytically by treating the singularity of the Hankel function at R = 0, based on the proper expression of the Green's function and the addition theorem of the Hankel function. No far-field and Born approximation is made. We investigate the scattering of surface waves propagating in layered reference models imbedding a heterogeneity with different density, as well as Lamé constant contrasts, both in frequency and time domains, for incident plane waves and point sources.
Holst, Michael; Meier, Caleb; Tsogtgerel, G.
2017-11-01
In this article we continue our effort to do a systematic development of the solution theory for conformal formulations of the Einstein constraint equations on compact manifolds with boundary. By building in a natural way on our recent work in Holst and Tsogtgerel (Class Quantum Gravity 30:205011, 2013), and Holst et al. (Phys Rev Lett 100(16):161101, 2008, Commun Math Phys 288(2):547-613, 2009), and also on the work of Maxwell (J Hyperbolic Differ Eqs 2(2):521-546, 2005a, Commun Math Phys 253(3):561-583, 2005b, Math Res Lett 16(4):627-645, 2009) and Dain (Class Quantum Gravity 21(2):555-573, 2004), under reasonable assumptions on the data we prove existence of both near- and far-from-constant mean curvature (CMC) solutions for a class of Robin boundary conditions commonly used in the literature for modeling black holes, with a third existence result for CMC appearing as a special case. Dain and Maxwell addressed initial data engineering for space-times that evolve to contain black holes, determining solutions to the conformal formulation on an asymptotically Euclidean manifold in the CMC setting, with interior boundary conditions representing excised interior black hole regions. Holst and Tsogtgerel compiled the interior boundary results covered by Dain and Maxwell, and then developed general interior conditions to model the apparent horizon boundary conditions of Dainand Maxwell for compact manifolds with boundary, and subsequently proved existence of solutions to the Lichnerowicz equation on compact manifolds with such boundary conditions. This paper picks up where Holst and Tsogtgerel left off, addressing the general non-CMC case for compact manifolds with boundary. As in our previous articles, our focus here is again on low regularity data and on the interaction between different types of boundary conditions. While our work here serves primarily to extend the solution theory for the compact with boundary case, we also develop several technical tools that have
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.
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Jinhua Wang
2010-01-01
Full Text Available We consider the existence and uniqueness of positive solution to nonzero boundary values problem for a coupled system of fractional differential equations. The differential operator is taken in the standard Riemann-Liouville sense. By using Banach fixed point theorem and nonlinear differentiation of Leray-Schauder type, the existence and uniqueness of positive solution are obtained. Two examples are given to demonstrate the feasibility of the obtained results.
Multi-point boundary value problems for linear functional-differential equations
Czech Academy of Sciences Publication Activity Database
Domoshnitsky, A.; Hakl, Robert; Půža, Bedřich
2017-01-01
Roč. 24, č. 2 (2017), s. 193-206 ISSN 1072-947X Institutional support: RVO:67985840 Keywords : boundary value problems * linear functional-differential equations * functional-differential inequalities Subject RIV: BA - General Mathematics Impact factor: 0.290, year: 2016 https://www.degruyter.com/view/j/gmj.2017.24.issue-2/gmj-2016-0076/gmj-2016-0076. xml
Moving-boundary problems for the time-fractional diffusion equation
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Sabrina D. Roscani
2017-02-01
Full Text Available We consider a one-dimensional moving-boundary problem for the time-fractional diffusion equation. The time-fractional derivative of order $\\alpha\\in (0,1$ is taken in the sense of Caputo. We study the asymptotic behaivor, as t tends to infinity, of a general solution by using a fractional weak maximum principle. Also, we give some particular exact solutions in terms of Wright functions.
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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.
On Riemann boundary value problems for null solutions of the two dimensional Helmholtz equation
Bory Reyes, Juan; Abreu Blaya, Ricardo; Rodríguez Dagnino, Ramón Martin; Kats, Boris Aleksandrovich
2018-01-01
The Riemann boundary value problem (RBVP to shorten notation) in the complex plane, for different classes of functions and curves, is still widely used in mathematical physics and engineering. For instance, in elasticity theory, hydro and aerodynamics, shell theory, quantum mechanics, theory of orthogonal polynomials, and so on. In this paper, we present an appropriate hyperholomorphic approach to the RBVP associated to the two dimensional Helmholtz equation in R^2 . Our analysis is based on a suitable operator calculus.
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.
Thompson, J. F.; Warsi, Z. U. A.; Mastin, C. W.
1982-01-01
A comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given. Some general mathematical framework and error analysis common to such coordinate systems is also included. The general categories of generating systems are those based on conformal mapping, orthogonal systems, nearly orthogonal systems, systems produced as the solution of elliptic and hyperbolic partial differential equations, and systems generated algebraically by interpolation among the boundaries. Also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation. Dynamically adaptive coordinate systems, coupled with the physical solution, and time-dependent systems that follow moving boundaries are treated. References reporting experience using such coordinate systems are reviewed as well as those covering the system development.
Zhang, Jialin; Chen, Qian; Li, Jiaji; Zuo, Chao
2017-02-01
The transport of intensity equation (TIE) is a powerful tool for direct quantitative phase retrieval in microscopy imaging. However, there may be some problems when dealing with the boundary condition of the TIE. The previous work introduces a hard-edged aperture to the camera port of the traditional bright field microscope to generate the boundary signal for the TIE solver. Under this Neumann boundary condition, we can obtain the quantitative phase without any assumption or prior knowledge about the test object and the setup. In this paper, we will demonstrate the effectiveness of this method based on some experiments in practice. The micro lens array will be used for the comparison of two TIE solvers results based on introducing the aperture or not and this accurate quantitative phase imaging technique allows measuring cell dry mass which is used in biology to follow cell cycle, to investigate cell metabolism, or to address effects of drugs.
The LEM exponential integrator for advection-diffusion-reaction equations
Caliari, Marco; Vianello, Marco; Bergamaschi, Luca
2007-12-01
We implement a second-order exponential integrator for semidiscretized advection-diffusion-reaction equations, obtained by coupling exponential-like Euler and Midpoint integrators, and computing the relevant matrix exponentials by polynomial interpolation at Leja points. Numerical tests on 2D models discretized in space by finite differences or finite elements, show that the Leja-Euler-Midpoint (LEM) exponential integrator can be up to 5 times faster than a classical second-order implicit solver.
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...
On a nonlocal boundary value problem for the two-term time-fractional diffusion-wave equation
Bazhlekova, E.
2013-10-01
We study a nonlocal boundary value problem for the spatially one-dimensional diffusion-wave equation with two fractional Caputo time-derivatives of different orders α and β, where 1 telegraph equation. In the limiting case of one time-derivative of order 2 (wave equation) the oscillation amplitude of the system increases infinitely with time in the absence of forcing terms in the formulation of the problem. We first give an explanation for this resonant behaviour. Then we study the damping effect due to decreasing of the order of differentiation from 2 to αɛ (1,2) and/or adding a second fractional term. Our considerations are based on the generalized eigenfunction expansion of the solution. The time-dependent components in this expansion are studied in detail. Their properties are derived from the representation as a Laplace inverse integral. It appears that the time-dependent components exhibit oscillations with decreasing amplitude for sufficiently large t, and algebraic decay for t → ∞, except in the case α = 2,β = 1 (telegraph equation), when this decay is exponential. To illustrate the analytical formulas, results of numerical calculations and plots are presented.
Modified decomposition method for nonlinear Volterra-Fredholm integral equations
Energy Technology Data Exchange (ETDEWEB)
Bildik, Necdet [Department of Mathematics, Celal Bayar University, 45030 Manisa (Turkey)]. E-mail: necdet.bildik@bayar.edu.tr; Inc, Mustafa [Department of Mathematics, Firat University, 23119 Elazig (Turkey)]. E-mail: minc@firat.edu.tr
2007-07-15
In this paper, the nonlinear Volterra-Fredholm integral equations are solved by using the modified decomposition method (MDM). The approximate solution of this equation is calculated in the form of a series with easily computable components. The accuracy of the proposed numerical scheme is examined by comparison with other analytical and numerical results. Two test problems are presented to illustrate the reliability and the performance of the modified decomposition method.
Directory of Open Access Journals (Sweden)
Guo Chun Wen
2009-05-01
Full Text Available This article concerns the oblique derivative problems for second-order quasilinear degenerate equations of mixed type with several characteristic boundaries, which include the Tricomi problem as a special case. First we formulate the problem and obtain estimates of its solutions, then we show the existence of solutions by the successive iterations and the Leray-Schauder theorem. We use a complex analytic method: elliptic complex functions are used in the elliptic domain, and hyperbolic complex functions in the hyperbolic domain, such that second-order equations of mixed type with degenerate curve are reduced to the first order mixed complex equations with singular coefficients. An application of the complex analytic method, solves (1.1 below with $m=n=1$, $a=b=0$, which was posed as an open problem by Rassias.
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.
Symbolic-Numeric Integration of the Dynamical Cosserat Equations
Lyakhov, Dmitry A.
2017-08-29
We devise a symbolic-numeric approach to the integration of the dynamical part of the Cosserat equations, a system of nonlinear partial differential equations describing the mechanical behavior of slender structures, like fibers and rods. This is based on our previous results on the construction of a closed form general solution to the kinematic part of the Cosserat system. Our approach combines methods of numerical exponential integration and symbolic integration of the intermediate system of nonlinear ordinary differential equations describing the dynamics of one of the arbitrary vector-functions in the general solution of the kinematic part in terms of the module of the twist vector-function. We present an experimental comparison with the well-established generalized \\\\alpha -method illustrating the computational efficiency of our approach for problems in structural mechanics.
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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.
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Mehriban Imanova Natiq
2012-03-01
Full Text Available Normal 0 false false false EN-US X-NONE X-NONE As is known, many problems of natural science are reduced mainly to the solution of nonlinear Volterra integral equations. The method of quadratures that was first applied by Volterra to solving variable boundary integral equations is popular among numerical methods for the solution of such equations. At present, there are different modifications of the method of quadratures that have bounded accuracies. Here we suggest a second derivative multistep method for constructing more exact methods.
Master equations and the theory of stochastic path integrals.
Weber, Markus F; Frey, Erwin
2017-04-01
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a 'generating functional', which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a 'forward' and a 'backward' path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon
Master equations and the theory of stochastic path integrals
Weber, Markus F.; Frey, Erwin
2017-04-01
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a ‘generating functional’, which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a ‘forward’ and a ‘backward’ path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from
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.
Solvability of Nonlinear Integral Equations of Volterra Type
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Zeqing Liu
2012-01-01
Full Text Available This paper deals with the existence of continuous bounded solutions for a rather general nonlinear integral equation of Volterra type and discusses also the existence and asymptotic stability of continuous bounded solutions for another nonlinear integral equation of Volterra type. The main tools used in the proofs are some techniques in analysis and the Darbo fixed point theorem via measures of noncompactness. The results obtained in this paper extend and improve essentially some known results in the recent literature. Two nontrivial examples that explain the generalizations and applications of our results are also included.
Investigation of carbon nanotube antennas using thin wire integral equations
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N. Fichtner
2008-05-01
Full Text Available In this paper the characteristics of small carbon nanotube (CNT dipole antennas are investigated on the basis of the thin wire Hallén integral equation (IE. A surface impedance model for the CNT is adopted to account for the specific material properties resulting in a modified kernel function for the integral equation. A numerical solution for the IE gives the current distribution along the CNT. From the current distribution the antenna driving point impedance and the antenna efficiency are computed. The presented numerical examples demonstrate the strong dependence of the antenna characteristics on the used material and show the limitations of nanoscale antennas.
An overview of integration methods for hypersingular boundary integrals
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Lutz, E.; Ingraffea, A.R. (Cornell Univ., Ithaca, NY (United States)); Gray, L.J. (Oak Ridge National Lab., TN (United States))
1991-01-01
Several methods of analyzing the hypersignular gradient BIE have been developed recently. This paper is a review highlighting the numerous common aspects and several differences among the methods. Significant common aspects include (a) a regularization of constant and linear terms, (b) analysis of integration points near rather than on the surface, and (c) analysis of the neighborhood of the singular point rather than of individual elements. 26 refs.
Musharbash, Eleonora; Nobile, Fabio
2018-02-01
In this paper we propose a method for the strong imposition of random Dirichlet boundary conditions in the Dynamical Low Rank (DLR) approximation of parabolic PDEs and, in particular, incompressible Navier Stokes equations. We show that the DLR variational principle can be set in the constrained manifold of all S rank random fields with a prescribed value on the boundary, expressed in low rank format, with rank smaller then S. We characterize the tangent space to the constrained manifold by means of a Dual Dynamically Orthogonal (Dual DO) formulation, in which the stochastic modes are kept orthonormal and the deterministic modes satisfy suitable boundary conditions, consistent with the original problem. The Dual DO formulation is also convenient to include the incompressibility constraint, when dealing with incompressible Navier Stokes equations. We show the performance of the proposed Dual DO approximation on two numerical test cases: the classical benchmark of a laminar flow around a cylinder with random inflow velocity, and a biomedical application for simulating blood flow in realistic carotid artery reconstructed from MRI data with random inflow conditions coming from Doppler measurements.
Integral equations for the four-body problem; Equations integrales pour le probleme a quatre corps
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Mora, Ch. [Laboratoire Pierre-Aigrain, Ecole normale superieure, CNRS and Paris 7 Diderot, 24, rue Lhomond, 75231 Paris cedex 05 (France); Castin, Y. [Laboratoire Kastler Brossel, Ecole normale superieure, CNRS and UPMC, 24, rue Lhomond, 75231 Paris cedex 05 (France); Pricoupenko, L. [Laboratoire de physique theorique de la matiere condensee, UPMC and CNRS, 4, place Jussieu, 75252 Paris cedex 05 (France)
2011-01-15
We consider the four-boson and 3+1 fermionic problems with a model Hamiltonian which encapsulates the mechanism of the Feshbach resonance involving the coherent coupling of two atoms in the open channel and a molecule in the closed channel. The model includes also the pair-wise direct interaction between atoms in the open channel and in the bosonic case, the direct molecule-molecule interaction in the closed channel. Interactions are modeled by separable potentials which makes it possible to reduce the four-body problem to the study of a single integral equation. We take advantage of the rotational symmetry and parity invariance of the Hamiltonian to reduce the general eigenvalue equation in each angular momentum sector to an integral equation for functions of three real variables only. A first application of this formalism in the zero-range limit is given elsewhere [Y. Castin, C. Mora, L. Pricoupenko, Phys. Rev. Lett. 105 (2010) 223201]. (authors)
Numerical treatments for solving nonlinear mixed integral equation
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M.A. Abdou
2016-12-01
Full Text Available We consider a mixed type of nonlinear integral equation (MNLIE of the second kind in the space C[0,T]×L2(Ω,T<1. The Volterra integral terms (VITs are considered in time with continuous kernels, while the Fredholm integral term (FIT is considered in position with singular general kernel. Using the quadratic method and separation of variables method, we obtain a nonlinear system of Fredholm integral equations (NLSFIEs with singular kernel. A Toeplitz matrix method, in each case, is then used to obtain a nonlinear algebraic system. Numerical results are calculated when the kernels take a logarithmic form or Carleman function. Moreover, the error estimates, in each case, are then computed.
Fredholm-Volterra integral equation with potential kernel
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M. A. Abdou
2001-01-01
Full Text Available A method is used to solve the Fredholm-Volterra integral equation of the first kind in the space L2(Ω×C(0,T, Ω={(x,y:x2+y2≤a}, z=0, and T<∞. The kernel of the Fredholm integral term considered in the generalized potential form belongs to the class C([Ω]×[Ω], while the kernel of Volterra integral term is a positive and continuous function that belongs to the class C[0,T]. Also in this work the solution of Fredholm integral equation of the second and first kind with a potential kernel is discussed. Many interesting cases are derived and established in the paper.
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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.
An integrable semi-discretization of the Boussinesq equation
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Zhang, Yingnan, E-mail: ynzhang@njnu.edu.cn [Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu (China); Tian, Lixin, E-mail: tianlixin@njnu.edu.cn [Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu (China); Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu (China)
2016-10-23
Highlights: • A new integrable semi-discretization of the Boussinesq equation is present. • A Bäcklund transformation and a Lax pair for the differential-difference system is derived by using Hirota's bilinear method. • The soliton solutions of 'good' Boussinesq equation and numerical algorithms are investigated. - Abstract: In this paper, we present an integrable semi-discretization of the Boussinesq equation. Different from other discrete analogues, we discretize the ‘time’ variable and get an integrable differential-difference system. Under a standard limitation, the differential-difference system converges to the continuous Boussinesq equation such that the discrete system can be used to design numerical algorithms. Using Hirota's bilinear method, we find a Bäcklund transformation and a Lax pair of the differential-difference system. For the case of ‘good’ Boussinesq equation, we investigate the soliton solutions of its discrete analogue and design numerical algorithms. We find an effective way to reduce the phase shift caused by the discretization. The numerical results coincide with our analysis.
The pentabox Master Integrals with the Simplified Differential Equations approach
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Papadopoulos, Costas G. [Institute of Nuclear and Particle Physics, NCSR ‘Demokritos’,PO Box 60037, Agia Paraskevi, 15310 (Greece); University of Debrecen and MTA-DE Particle Physics Research Group,PO Box 105, H-4010 Debrecen (Hungary); Tommasini, Damiano [Institute of Nuclear and Particle Physics, NCSR ‘Demokritos’,PO Box 60037, Agia Paraskevi, 15310 (Greece); Wever, Christopher [Institute of Nuclear and Particle Physics, NCSR ‘Demokritos’,PO Box 60037, Agia Paraskevi, 15310 (Greece); Institute for Theoretical Particle Physics (TTP), Karlsruhe Institute of Technology,Engesserstraße 7, D-76128 Karlsruhe (Germany); Institute for Nuclear Physics (IKP), Karlsruhe Institute of Technology,Hermann-von-Helmholtz-Platz 1, D-76344 Eggenstein-Leopoldshafen (Germany)
2016-04-13
We present the calculation of massless two-loop Master Integrals relevant to five-point amplitudes with one off-shell external leg and derive the complete set of planar Master Integrals with five on-mass-shell legs, that contribute to many 2→3 amplitudes of interest at the LHC, as for instance three jet production, γ,V,H+2 jets etc., based on the Simplified Differential Equations approach.
A Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion Equation
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Wenyuan Liao
2015-01-01
Full Text Available The semidiscrete ordinary differential equation (ODE system resulting from compact higher-order finite difference spatial discretization of a nonlinear parabolic partial differential equation, for instance, the reaction-diffusion equation, is highly stiff. Therefore numerical time integration methods with stiff stability such as implicit Runge-Kutta methods and implicit multistep methods are required to solve the large-scale stiff ODE system. However those methods are computationally expensive, especially for nonlinear cases. Rosenbrock method is efficient since it is iteration-free; however it suffers from order reduction when it is used for nonlinear parabolic partial differential equation. In this work we construct a new fourth-order Rosenbrock method to solve the nonlinear parabolic partial differential equation supplemented with Dirichlet or Neumann boundary condition. We successfully resolved the phenomena of order reduction, so the new method is fourth-order in time when it is used for nonlinear parabolic partial differential equations. Moreover, it has been shown that the Rosenbrock method is strongly A-stable hence suitable for the stiff ODE system obtained from compact finite difference discretization of the nonlinear parabolic partial differential equation. Several numerical experiments have been conducted to demonstrate the efficiency, stability, and accuracy of the new method.
Higher-Order Integral Equation Methods in Computational Electromagnetics
DEFF Research Database (Denmark)
Jørgensen, Erik; Meincke, Peter
Higher-order integral equation methods have been investigated. The study has focused on improving the accuracy and efficiency of the Method of Moments (MoM) applied to electromagnetic problems. A new set of hierarchical Legendre basis functions of arbitrary order is developed. The new basis...
Weighted norms and Volterra integral equations in LP spaces
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Jaroslaw Kwapisz
1991-01-01
Full Text Available A new simple proof of existence and uniqueness of solutions of the Volterra integral equation in Lebesque spaces is given. It is shown that the weighted norm technique and the Banach contraction mapping principle can be applied (as in the case of continuous functions space.
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
Yoon, Gangjoon; Min, Chohong
2017-11-01
The Shortley-Weller method is a standard finite difference method for solving the Poisson equation with Dirichlet boundary condition. Unless the domain is rectangular, the method meets an inevitable problem that some of the neighboring nodes may be outside the domain. In this case, an usual treatment is to extrapolate the function values at outside nodes by quadratic polynomial. The extrapolation may become unstable in the sense that some of the extrapolation coefficients increase rapidly when the grid nodes are getting closer to the boundary. A practical remedy, which we call artificial perturbation, is to treat grid nodes very near the boundary as boundary points. The aim of this paper is to reveal the adverse effects of the artificial perturbation on solving the linear system and the convergence of the solution. We show that the matrix is nearly symmetric so that the ratio of its minimum and maximum eigenvalues is an important factor in solving the linear system. Our analysis shows that the artificial perturbation results in a small enhancement of the eigenvalue ratio from O (1 / (h ṡhmin) to O (h-3) and triggers an oscillatory order of convergence. Instead, we suggest using Jacobi or ILU-type preconditioner on the matrix without applying the artificial perturbation. According to our analysis, the preconditioning not only reduces the eigenvalue ratio from O (1 / (h ṡhmin) to O (h-2), but also keeps the sharp second order convergence.
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Sobhan Mosayebidorcheh
2013-01-01
Full Text Available The boundary layer equation of the pseudoplastic fluid over a flat plate is considered. This equation is a boundary value problem (BVP with the high nonlinearity and a boundary condition at infinity. To solve such problems, powerful numerical techniques are usually used. Here, through using differential transform method (DTM, the BVP is replaced by two initial value problems (IVP and then multi-step differential transform method (MDTM is applied to solve them. The differential equation and its boundary conditions are transformed to a set of algebraic equations, and the Taylor series of solution is calculated in every sub domain. In this solution, there is no need for restrictive assumptions or linearization. Finally, DTM results are compared with the numerical solution of the problem, and a good accuracy of the proposed method is observed.
Nasertayoob, Payam; Vaezpour, S. Mansour; Baleanu, Dumitru
2014-01-01
The aim of this paper is to study the solvability for a coupled system of fractional integrodifferential equations with multipoint fractional boundary value problems on the half-line. An example is given to demonstrate the validity of our assumptions.
Coercive solvability of the nonlocal boundary value problem for parabolic differential equations
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P. E. Sobolevskii
2001-06-01
Full Text Available The nonlocal boundary value problem, vÃ¢Â€Â²(t+Av(t=f(t(0Ã¢Â‰Â¤tÃ¢Â‰Â¤1,v(0=v(ÃŽÂ»+ÃŽÂ¼(0<ÃŽÂ»Ã¢Â‰Â¤1, in an arbitrary Banach space E with the strongly positive operator A, is considered. The coercive stability estimates in HÃƒÂ¶lder norms for the solution of this problem are proved. The exact Schauder's estimates in HÃƒÂ¶lder norms of solutions of the boundary value problem on the range {0Ã¢Â‰Â¤tÃ¢Â‰Â¤1,xÃ¢Â„ÂÃ¢Â€Â‰n} for 2m-order multidimensional parabolic equations are obtaine.
A Monte Carlo method for solving the one-dimensional telegraph equations with boundary conditions
Acebrón, Juan A.; Ribeiro, Marco A.
2016-01-01
A Monte Carlo algorithm is derived to solve the one-dimensional telegraph equations in a bounded domain subject to resistive and non-resistive boundary conditions. The proposed numerical scheme is more efficient than the classical Kac's theory because it does not require the discretization of time. The algorithm has been validated by comparing the results obtained with theory and the Finite-difference time domain (FDTD) method for a typical two-wire transmission line terminated at both ends with general boundary conditions. We have also tested transmission line heterogeneities to account for wave propagation in multiple media. The algorithm is inherently parallel, since it is based on Monte Carlo simulations, and does not suffer from the numerical dispersion and dissipation issues that arise in finite difference-based numerical schemes on a lossy medium. This allowed us to develop an efficient numerical method, capable of outperforming the classical FDTD method for large scale problems and high frequency signals.
Russell, John
2000-11-01
A modified Orr-Sommerfeld equation that applies to the asymptotic suction boundary layer was reported by Bussmann & Münz in a wartime report dated 1942 and by Hughes & Reid in J.F.M. ( 23, 1965, p715). Fundamental systems of exact solutions of the Orr-Sommerfeld equation for this mean velocity distribution were reported by D. Grohne in an unpublished typescript dated 1950. Exact solutions of the equation of Bussmann, Münz, Hughes, & Reid were reported by P. Baldwin in Mathematika ( 17, 1970, p206). Grohne and Baldwin noticed that these exact solutions may be expressed either as Barnes integrals or as convolution integrals. In a later paper (Phil. Trans. Roy. Soc. A, 399, 1985, p321), Baldwin applied the convolution integrals in the contruction of large-Reynolds number asymptotic approximations that hold uniformly. The present talk discusses the subtleties that arise in the construction of such convolution integrals, including several not reported by Grohne or Baldwin. The aim is to recover the full set of seven solutions (one well balanced, three balanced, and three dominant-recessive) postulated by W.H. Reid in various works on the uniformly valid solutions.
Maksimova, N.V.; Akhmetov, R. G.
2013-01-01
The work deals with a boundary value problem for a quasilinear partial elliptical equation. The equation describes a stationary process of convective diffusion near a cylinder and takes into account the value of a chemical reaction for large Peclet numbers and for large constant of chemical reaction. The quantity the rate constant of the chemical reaction and Peclet number is assumed to have a constant value. The leading term of the asymptotics of the solution is constructed in the boundary l...
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Handlovičová Angela
2016-01-01
Full Text Available In this paper, the numerical solution to the Helmholtz equation with impedance boundary condition, based on the Finite volume method, is discussed. We used the Robin boundary condition using exterior points. Properties of the weak solution to the Helmholtz equation and numerical solution are presented. Further the numerical experiments, comparing the numerical solution with the exact one, and the computation of the experimental order of convergence are presented.
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Aziz Khan
2017-01-01
Full Text Available We discuss existence, uniqueness, and Hyers-Ulam stability of solutions for coupled nonlinear fractional order differential equations (FODEs with boundary conditions. Using generalized metric space, we obtain some relaxed conditions for uniqueness of positive solutions for the mentioned problem by using Perov’s fixed point theorem. Moreover, necessary and sufficient conditions are obtained for existence of at least one solution by Leray-Schauder-type fixed point theorem. Further, we also develop some conditions for Hyers-Ulam stability. To demonstrate our main result, we provide a proper example.
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
Nonlinear convection in reaction-diffusion equations under dynamical boundary conditions
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Gaelle Pincet Mailly
2013-01-01
Full Text Available We study the blow-up phenomena for positive solutions of nonlinear reaction-diffusion equations including a nonlinear convection term $partial_t u = Delta u - g(u cdot abla u + f(u$ in a bounded domain of $mathbb{R}^N$ under the dissipative dynamical boundary conditions $sigma partial_t u + partial_u u =0$. Some conditions on g and f are discussed to state if the positive solutions blow up in finite time or not. Moreover, for certain classes of nonlinearities, an upper-bound for the blow-up time can be derived and the blow-up rate can be determined.
Nonexistence of Global Solutions to an Elliptic Equation with a Dynamical Boundary Condition
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Stanislav I. POHOZAEV
2004-01-01
Full Text Available We consider the equation Delta u = 0 posed in Q := (0;+inftyimes Omega, Omega:= { x=(x',x_N/ x'in R^{N-1}, x_N >0}, with the dynamical boundary conditionB(t,x',0u_{tt} + A(t,x',0u_t - u_{x_N} geq D(t, x',0|u|^q on Sigma := (0,+infty imes R^{N-1} imes {0} and give conditions on the coeﬃcient functions A(t,x',0; B(t,x',0 and D(t;x',0 for the nonexistence of global solutions.
Asymptotically linear Schrodinger equation with zero on the boundary of the spectrum
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Dongdong Qin
2015-08-01
Full Text Available This article concerns the Schr\\"odinger equation $$\\displaylines{ -\\Delta u+V(xu=f(x, u, \\quad \\text{for } x\\in\\mathbb{R}^N,\\cr u(x\\to 0, \\quad \\text{as } |x| \\to \\infty, }$$ where V and f are periodic in x, and 0 is a boundary point of the spectrum $\\sigma(-\\Delta+V$. Assuming that f(x,u is asymptotically linear as $|u|\\to\\infty$, existence of a ground state solution is established using some new techniques.
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Nicolae Tarfulea
2009-10-01
Full Text Available We investigate the existence of weak solutions to a class of quasilinear elliptic equations with nonlinear Neumann boundary conditions in exterior domains. Problems of this kind arise in various areas of science and technology. An important model case related to the initial data problem in general relativity is presented. As an application of our main result, we deduce the existence of the conformal factor for the Hamiltonian constraint in general relativity in the presence of multiple black holes. We also give a proof for uniqueness in this case.
Integral equations for the microstructures of supercritical fluids
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Lee, L.L. [Univ. of Oklahoma, Norman, OK (United States). School of Chemical Engineering and Materials Science; Cochran, H.D. [Oak Ridge National Lab., TN (United States)
1993-11-01
Molecular interactions and molecular distributions are at the heart of the supercritical behavior of fluid mixtures. The distributions, i.e. structure, can be obtained through any of the three routes: (1) scattering experiments, (2) Monte Carlo or molecular dynamics simulation, and (3) integral equations that govern the relation between the molecular interactions u(r) and the probability distributions g{sub ij}(r). Most integral equations are based on the Ornstein-Zernike relation connecting the total correlation to the direct correlation. The OZ relation requires a {open_quotes}closure{close_quotes} equation to be solvable. Thus the Percus-Yevick, hypernetted chain, and mean spherical approximations have been proposed. The authors outline the numerical methods of solution for these integral equations, including the Picard, Labik-Gillan, and Baxter methods. Solution of these equations yields the solvent-solute, solvent-solvent, and solute-solute pair correlation functions (pcf`s). Interestingly, these pcf`s exhibit characteristical signatures for supercritical mixtures that are classified as {open_quotes}attractive{close_quotes} or {open_quotes}repulsive{close_quotes} in nature. Close to the critical locus, the pcf shows enhanced first neighbor peaks with concomitant long-range build-ups (sic attractive behavior) or reduced first peaks plus long-range depletion (sic repulsive behavior) of neighbors. For ternary mixtures with entrainers, there are synergistic effects between solvent and cosolvent, or solute and cosolute. These are also detectable on the distribution function level. The thermodynamic consequences are deciphered through the Kirkwood-Buff fluctuation integrals (G{sub ij}) and their matrix inverses: the direct correlation function integrals (DCFI`s). These quantities connect the correlation functions to the chemical potential derivatives (macroscopic variables) thus acting as {open_quotes}bridges{close_quotes} between the two Weltanschauungen.
Darboux integrability of trapezoidal H 4 and H 4 families of lattice equations I: first integrals
Gubbiotti, G.; Yamilov, R. I.
2017-08-01
In this paper we prove that the trapezoidal H4 and the H6 families of quad-equations are Darboux integrable by constructing their first integrals. This result explains why the rate of growth of the degrees of the iterates of these equations is linear (Gubbiotti et al 2016 J. Nonlinear Math. Phys. 23 507-43), which according to the algebraic entropy conjecture implies linearizability. We conclude by showing how first integrals can be used to obtain general solutions.
A High-Order Direct Solver for Helmholtz Equations with Neumann Boundary Conditions
Sun, Xian-He; Zhuang, Yu
1997-01-01
In this study, a compact finite-difference discretization is first developed for Helmholtz equations on rectangular domains. Special treatments are then introduced for Neumann and Neumann-Dirichlet boundary conditions to achieve accuracy and separability. Finally, a Fast Fourier Transform (FFT) based technique is used to yield a fast direct solver. Analytical and experimental results show this newly proposed solver is comparable to the conventional second-order elliptic solver when accuracy is not a primary concern, and is significantly faster than that of the conventional solver if a highly accurate solution is required. In addition, this newly proposed fourth order Helmholtz solver is parallel in nature. It is readily available for parallel and distributed computers. The compact scheme introduced in this study is likely extendible for sixth-order accurate algorithms and for more general elliptic equations.
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Alexander Domoshnitsky
2014-01-01
Full Text Available The impulsive delay differential equation is considered (Lx(t=x′(t+∑i=1mpi(tx(t-τi(t=f(t, t∈[a,b], x(tj=βjx(tj-0, j=1,…,k, a=t0
Algebro-Geometric Solutions for a Discrete Integrable Equation
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Mengshuang Tao
2017-01-01
Full Text Available With the assistance of a Lie algebra whose element is a matrix, we introduce a discrete spectral problem. By means of discrete zero curvature equation, we obtain a discrete integrable hierarchy. According to decomposition of the discrete systems, the new differential-difference integrable systems with two-potential functions are derived. By constructing the Abel-Jacobi coordinates to straighten the continuous and discrete flows, the Riemann theta functions are proposed. Based on the Riemann theta functions, the algebro-geometric solutions for the discrete integrable systems are obtained.
The Lyapunov stabilization of satellite equations of motion using integrals
Nacozy, P. E.
1973-01-01
A method is introduced that weakens the Lyapunov or in track instability of satellite equations of motion. The method utilizes a linearized energy integral of satellite motion as a constraint on solutions obtained by numerical integration. The procedure prevents local numerical error from altering the frequency associated with the fast angular variable and thereby reduces the Lyapunov instability and the global numerical error. Applications of the method to satellite motion show accuracy improvements of two to three orders of magnitude in position and velocity after 50 revolutions. A modification of the method is presented that allows the use of slowly varying integrals of motion.
Energy Technology Data Exchange (ETDEWEB)
Neufeld, E [Foundation for Research on Information Technologies in Society (IT' IS), ETH Zurich, 8092 Zurich (Switzerland); Chavannes, N [Foundation for Research on Information Technologies in Society (IT' IS), ETH Zurich, 8092 Zurich (Switzerland); Samaras, T [Radiocommunications Laboratory, Aristotle University of Thessaloniki, GR-54124 Thessaloniki (Greece); Kuster, N [Foundation for Research on Information Technologies in Society (IT' IS), ETH Zurich, 8092 Zurich (Switzerland)
2007-08-07
The modeling of thermal effects, often based on the Pennes Bioheat Equation, is becoming increasingly popular. The FDTD technique commonly used in this context suffers considerably from staircasing errors at boundaries. A new conformal technique is proposed that can easily be integrated into existing implementations without requiring a special update scheme. It scales fluxes at interfaces with factors derived from the local surface normal. The new scheme is validated using an analytical solution, and an error analysis is performed to understand its behavior. The new scheme behaves considerably better than the standard scheme. Furthermore, in contrast to the standard scheme, it is possible to obtain with it more accurate solutions by increasing the grid resolution.
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Ahmed Alsaedi
2016-08-01
Full Text Available In this article, we study a boundary value problem of coupled systems of nonlinear Riemann-Liouvillle fractional integro-differential equations supplemented with nonlocal Riemann-Liouvillle fractional integro-differential boundary conditions. Our results rely on some standard tools of the fixed point theory. An illustrative example is also discussed.
Kwong-Wong-type integral equation on time scales
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Baoguo Jia
2011-09-01
Full Text Available Consider the second-order nonlinear dynamic equation $$ [r(tx^Delta(ho(t]^Delta+p(tf(x(t=0, $$ where $p(t$ is the backward jump operator. We obtain a Kwong-Wong-type integral equation, that is: If $x(t$ is a nonoscillatory solution of the above equation on $[T_0,infty$, then the integral equation $$ frac{r^sigma(tx^Delta(t}{f(x^sigma(t} =P^sigma(t+int^infty_{sigma(t}frac{r^sigma(s [int^1_0f'(x_h(sdh][x^Delta(s]^2}{f(x(s f(x^sigma(s}Delta s $$ is satisfied for $tgeq T_0$, where $P^sigma(t=int^infty_{sigma(t}p(sDelta s$, and $x_h(s=x(s+hmu(sx^Delta(s$. As an application, we show that the superlinear dynamic equation $$ [r(tx^{Delta}(ho(t]^Delta+p(tf(x(t=0, $$ is oscillatory, under certain conditions.
Integrable boundary conditions for a non-Abelian anyon chain with D(D{sub 3}) symmetry
Energy Technology Data Exchange (ETDEWEB)
Dancer, K.A.; Finch, P.E.; Isaac, P.S. [The University of Queensland, Centre for Mathematical Physics, School of Physical Sciences, 4072 (Australia); Links, J. [The University of Queensland, Centre for Mathematical Physics, School of Physical Sciences, 4072 (Australia)], E-mail: jrl@maths.uq.edu.au
2009-05-11
A general formulation of the Boundary Quantum Inverse Scattering Method is given which is applicable in cases where R-matrix solutions of the Yang-Baxter equation do not have the property of crossing unitarity. Suitably modified forms of the reflection equations are presented which permit the construction of a family of commuting transfer matrices. As an example, we apply the formalism to determine the most general solutions of the reflection equations for a solution of the Yang-Baxter equation with underlying symmetry given by the Drinfeld double D(D{sub 3}) of the dihedral group D{sub 3}. This R-matrix does not have the crossing unitarity property. In this manner we derive integrable boundary conditions for an open chain model of interacting non-Abelian anyons.
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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.
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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
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Hassan A. Zedan
2017-01-01
Full Text Available Chebyshev spectral method based on operational matrix is applied to both systems of fractional integro-differential equations and Abel’s integral equations. Some test problems, for which the exact solution is known, are considered. Numerical results with comparisons are made to confirm the reliability of the method. Chebyshev spectral method may be considered as alternative and efficient technique for finding the approximation of system of fractional integro-differential equations and Abel’s integral equations.
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.
Yu Zhang
2014-01-01
The initial-boundary value problems for the local fractional differential equation are investigated in this paper. The local fractional Fourier series solutions with the nondifferential terms are obtained. Two illustrative examples are given to show efficiency and accuracy of the presented method to process the local fractional differential equations.
LU, Qi
2015-01-01
We generalize the concept "well-posed linear system" to stochastic linear control systems and study some basic properties of such kind systems. Under our generalized definition, we show the well-posedness of the stochastic heat equation and the stochastic Schr\\"odinger equation with suitable boundary control and observation operators, respectively.
Kreienkamp, Amelia B.; Liu, Lucy Y.; Minkara, Mona S.; Knepley, Matthew G.; Bardhan, Jaydeep P.; Radhakrishnan, Mala L.
2013-01-01
We analyze and suggest improvements to a recently developed approximate continuum-electrostatic model for proteins. The model, called BIBEE/I (boundary-integral based electrostatics estimation with interpolation), was able to estimate electrostatic solvation free energies to within a mean unsigned error of 4% on a test set of more than 600 proteins—a significant improvement over previous BIBEE models. In this work, we tested the BIBEE/I model for its capability to predict residue-by-residue interactions in protein–protein binding, using the widely studied model system of trypsin and bovine pancreatic trypsin inhibitor (BPTI). Finding that the BIBEE/I model performs surprisingly less well in this task than simpler BIBEE models, we seek to explain this behavior in terms of the models’ differing spectral approximations of the exact boundary-integral operator. Calculations of analytically solvable systems (spheres and tri-axial ellipsoids) suggest two possibilities for improvement. The first is a modified BIBEE/I approach that captures the asymptotic eigenvalue limit correctly, and the second involves the dipole and quadrupole modes for ellipsoidal approximations of protein geometries. Our analysis suggests that fast, rigorous approximate models derived from reduced-basis approximation of boundary-integral equations might reach unprecedented accuracy, if the dipole and quadrupole modes can be captured quickly for general shapes. PMID:24466561
Symmetries and first integrals of some differential equations of dynamics
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Eliezer, C.J.
1979-10-01
The theme of this paper is the study of the symmetry properties of some differential equations of dynamics, and of the construction of first integrals. For the time-dependent harmonic oscillator the Lewis invariant provides a quadratic function which is a constant of motion. Different derivatives are considered with a view to assigning some physical meaning to the invariant and to the function rho(t) in terms of which the invariant is expressed. Lie's theory of differential equations, which until recently has been sadly neglected in comparison with his other pioneering works, is applied to consider groups of point transformations which leave invariant the equations of motion. For the time-dependent oscillator, an eight-parameter Lie group is obtained. A five-parameter Noether sub-group leaves also the action function invariant. Some results concerning the symmetries of the Kepler problem are also reported. Dynamical symmetries, not covered by point transformations, are briefly discussed.
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.
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D. Goos
2015-01-01
Full Text Available We consider the time-fractional derivative in the Caputo sense of order α∈(0, 1. Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function in R+, two different initial-boundary-value problems for the time-fractional diffusion equation on the real positive semiaxis are solved. Moreover, the limit when α↗1 of the respective solutions is analyzed, recovering the solutions of the classical boundary-value problems when α = 1, and the fractional diffusion equation becomes the heat equation.
Integral equations with difference kernels on finite intervals
Sakhnovich, Lev A
2015-01-01
This book focuses on solving integral equations with difference kernels on finite intervals. The corresponding problem on the semiaxis was previously solved by N. Wiener–E. Hopf and by M.G. Krein. The problem on finite intervals, though significantly more difficult, may be solved using our method of operator identities. This method is also actively employed in inverse spectral problems, operator factorization and nonlinear integral equations. Applications of the obtained results to optimal synthesis, light scattering, diffraction, and hydrodynamics problems are discussed in this book, which also describes how the theory of operators with difference kernels is applied to stable processes and used to solve the famous M. Kac problems on stable processes. In this second edition these results are extensively generalized and include the case of all Levy processes. We present the convolution expression for the well-known Ito formula of the generator operator, a convolution expression that has proven to be fruitful...
TBA-like integral equations from quantized mirror curves
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Okuyama, Kazumi [Department of Physics, Shinshu University,Matsumoto 390-8621 (Japan); Zakany, Szabolcs [Département de Physique Théorique, Université de Genève,Genève, CH-1211 (Switzerland)
2016-03-15
Quantizing the mirror curve of certain toric Calabi-Yau (CY) three-folds leads to a family of trace class operators. The resolvent function of these operators is known to encode topological data of the CY. In this paper, we show that in certain cases, this resolvent function satisfies a system of non-linear integral equations whose structure is very similar to the Thermodynamic Bethe Ansatz (TBA) systems. This can be used to compute spectral traces, both exactly and as a semiclassical expansion. As a main example, we consider the system related to the quantized mirror curve of local ℙ{sup 2}. According to a recent proposal, the traces of this operator are determined by the refined BPS indices of the underlying CY. We use our non-linear integral equations to test that proposal.
Local asymptotic stability for nonlinear quadratic functional integral equations
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Bapurao Dhage
2008-03-01
Full Text Available In the present study, using the characterizations of measures of noncompactness we prove a theorem on the existence and local asymptotic stability of solutions for a quadratic functional integral equation via a fixed point theorem of Darbo. The investigations are placed in the Banach space of real functions defined, continuous and bounded on an unbounded interval. An example is indicated to demonstrate the natural realizations of abstract result presented in the paper.
Upper and lower bounds of solutions for fractional integral equations
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Shaher Momani
2008-03-01
Full Text Available In this paper we consider the integral equation offractional order in sense of Riemann-Liouville operatorum(t = a(t Iα [b(tu(t]+f(twith m ≥ 1, t ∈ [0, T], T < ∞ and 0< α <1. We discuss the existence, uniqueness, maximal, minimal and the upper and lower bounds of the solutions. Also we illustrate our results with examples.
Towards the automatized evaluation of Feynman integrals with differential equations
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Meyer, Christoph; Uwer, Peter [HU Berlin, Berlin (Germany)
2016-07-01
In the past years the method of differential equations has proven itself to be a powerful tool for the computation of multi-loop Feynman integrals. This method relies on the choice of a basis of master integrals in which the dependence on the dimensional regulator factorizes. We present an algorithm which automatizes the transformation to such a basis, starting from a given reduction basis. The algorithm only requires some mild assumptions about the basis. It applies to problems with multiple scales of which we will present some examples.
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
Oxygen boundary crossing probabilities.
Busch, N A; Silver, I A
1987-01-01
The probability that an oxygen particle will reach a time dependent boundary is required in oxygen transport studies involving solution methods based on probability considerations. A Volterra integral equation is presented, the solution of which gives directly the boundary crossing probability density function. The boundary crossing probability is the probability that the oxygen particle will reach a boundary within a specified time interval. When the motion of the oxygen particle may be described as strongly Markovian, then the Volterra integral equation can be rewritten as a generalized Abel equation, the solution of which has been widely studied.
A New time Integration Scheme for Cahn-hilliard Equations
Schaefer, R.
2015-06-01
In this paper we present a new integration scheme that can be applied to solving difficult non-stationary non-linear problems. It is obtained by a successive linearization of the Crank- Nicolson scheme, that is unconditionally stable, but requires solving non-linear equation at each time step. We applied our linearized scheme for the time integration of the challenging Cahn-Hilliard equation, modeling the phase separation in fluids. At each time step the resulting variational equation is solved using higher-order isogeometric finite element method, with B- spline basis functions. The method was implemented in the PETIGA framework interfaced via the PETSc toolkit. The GMRES iterative solver was utilized for the solution of a resulting linear system at every time step. We also apply a simple adaptivity rule, which increases the time step size when the number of GMRES iterations is lower than 30. We compared our method with a non-linear, two stage predictor-multicorrector scheme, utilizing a sophisticated step length adaptivity. We controlled the stability of our simulations by monitoring the Ginzburg-Landau free energy functional. The proposed integration scheme outperforms the two-stage competitor in terms of the execution time, at the same time having a similar evolution of the free energy functional.
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.
Field, J. H.
2011-01-01
It is shown how the time-dependent Schrodinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton-Jacobi equation of classical mechanics. Schrodinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton-Jacobi…
Integrability of a deterministic cellular automaton driven by stochastic boundaries
Prosen, Tomaž; Mejía-Monasterio, Carlos
2016-05-01
We propose an interacting many-body space-time-discrete Markov chain model, which is composed of an integrable deterministic and reversible cellular automaton (rule 54 of Bobenko et al 1993 Commun. Math. Phys. 158 127) on a finite one-dimensional lattice {({{{Z}}}2)}× n, and local stochastic Markov chains at the two lattice boundaries which provide chemical baths for absorbing or emitting the solitons. Ergodicity and mixing of this many-body Markov chain is proven for generic values of bath parameters, implying the existence of a unique nonequilibrium steady state. The latter is constructed exactly and explicitly in terms of a particularly simple form of matrix product ansatz which is termed a patch ansatz. This gives rise to an explicit computation of observables and k-point correlations in the steady state as well as the construction of a nontrivial set of local conservation laws. The feasibility of an exact solution for the full spectrum and eigenvectors (decay modes) of the Markov matrix is suggested as well. We conjecture that our ideas can pave the road towards a theory of integrability of boundary driven classical deterministic lattice systems.
INSTANTANEOUS AND INTEGRAL POWER EQUATIONS OF NONSINUSOIDAL 3-PHASE PROCESSES
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Iu.A. Sirotin
2016-03-01
Full Text Available Purpose. To identify the mathematical relationship between the instantaneous powers (classical and vectorial and integral powers in non-sinusoidal mode and to get complex form of instantaneous powers in 3-phase 4-wire power supply in terms of the spectral approach. Methodology. We have applied the vector approach with one voice allows you to analyze the energy characteristics of 3-phase power supply circuits (for 4-wire and 3-wire circuits in sinusoidal and non–sinusoidal mode, both the time domain and frequency domain. We have used 3-dimensional representation of the energy waveforms with the complex multi-dimensional Fourier series. Results. For 4-wire network with a non-sinusoidal (regardless of their symmetry processes, we have developed the mathematical model one-dimensional representations of the complex form for the active (scalar instantaneous power (IP and 3-dimensional form (inactive vectorial IP. It is possible to obtain two dual integral power equations for complex scalar and vector integrated power of non-sinusoidal modes. The power equations generalize generalizes the equations of sinusoidal modes for 4-wire network. Originality. In addition to the classification of energy local regimes in the time domain for the first time we spent the classification of non-sinusoidal modes in the spectral region and showed the value and importance of the classification of regimes based on the instantaneous powers. Practical value. The practical value the obtained equations is the possibility of their use for improving the quality of electricity supply and the quality electricity consumption.
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Xue-Lian Jin
2017-01-01
Full Text Available The exponential stability of the monotubular heat exchanger equation with boundary observation possessing a time delay and inner control was investigated. Firstly, the close-loop system was translated into an abstract Cauchy problem in the suitable state space. A uniformly bounded C0-semigroup generated by the close-loop system, which implies that the unique solution of the system exists, was shown. Secondly, the spectrum configuration of the closed-loop system was analyzed and the eventual differentiability and the eventual compactness of the semigroup were shown by the resolvent estimates on some resolvent sets. This implies that the spectrum-determined growth assumption holds. Finally, a sufficient condition, which is related to the physical parameters in the system and is independent of the time delay, of the exponential stability of the closed-loop system was given.
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Josefa Caballero
2014-01-01
Full Text Available We study an existence result for the following coupled system of nonlinear fractional hybrid differential equations with homogeneous boundary conditions D0+α[x(t/f(t,x(t,y(t]=g(t,x(t,y(t,D0+αy(t/f(t,y(t,x(t=g(t,y(t,x(t, 0
Multiple solutions to a boundary value problem for an n-th order nonlinear difference equation
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Susan D. Lauer
1998-11-01
Full Text Available We seek multiple solutions to the n-th order nonlinear difference equation $$Delta^n x(t= (-1^{n-k} f(t,x(t,quad t in [0,T]$$ satisfying the boundary conditions $$x(0 = x(1 = cdots = x(k - 1 = x(T + k + 1 = cdots = x(T+ n = 0,.$$ Guo's fixed point theorem is applied multiple times to an operator defined on annular regions in a cone. In addition, the hypotheses invoked to obtain multiple solutions to this problem involves the condition (A $f:[0,T] imes {mathbb R}^+ o {mathbb R}^+$ is continuous in $x$, as well as one of the following: (B $f$ is sublinear at $0$ and superlinear at $infty$, or (C $f$ is superlinear at $0$ and sublinear at $infty$.
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Gülden Gün
2013-01-01
Full Text Available We analyze Noether and -symmetries of the path equation describing the minimum drag work. First, the partial Lagrangian for the governing equation is constructed, and then the determining equations are obtained based on the partial Lagrangian approach. For specific altitude functions, Noether symmetry classification is carried out and the first integrals, conservation laws and group invariant solutions are obtained and classified. Then, secondly, by using the mathematical relationship with Lie point symmetries we investigate -symmetry properties and the corresponding reduction forms, integrating factors, and first integrals for specific altitude functions of the governing equation. Furthermore, we apply the Jacobi last multiplier method as a different approach to determine the new forms of -symmetries. Finally, we compare the results obtained from different classifications.
Liu, Hualin; Zhao, Wenwen; Chen, Weifang
2016-11-01
Gas or liquid flow through small channels has become more and more popular due to the micro-electro-mechanical systems (MEMS) fabrication technologies such as micro-motors, electrostatic comb-drive, micro-chromatographs, micro-actuators, micro-turbines and micro-pumps, etc. The flow conditions in and around these systems are always recognized as typical transitional regimes. Under these conditions, the mean free path of gas molecules approaches the characteristic scale of the micro-devices itself, and due to the little collisions the heat and momentum cannot equilibrate between the wall and fluids quickly. Couette flow is a simple and critical model in fluid dynamics which focuses on the mechanism of the heat transfer in shear-driven micro-cavities or micro-channels. Despite numerous work on the numerical solutions of the Couette flow, how to propose stable and accurate slip boundary conditions in rarefied flow conditions still remains to be elucidated. In this paper, converged solutions for steady-state micro Couette flows are obtained by using conventional Burnett equations with a set of modified slip boundary conditions. Instead of using the physical variables at the wall, the modified slip conditions use the variables at the edge of the Knudsen layer based on a physically plausible assumption in literature that Knudsen layer has a thickness only in the order of a mean free path and molecules are likely to travel without collision in this layer. Numerical results for non-dimensional wall shear stress and heat flux are compared with those of the DSMC solutions. Although there are not much improvement in the accuracy by using this modified slip conditions, the modified conditions perform much better than the unmodified slip conditions for numerical stabilization. All results show that the set of conventional Burnett equations with second order modified conditions are proved to be an appropriate model for the micro-Couette flows.
Numerical integration of relativistic equations of motion for Earth satellites
San Miguel, A.
2009-01-01
The equations of motion proposed by Brumberg for an artificial satellite around the Earth (Celest Mech Dyn Astron 88:209, 2004), in which the relativistic effects due to the Earth’s oblatness and the gravitational action caused by a third body are added to those perturbations considered in the International Earth Rotation and Reference System Service (2003) convention, are here integrated numerically. To compute the solution of the time-dependent Langrangian system for a gravitational satellite Earth Sun model we consider a six-order partitioned Runge Kutta integrator, whose coefficients satisfy the condition of symplecticity. A comparison with the classical Adams Basforth Moulton method allows to verify the good-performance of the partitioned Runge Kutta method both in the description of the evolution of the satellite energy and in the efficiency of the method when applied to a long-term integration.
On the Energetics of a Damped Beam-Like Equation for Different Boundary Conditions
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SAJAD HUSSAIN SANDILO
2017-04-01
Full Text Available In this paper, the energy estimates for a damped linear homogeneous beam-like equation will be considered. The energy estimates will be studied for different BCs (Boundary Conditions for the axially moving continuum. The problem has physical and engineering application. The applications are mostly occurring in models of conveyor belts and band-saw blades. The research study is focused on the Dirichlet, the Neumann and the Robin type of BCs. From physical point of view, the considered mathematical model expounds the transversal vibrations of a moving belt system or moving band-saw blade. It is assumed that a viscous damping parameter and the horizontal velocity are positive and constant. It will be shown in this paper that change in geometry or the physics of the boundaries can affect the stability properties of the system in general and stability depends on the axial direction of the motion. In all cases of the BCs, it will be shown that there is energy decay due to viscous damping parameter and it will also be shown that in some cases there is no conclusion whether the beam energy decreases or increases. The detailed physical interpretation of all terms and expressions is provided and studied in detail.
The classical nonlinear Schr\\"odinger model with a new integrable boundary
Zambon, Cristina
2014-01-01
A new integrable boundary for the classical nonlinear Schr\\"odinger model is derived by dressing a boundary with a defect. A complete investigation of the integrability of the new boundary is carried out in the sense that the boundary ${\\cal K}$ matrix is derived and the integrability is proved via the classical $r$-matrix. The issue of conserved charges is also discussed. The key point in proving the integrability of the new boundary is the use of suitable modified Poisson brackets. Finally,...
Scalable smoothing strategies for a geometric multigrid method for the immersed boundary equations
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Bhalla, Amneet Pal Singh [Univ. of North Carolina, Chapel Hill, NC (United States); Knepley, Matthew G. [Rice Univ., Houston, TX (United States); Adams, Mark F. [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Guy, Robert D. [Univ. of California, Davis, CA (United States); Griffith, Boyce E. [Univ. of North Carolina, Chapel Hill, NC (United States)
2016-12-20
The immersed boundary (IB) method is a widely used approach to simulating fluid-structure interaction (FSI). Although explicit versions of the IB method can suffer from severe time step size restrictions, these methods remain popular because of their simplicity and generality. In prior work (Guy et al., Adv Comput Math, 2015), some of us developed a geometric multigrid preconditioner for a stable semi-implicit IB method under Stokes flow conditions; however, this solver methodology used a Vanka-type smoother that presented limited opportunities for parallelization. This work extends this Stokes-IB solver methodology by developing smoothing techniques that are suitable for parallel implementation. Specifically, we demonstrate that an additive version of the Vanka smoother can yield an effective multigrid preconditioner for the Stokes-IB equations, and we introduce an efficient Schur complement-based smoother that is also shown to be effective for the Stokes-IB equations. We investigate the performance of these solvers for a broad range of material stiffnesses, both for Stokes flows and flows at nonzero Reynolds numbers, and for thick and thin structural models. We show here that linear solver performance degrades with increasing Reynolds number and material stiffness, especially for thin interface cases. Nonetheless, the proposed approaches promise to yield effective solution algorithms, especially at lower Reynolds numbers and at modest-to-high elastic stiffnesses.
Simulation of Thin Film Equations on an Eye-Shaped Domain with Moving Boundary
Brosch, Joseph; Driscoll, Tobin; Braun, Richard
During a normal eye blink, the upper lid moves, and during the upstroke the lid paints a thin tear film over the exposed corneal and conjunctival surfaces. This thin tear film may be modeled by a nonlinear fourth-order PDE derived from lubrication theory. A major stumbling block in the numerical simulation of this model is to include both the geometry of the eye and the movement of the eyelid. Using a pair of orthogonal and conformal maps, we transform a computational box into a rough representation of a human eye where we proceed to simulate the thin tear film equations. Although we give up some realism, we gain spectrally accurate numerical methods on the computational box. We have applied this method to the heat equation on the blinking domain with both Dirichlet and no-flux boundary conditions, in each case demonstrating at least 10 digits of accuracy.. We are able to perform these simulations very quickly (generally in under a minute) using a desktop version of MATLAB. This project was supported by Grant 1022706 (R.J.B., T.A.D., J.K.B.) from the NSF.
Integral geometry and inverse problems for hyperbolic equations
Romanov, V G
1974-01-01
There are currently many practical situations in which one wishes to determine the coefficients in an ordinary or partial differential equation from known functionals of its solution. These are often called "inverse problems of mathematical physics" and may be contrasted with problems in which an equation is given and one looks for its solution under initial and boundary conditions. Although inverse problems are often ill-posed in the classical sense, their practical importance is such that they may be considered among the pressing problems of current mathematical re search. A. N. Tihonov showed [82], [83] that there is a broad class of inverse problems for which a particular non-classical definition of well-posed ness is appropriate. This new definition requires that a solution be unique in a class of solutions belonging to a given subset M of a function space. The existence of a solution in this set is assumed a priori for some set of data. The classical requirement of continuous dependence of the solutio...
Integrability of the higher-order nonlinear Schrödinger equation revisited
Sakovich, S Yu
1999-01-01
Only the known integrable cases of the Kodama-Hasegawa higher-order nonlinear Schroedinger equation pass the Painleve test. Recent results of Ghosh and Nandy add no new integrable cases of this equation.
Pskhu, A. V.
2017-12-01
We solve the first boundary-value problem in a non-cylindrical domain for a diffusion-wave equation with the Dzhrbashyan– Nersesyan operator of fractional differentiation with respect to the time variable. We prove an existence and uniqueness theorem for this problem, and construct a representation of the solution. We show that a sufficient condition for unique solubility is the condition of Hölder smoothness for the lateral boundary of the domain. The corresponding results for equations with Riemann– Liouville and Caputo derivatives are particular cases of results obtained here.
Computational Methods for Solving Linear Fuzzy Volterra Integral Equation
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Jihan Hamaydi
2017-01-01
Full Text Available Two numerical schemes, namely, the Taylor expansion and the variational iteration methods, have been implemented to give an approximate solution of the fuzzy linear Volterra integral equation of the second kind. To display the validity and applicability of the numerical methods, one illustrative example with known exact solution is presented. Numerical results show that the convergence and accuracy of these methods were in a good agreement with the exact solution. However, according to comparison of these methods, we conclude that the variational iteration method provides more accurate results.
On global attractivity of solutions of a functional-integral equation
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Mohamed Darwish
2007-10-01
Full Text Available We prove an existence theorem for a quadratic functional-integral equation of mixed type. The functional-integral equation studied below contains as special cases numerous integral equations encountered in nonlinear analysis. With help of a suitable measure of noncompactness, we show that the functional integral equation of mixed type has solutions being continuous and bounded on the interval $[0,\\infty$ and those solutions are globally attractive.
Bhattacharya, Amitabh; Kesarkar, Tejas
2016-10-01
A combination of finite difference (FD) and boundary integral (BI) methods is used to formulate an efficient solver for simulating unsteady Stokes flow around particles. The two-dimensional (2D) unsteady Stokes equation is being solved on a Cartesian grid using a second order FD method, while the 2D steady Stokes equation is being solved near the particle using BI method. The two methods are coupled within the viscous boundary layer, a few FD grid cells away from the particle, where solutions from both FD and BI methods are valid. We demonstrate that this hybrid method can be used to accurately solve for the flow around particles with irregular shapes, even though radius of curvature of the particle surface is not resolved by the FD grid. For dilute particle concentrations, we construct a virtual envelope around each particle and solve the BI problem for the flow field located between the envelope and the particle. The BI solver provides velocity boundary condition to the FD solver at "boundary" nodes located on the FD grid, adjacent to the particles, while the FD solver provides the velocity boundary condition to the BI solver at points located on the envelope. The coupling between FD method and BI method is implicit at every time step. This method allows us to formulate an O(N) scheme for dilute suspensions, where N is the number of particles. For semidilute suspensions, where particles may cluster, an envelope formation method has been formulated and implemented, which enables solving the BI problem for each individual particle cluster, allowing efficient simulation of hydrodynamic interaction between particles even when they are in close proximity. The method has been validated against analytical results for flow around a periodic array of cylinders and for Jeffrey orbit of a moving ellipse in shear flow. Simulation of multiple force-free irregular shaped particles in the presence of shear in a 2D slit flow has been conducted to demonstrate the robustness of
Explicit solution of Calderon preconditioned time domain integral equations
Ulku, Huseyin Arda
2013-07-01
An explicit marching on-in-time (MOT) scheme for solving Calderon-preconditioned time domain integral equations is proposed. The scheme uses Rao-Wilton-Glisson and Buffa-Christiansen functions to discretize the domain and range of the integral operators and a PE(CE)m type linear multistep to march on in time. Unlike its implicit counterpart, the proposed explicit solver requires the solution of an MOT system with a Gram matrix that is sparse and well-conditioned independent of the time step size. Numerical results demonstrate that the explicit solver maintains its accuracy and stability even when the time step size is chosen as large as that typically used by an implicit solver. © 2013 IEEE.
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Goncalez, Tifani T. [Universidade Federal do Rio Grande do Sul (PROMEC/UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica; Segatto, Cynthia F.; Vilhena, Marco Tullio, E-mail: csegatto@pq.cnpq.b, E-mail: vilhena@pq.cnpq.b [Universidade Federal do Rio Grande do Sul (DMPA/UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Matematica Aplicada
2011-07-01
In this work, we report an analytical solution for the set of S{sub N} equations for the angular flux, in a rectangle, using the double Laplace transform technique. Its main idea comprehends the steps: application of the Laplace transform in one space variable, solution of the resulting equation by the LTS{sub N} method and reconstruction of the double Laplace transformed angular flux using the inversion theorem of the Laplace transform. We must emphasize that we perform the Laplace inversion by the LTS{sub N} method in the x direction, meanwhile we evaluate the inversion in the y direction performing the calculation of the corresponding line integral solution by the Stefest method. We have also to figure out that the application of Laplace transform to this type of boundary value problem introduces additional unknown functions associated to the partial derivatives of the angular flux at boundary. Based on the good results attained by the nodal LTS{sub N} method, we assume that the angular flux at boundary is also approximated by an exponential function. By analytical we mean that no approximation is done along the solution derivation except for the exponential hypothesis for the exiting angular flux at boundary. For sake of completeness, we report numerical comparisons of the obtained results against the ones of the literature. (author)
Hierarchical Matrices Method and Its Application in Electromagnetic Integral Equations
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Han Guo
2012-01-01
Full Text Available Hierarchical (H- matrices method is a general mathematical framework providing a highly compact representation and efficient numerical arithmetic. When applied in integral-equation- (IE- based computational electromagnetics, H-matrices can be regarded as a fast algorithm; therefore, both the CPU time and memory requirement are reduced significantly. Its kernel independent feature also makes it suitable for any kind of integral equation. To solve H-matrices system, Krylov iteration methods can be employed with appropriate preconditioners, and direct solvers based on the hierarchical structure of H-matrices are also available along with high efficiency and accuracy, which is a unique advantage compared to other fast algorithms. In this paper, a novel sparse approximate inverse (SAI preconditioner in multilevel fashion is proposed to accelerate the convergence rate of Krylov iterations for solving H-matrices system in electromagnetic applications, and a group of parallel fast direct solvers are developed for dealing with multiple right-hand-side cases. Finally, numerical experiments are given to demonstrate the advantages of the proposed multilevel preconditioner compared to conventional “single level” preconditioners and the practicability of the fast direct solvers for arbitrary complex structures.
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Liu Yang
2014-01-01
Full Text Available We consider the existence of infinitely many classical solutions to a class of impulsive differential equations with Dirichlet boundary value condition. Our main tools are based on variant fountain theorems and variational method. We study the case in which the nonlinearity is sublinear. Some recent results are extended and improved.
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Iftikhar Ahmed
2014-04-01
Full Text Available This paper deals with the evolution r-Laplacian equation with absorption and nonlinear boundary condition. By using differential inequality techniques, global existence and blow-up criteria of nonnegative solutions are determined. Moreover, upper bound of the blow-up time for the blow-up solution is obtained.
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Payam Nasertayoob
2014-01-01
Full Text Available The aim of this paper is to study the solvability for a coupled system of fractional integrodifferential equations with multipoint fractional boundary value problems on the half-line. An example is given to demonstrate the validity of our assumptions.
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Neda Khodabakhshi
2013-12-01
Full Text Available This paper deals with the existence results for solutions of coupled system of nonlinear fractional differential equations with boundary value problems on an unbounded domain. Also, we give an illustrative example in order to indicate the validity of our assumptions.
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Huina Zhang
2014-01-01
Full Text Available This paper studies the existence and uniqueness of solutions for a coupled system of nonlinear fractional differential equations of order α,β∈(4,5] with antiperiodic boundary conditions. Our results are based on the nonlinear alternative of Leray-Schauder type and the contraction mapping principle. Two illustrative examples are also presented.
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Guotao Wang
2012-01-01
Full Text Available This paper investigates the existence of solutions for a coupled system of nonlinear fractional differential equations with m-point fractional boundary conditions on an unbounded domain. Some standard fixed point theorems are applied to obtain the main results. The paper concludes with two illustrative examples.
Guotao Wang; Bashir Ahmad; Lihong Zhang
2012-01-01
This paper investigates the existence of solutions for a coupled system of nonlinear fractional differential equations with m-point fractional boundary conditions on an unbounded domain. Some standard fixed point theorems are applied to obtain the main results. The paper concludes with two illustrative examples.
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Lingling Cheng
2014-02-01
Full Text Available In this article, we discuss the existence of solutions to boundary-value problems for a coupled system of fractional differential equations with p-Laplacian operator at resonance. We prove the existence of solutions when $\\dim \\ker L\\geq 2$, using the coincidence degree theory by Mawhin.
Liu, Yang; Xie, Dapeng; Yang, Dandan; Bai, Chuanzhi
2017-01-01
In this paper, we investigate the existence of positive solutions for the boundary value problem of nonlinear fractional differential equation with mixed fractional derivatives and p-Laplacian operator. Then we establish two smart generalizations of Lyapunov-type inequalities. Some applications are given to demonstrate the effectiveness of the new results.
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Mitsuhiro Nakao
2014-01-01
Full Text Available We prove the existence and uniqueness of a global decaying solution to the initial boundary value problem for the quasilinear wave equation with Kelvin-Voigt dissipation and a derivative nonlinearity. To derive the required estimates of the solutions we employ a 'loan' method and use a difference inequality on the energy.
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Juergen Saal
2007-02-01
Full Text Available It is proved under mild regularity assumptions on the data that the Navier-Stokes equations in bounded and unbounded noncylindrical regions admit a unique local-in-time strong solution. The result is based on maximal regularity estimates for the in spatial regions with a moving boundary obtained in [16] and the contraction mapping principle.
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Abdulla Ugur G
2005-01-01
Full Text Available This paper establishes necessary and sufficient condition for the regularity of a characteristic top boundary point of an arbitrary open subset of ( for the diffusion (or heat equation. The result implies asymptotic probability law for the standard -dimensional Brownian motion.
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Giai Giang Vo
2015-01-01
Full Text Available This paper is devoted to the study of a wave equation with a boundary condition of many-point type. The existence of weak solutions is proved by using the Galerkin method. Also, the uniqueness and the stability of solutions are established.
Shuba, M. V.; Melnikov, A. V.; Kuzhir, P. P.; Maksimenko, S. A.; Slepyan, G. Y.; Boag, A.; Conte, A. Mosca; Pulci, O.; Bellucci, S.
2017-11-01
We present the electromagnetic scattering theory for a finite-length nanowire with an embedded mesoscopic object. The theory is based on a synthesis of the integral equation technique of classical electrodynamics and the quantum transport formalism. We formulate Hallén-type integral equations, where the canonical integral operators from wire antenna theory are combined with special terms responsible for the mesoscopic structure. The theory is applied to calculate the polarizability of a finite-length single-walled carbon nanotube (CNT) with a short low-conductive section (LCS) in the microwave and subterahertz ranges. The LCS is modeled as a multichannel two-electrode mesoscopic system. The effective resistive sheet impedance boundary conditions for the scattered field are applied on the CNT surface. It is shown that the imaginary part of the polarizability spectrum has three peaks. Two of them are in the terahertz range, while the third is in the gigahertz range. The polarizability spectrum of the CNT with many LCSs has only one gigahertz peak, which shifts to low frequencies as the number of LCSs increases. The physical nature of these peaks is explained, and potential applications of nanoantennas are proposed.
Said-Houari, Belkacem
2012-09-01
The goal of this work is to study a model of the viscoelastic wave equation with nonlinear boundary/interior sources and a nonlinear interior damping. First, applying the Faedo-Galerkin approximations combined with the compactness method to obtain existence of regular global solutions to an auxiliary problem with globally Lipschitz source terms and with initial data in the potential well. It is important to emphasize that it is not possible to consider density arguments to pass from regular to weak solutions if one considers regular solutions of our problem where the source terms are locally Lipschitz functions. To overcome this difficulty, we use an approximation method involving truncated sources and adapting the ideas in [13] to show that the existence of weak solutions can still be obtained for our problem. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term, then the solution ceases to exist and blows up in finite time provided that the initial data are large enough.
Kleinert, H; Zatloukal, V
2013-11-01
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.
Wang, Ya-Guang; Yin, Jierong; Zhu, Shiyong
2017-10-01
In this paper, we will analyze the vanishing viscosity limit of the incompressible Navier-Stokes equations with the Navier slip boundary conditions in a bounded domain of R2. When the slip length is smaller than or equal to the order of viscosity, by using an energy method and developing Kato's approach given in the work of Kato [Math. Sci. Res. Inst. Publ. 2, 85-98 (1984)], we obtain several conditions to guarantee that the solution of the Navier-Stokes equations with the Navier slip boundary conditions goes to the solution of the associated problem of the Euler equations in the energy space L2 uniformly in time, as the viscosity goes to zero.
Kuehl, Joseph
2016-11-01
The parabolized stability equations (PSE) have been developed as an efficient and powerful tool for studying the stability of advection-dominated laminar flows. In this work, a new "wavepacket" formulation of the PSE is presented. This method accounts for the influence of finite-bandwidth-frequency distributions on nonlinear stability calculations. The methodology is motivated by convolution integrals and is found to appropriately represent nonlinear energy transfer between primary modes and harmonics, in particular nonlinear feedback, via a "nonlinear coupling coefficient." It is found that traditional discrete mode formulations overestimate nonlinear feedback by approximately 70%. This results in smaller maximum disturbance amplitudes than those observed experimentally. The new formulation corrects this overestimation, accounts for the generation of side lobes responsible for spectral broadening and results in disturbance saturation amplitudes consistent with experiment. A Mach 6 flared-cone example is presented. Support from the AFOSR Young Investigator Program via Grant FA9550-15-1-0129 is gratefully acknowledges.
An integral equation model for warm and hot dense mixtures
Starrett, C E; Daligault, J; Hamel, S
2014-01-01
In Starrett and Saumon [Phys. Rev. E 87, 013104 (2013)] a model for the calculation of electronic and ionic structures of warm and hot dense matter was described and validated. In that model the electronic structure of one "atom" in a plasma is determined using a density functional theory based average-atom (AA) model, and the ionic structure is determined by coupling the AA model to integral equations governing the fluid structure. That model was for plasmas with one nuclear species only. Here we extend it to treat plasmas with many nuclear species, i.e. mixtures, and apply it to a carbon-hydrogen mixture relevant to inertial confinement fusion experiments. Comparison of the predicted electronic and ionic structures with orbital-free and Kohn-Sham molecular dynamics simulations reveals excellent agreement wherever chemical bonding is not significant.
Comparison of four stable numerical methods for Abel's integral equation
Murio, Diego A.; Mejia, Carlos E.
1991-01-01
The 3-D image reconstruction from cone-beam projections in computerized tomography leads naturally, in the case of radial symmetry, to the study of Abel-type integral equations. If the experimental information is obtained from measured data, on a discrete set of points, special methods are needed in order to restore continuity with respect to the data. A new combined Regularized-Adjoint-Conjugate Gradient algorithm, together with two different implementations of the Mollification Method (one based on a data filtering technique and the other on the mollification of the kernal function) and a regularization by truncation method (initially proposed for 2-D ray sample schemes and more recently extended to 3-D cone-beam image reconstruction) are extensively tested and compared for accuracy and numerical stability as functions of the level of noise in the data.
Solution of nonlinear Volterra-Hammerstein integral equations via single-term Walsh series method
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Sepehrian B.
2005-01-01
Full Text Available Single-term Walsh series are developed to approximate the solutions of nonlinear Volterra-Hammerstein integral equations. Properties of single-term Walsh series are presented and are utilized to reduce the computation of integral equations to some algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.
Galenko, Peter K; Alexandrov, Dmitri V; Titova, Ekaterina A
2018-02-28
The boundary integral method for propagating solid/liquid interfaces is detailed with allowance for the thermo-solutal Stefan-type models. Two types of mass transfer mechanisms corresponding to the local equilibrium (parabolic-type equation) and local non-equilibrium (hyperbolic-type equation) solidification conditions are considered. A unified integro-differential equation for the curved interface is derived. This equation contains the steady-state conditions of solidification as a special case. The boundary integral analysis demonstrates how to derive the quasi-stationary Ivantsov and Horvay-Cahn solutions that, respectively, define the paraboloidal and elliptical crystal shapes. In the limit of highest Péclet numbers, these quasi-stationary solutions describe the shape of the area around the dendritic tip in the form of a smooth sphere in the isotropic case and a deformed sphere along the directions of anisotropy strength in the anisotropic case. A thermo-solutal selection criterion of the quasi-stationary growth mode of dendrites which includes arbitrary Péclet numbers is obtained. To demonstrate the selection of patterns, computational modelling of the quasi-stationary growth of crystals in a binary mixture is carried out. The modelling makes it possible to obtain selected structures in the form of dendritic, fractal or planar crystals.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'. © 2018 The Author(s).
Stück, Arthur
2015-11-01
Inconsistent discrete expressions in the boundary treatment of Navier-Stokes solvers and in the definition of force objective functionals can lead to discrete-adjoint boundary treatments that are not a valid representation of the boundary conditions to the corresponding adjoint partial differential equations. The underlying problem is studied for an elementary 1D advection-diffusion problem first using a node-centred finite-volume discretisation. The defect of the boundary operators in the inconsistently defined discrete-adjoint problem leads to oscillations and becomes evident with the additional insight of the continuous-adjoint approach. A homogenisation of the discretisations for the primal boundary treatment and the force objective functional yields second-order functional accuracy and eliminates the defect in the discrete-adjoint boundary treatment. Subsequently, the issue is studied for aerodynamic Reynolds-averaged Navier-Stokes problems in conjunction with a standard finite-volume discretisation on median-dual grids and a strong implementation of noslip walls, found in many unstructured general-purpose flow solvers. Going out from a base-line discretisation of force objective functionals which is independent of the boundary treatment in the flow solver, two improved flux-consistent schemes are presented; based on either body wall-defined or farfield-defined control-volumes they resolve the dual inconsistency. The behaviour of the schemes is investigated on a sequence of grids in 2D and 3D.
Zuo, Chao; Chen, Qian; Li, Hongru; Qu, Weijuan; Asundi, Anand
2014-07-28
Boundary conditions play a crucial role in the solution of the transport of intensity equation (TIE). If not appropriately handled, they can create significant boundary artifacts across the reconstruction result. In a previous paper [Opt. Express 22, 9220 (2014)], we presented a new boundary-artifact-free TIE phase retrieval method with use of discrete cosine transform (DCT). Here we report its experimental investigations with applications to the micro-optics characterization. The experimental setup is based on a tunable lens based 4f system attached to a non-modified inverted bright-field microscope. We establish inhomogeneous Neumann boundary values by placing a rectangular aperture in the intermediate image plane of the microscope. Then the boundary values are applied to solve the TIE with our DCT-based TIE solver. Experimental results on microlenses highlight the importance of boundary conditions that often overlooked in simplified models, and confirm that our approach effectively avoid the boundary error even when objects are located at the image borders. It is further demonstrated that our technique is non-interferometric, accurate, fast, full-field, and flexible, rendering it a promising metrological tool for the micro-optics inspection.
Integral method for the calculation of three-dimensional, laminar and turbulent boundary layers
Stock, H. W.
1978-01-01
The method for turbulent flows is a further development of an existing method; profile families with two parameters and a lag entrainment method replace the simple entrainment method and power profiles with one parameter. The method for laminar flows is a new development. Moment of momentum equations were used for the solution of the problem, the profile families were derived from similar solutions of boundary layer equations. Laminar and turbulent flows at the wings were calculated. The influence of wing tapering on the boundary layer development was shown. The turbulent boundary layer for a revolution ellipsoid is calculated for 0 deg and 10 deg incidence angles.
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Xuemei Zhang
2014-01-01
Full Text Available This paper investigates the expression and properties of Green’s function for a second-order singular boundary value problem with integral boundary conditions and delayed argument -x′′t-atx′t+btxt=ωtft, xαt, t∈0, 1; x′0=0, x1-∫01htxtdt=0, where a∈0, 1, 0, +∞, b∈C0, 1, 0, +∞ and, ω may be singular at t=0 or/and at t=1. Furthermore, several new and more general results are obtained for the existence of positive solutions for the above problem by using Krasnosel’skii’s fixed point theorem. We discuss our problems with a delayed argument, which may concern optimization issues of some technical problems. Moreover, the approach to express the integral equation of the above problem is different from earlier approaches. Our results cover a second-order boundary value problem without deviating arguments and are compared with some recent results.
Modern integral equation techniques for quantum reactive scattering theory
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Auerbach, Scott Michael [Univ. of California, Berkeley, CA (United States)
1993-11-01
Rigorous calculations of cross sections and rate constants for elementary gas phase chemical reactions are performed for comparison with experiment, to ensure that our picture of the chemical reaction is complete. We focus on the H/D+H_{2} → H_{2}/DH + H reaction, and use the time independent integral equation technique in quantum reactive scattering theory. We examine the sensitivity of H+H_{2} state resolved integral cross sections σ{sub v'j',vj}(E) for the transitions (v = 0,j = 0) to (v'} = 1,j' = 1,3), to the difference between the Liu-Siegbahn-Truhlar-Horowitz (LSTH) and double many body expansion (DMBE) ab initio potential energy surfaces (PES). This sensitivity analysis is performed to determine the origin of a large discrepancy between experimental cross sections with sharply peaked energy dependence and theoretical ones with smooth energy dependence. We find that the LSTH and DMBE PESs give virtually identical cross sections, which lends credence to the theoretical energy dependence.
A boundary integral method for two-dimensional (non)-Newtonian drops in slow viscous flow
Toose, E.M.; Geurts, Bernardus J.; Kuerten, Johannes G.M.
1995-01-01
A boundary integral method for the simulation of the time-dependent deformation of Newtonian or non-Newtonian drops suspended in a Newtonian fluid is developed. The boundary integral formulation for Stokes flow is used and the non-Newtonian stress is treated as a source term which yields an extra
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A. M. Shkhagapsoev
2017-11-01
Full Text Available We consider the boundary-value problem for a third-order equation of parabolic type with the fractional derivative of Caputo. By the method of energy inequalities an a priori estimate of the solution of the analogue of the second boundary value problem for an equation with multiple characteristics.
Analysis of transient plasmonic interactions using an MOT-PMCHWT integral equation solver
Uysal, Ismail Enes
2014-07-01
Device design involving metals and dielectrics at nano-scales and optical frequencies calls for simulation tools capable of analyzing plasmonic interactions. To this end finite difference time domain (FDTD) and finite element methods have been used extensively. Since these methods require volumetric meshes, the discretization size should be very small to accurately resolve fast-decaying fields in the vicinity of metal/dielectric interfaces. This can be avoided using integral equation (IE) techniques that discretize only on the interfaces. Additionally, IE solvers implicitly enforce the radiation condition and consequently do not need (approximate) absorbing boundary conditions. Despite these advantages, IE solvers, especially in time domain, have not been used for analyzing plasmonic interactions.
An arbitrary-order staggered time integrator for the linear acoustic wave equation
Lee, Jaejoon; Park, Hyunseo; Park, Yoonseo; Shin, Changsoo
2018-02-01
We suggest a staggered time integrator whose order of accuracy can arbitrarily be extended to solve the linear acoustic wave equation. A strategy to select the appropriate order of accuracy is also proposed based on the error analysis that quantitatively predicts the truncation error of the numerical solution. This strategy not only reduces the computational cost several times, but also allows us to flexibly set the modelling parameters such as the time step length, grid interval and P-wave speed. It is demonstrated that the proposed method can almost eliminate temporal dispersive errors during long term simulations regardless of the heterogeneity of the media and time step lengths. The method can also be successfully applied to the source problem with an absorbing boundary condition, which is frequently encountered in the practical usage for the imaging algorithms or the inverse problems.
Fu, Szu-Pei; Jiang, Shidong; Klöckner, Andreas; Ryham, Rolf; Wala, Matt; Young, Yuan-Nan
2017-11-01
In macroscopic model, the well-known Helfrich membrane model has been extensively utilized as it captures some macroscopic physical properties of a lipid bilayer membrane. However, some phenomena such as membrane fusion and micelle formation cannot be described in this macroscopic framework. Yet the immense molecular details of a lipid bilayer membrane are impossible to be included in a plausible physical model. Therefore, in order to include the salient molecular details, we study the dynamics of coarse-grained lipid bilayer membrane using Janus particle configurations to represent collections of lipids These coarse-grained lipid molecules interact with each other through an action field that describes their hydrophobic tail-tail interactions. For this action potential, we adopt the integral equation method on solving energy minimizer with specific boundary condition on each Janus particle. Both the QBX (quadrature by expansion) and fhe fast multipole method (FMM) are used to efficiently solve the integral equation. We also examine the numerical accuracy and qualitative observation from large system simulations.
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Gai Gongqi
2011-01-01
Full Text Available Abstract This article studies the boundary value problems for the third-order nonlinear singular difference equations Δ 3 u ( i - 2 + λ a ( i f ( i , u ( i = 0 , i ∈ [ 2 , T + 2 ] , satisfying five kinds of different boundary value conditions. This article shows the existence of positive solutions for positone and semi-positone type. The nonlinear term may be singular. Two examples are also given to illustrate the main results. The arguments are based upon fixed point theorems in a cone. MSC [2008]: 34B15; 39A10.
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Cercignani, C. [Politecnico di Milano, Milan (Italy)
1996-08-01
Recently R. Illner and the author proved that, under a physically realistic truncation on the collision kernel, the Boltzmann equation in the one-dimensional slab [0,1] with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense. Here it is proved that when the Maxwellians associated with the boundary conditions at x=0 and x = 1 are the same Maxwellian M{sub w}, then the solution is uniformly bounded and tends to M{sub w} for t {r_arrow}{infinity}.
Integral equations for different wavefunctions and their use in finding resonances
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Kapshai, Valery [Gomel State University, Department of Physics, 246019, Gomel (Belarus); Shilyaeva, Ksenia; Elander, Nils [Stockholm University, AlbaNova University Center, Department of Physics, SE-106 91, Stockholm (Sweden)], E-mail: kapshai@physto.se, E-mail: ksh@physto.se, E-mail: elander@physto.se
2009-02-28
Volterra integral equations for the regular and Jost solutions, widely used in the non-relativistic theory, do not have relativistic two-particle analogues. In the present work, the connections of such equations with each other and with the Fredholm equation, which has relativistic analogues, are discussed. It is shown that the Fredholm integral equation together with the complex scaling method can be used for finding a complex resonance spectrum. The formalism is applied to some analytical potentials.
Integral and Variational Formulations for the Helmholtz Equation Inverse Source Problem
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Marcelo L. Rainha
2012-01-01
Full Text Available The purpose of this paper is to explore the Hilbert space functional structure of the Helmholtz equation inverse source problem. An integral equation for the sources reconstruction based on the composition of the trace and Green's function operators is introduced and compared with the reciprocity source reconstruction methodologies. An equivalence theorem comparing the integral inverse source equation with the variational weak reciprocity gap functional equation is then demonstrated. Some examples on applications to the unitary disk are presented.
Riemann integral of a random function and the parabolic equation with a general stochastic measure
Radchenko, Vadym
2012-01-01
For stochastic parabolic equation driven by a general stochastic measure, the weak solution is obtained. The integral of a random function in the equation is considered as a limit in probability of Riemann integral sums. Basic properties of such integrals are studied in the paper.
1984-03-01
variational integ,:ral wa xnndas a -. e-thod of determining, whater thie finite-difference technrdiue or thie firnite-ele:.ent technmicue --ave a mrore...boundaries (Dirich- let boundary conditions). Other boundary conditions could produce a change in the integral that is to be extremized. A method of...t (n) HII!.( - , ___ i_(3.38) The difference between successive iterations was used to calculate the percent change in temperature I a")I t (n
Macomber, B.; Woollands, R. M.; Probe, A.; Younes, A.; Bai, X.; Junkins, J.
2013-09-01
Modified Chebyshev Picard Iteration (MCPI) is an iterative numerical method for approximating solutions of linear or non-linear Ordinary Differential Equations (ODEs) to obtain time histories of system state trajectories. Unlike other step-by-step differential equation solvers, the Runge-Kutta family of numerical integrators for example, MCPI approximates long arcs of the state trajectory with an iterative path approximation approach, and is ideally suited to parallel computation. Orthogonal Chebyshev Polynomials are used as basis functions during each path iteration; the integrations of the Picard iteration are then done analytically. Due to the orthogonality of the Chebyshev basis functions, the least square approximations are computed without matrix inversion; the coefficients are computed robustly from discrete inner products. As a consequence of discrete sampling and weighting adopted for the inner product definition, Runge phenomena errors are minimized near the ends of the approximation intervals. The MCPI algorithm utilizes a vector-matrix framework for computational efficiency. Additionally, all Chebyshev coefficients and integrand function evaluations are independent, meaning they can be simultaneously computed in parallel for further decreased computational cost. Over an order of magnitude speedup from traditional methods is achieved in serial processing, and an additional order of magnitude is achievable in parallel architectures. This paper presents a new MCPI library, a modular toolset designed to allow MCPI to be easily applied to a wide variety of ODE systems. Library users will not have to concern themselves with the underlying mathematics behind the MCPI method. Inputs are the boundary conditions of the dynamical system, the integrand function governing system behavior, and the desired time interval of integration, and the output is a time history of the system states over the interval of interest. Examples from the field of astrodynamics are
Solvability of an Integral Equation of Volterra-Wiener-Hopf Type
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Nurgali K. Ashirbayev
2014-01-01
Full Text Available The paper presents results concerning the solvability of a nonlinear integral equation of Volterra-Stieltjes type. We show that under some assumptions that equation has a continuous and bounded solution defined on the interval 0,∞ and having a finite limit at infinity. As a special case of the mentioned integral equation we obtain an integral equation of Volterra-Wiener-Hopf type. That fact enables us to formulate convenient and handy conditions ensuring the solvability of the equation in question in the class of functions defined and continuous on the interval 0,∞ and having finite limits at infinity.
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Stalin, S. [Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620024, Tamil Nadu (India); Senthilvelan, M., E-mail: velan@cnld.bdu.ac.in [Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620024, Tamil Nadu (India)
2011-10-17
In this Letter, we formulate an exterior differential system for the newly discovered cubically nonlinear integrable Camassa-Holm type equation. From the exterior differential system we establish the integrability of this equation. We then study Cartan prolongation structure of this equation. We also discuss the method of identifying conservation laws and Baecklund transformation for this equation from the identified exterior differential system. -- Highlights: → An exterior differential system for a cubic nonlinear integrable equation is given. → The conservation laws from the exterior differential system is derived. → The Baecklund transformation from the Cartan-Ehresmann connection is obtained.
Shi, Yan; Chan, Chi Hou
2010-02-01
In this paper, we present the multilevel Green's function interpolation method (MLGFIM) for analyses of three-dimensional doubly periodic structures consisting of dielectric media and conducting objects. The volume integral equation (VIE) and surface integral equation (SIE) are adopted, respectively, for the inhomogeneous dielectric and conducting objects in a unit cell. Conformal basis functions defined on curvilinear hexahedron and quadrilateral elements are used to solve the volume/surface integral equation (VSIE). Periodic boundary conditions are introduced at the boundaries of the unit cell. Computation of the space-domain Green's function is accelerated by means of Ewald's transformation. A periodic octary-cube-tree scheme is developed to allow adaptation of the MLGFIM for analyses of doubly periodic structures. The proposed algorithm is first validated by comparison with published data in the open literature. More complex periodic structures, such as dielectric coated conducting shells, folded dielectric structures, photonic bandgap structures, and split ring resonators (SRRs), are then simulated to illustrate that the MLGFIM has a computational complexity of O(N) when applied to periodic structures.
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Puskar Raj SHARMA
2012-01-01
Full Text Available Aim of the paper is to investigate solution of twodimensional linear parabolic partial differential equation with non-local boundary conditions using Homotopy Perturbation Method (HPM. This method is not only reliable in obtaining solution of such problems in series form with high accuracy but it also guarantees considerable saving of the calculation volume and time as compared to other methods. The application of the method has been illustrated through an example
Solution of nonlinear Volterra-Hammerstein integral equations via rationalized Haar functions
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Razzaghi M.
2001-01-01
Full Text Available Rationalized Haar functions are developed to approximate the solutions of the nonlinear Volterra-Hammerstein integral equations. Properties of Rationalized Haar functions are first presented, and the operational matrix of integration together with the product operational matrix are utilized to reduce the computation of integral equations to into some algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.
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Josefa Caballero
2006-05-01
Full Text Available We give an existence theorem for some functional-integral equations which includes many key integral and functional equations that arise in nonlinear analysis and its applications. In particular, we extend the class of characteristic functions appearing in Chandrasekhar's classical integral equation from astrophysics and retain existence of its solutions. Extensive use is made of measures of noncompactness and abstract fixed point theorems such as Darbo's theorem.
Optimal homotopy asymptotic method for solving Volterra integral equation of first kind
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N. Khan
2014-09-01
Full Text Available In this paper, authors demonstrate the efficiency of optimal homotopy asymptotic method (OHAM. This is done by solving nonlinear Volterra integral equation of first kind. OHAM is applied to Volterra integral equations which involves exponential, trigonometric function as their kernels. It is observed that solution obtained by OHAM is more accurate than existing techniques, which proves its validity and stability for solving Volterra integral equation of first kind.
Optimal homotopy asymptotic method for solving Volterra integral equation of first kind
Khan, N; Hashmi, M.S.; Iqbal, S.; Mahmood, T
2014-01-01
In this paper, authors demonstrate the efficiency of optimal homotopy asymptotic method (OHAM). This is done by solving nonlinear Volterra integral equation of first kind. OHAM is applied to Volterra integral equations which involves exponential, trigonometric function as their kernels. It is observed that solution obtained by OHAM is more accurate than existing techniques, which proves its validity and stability for solving Volterra integral equation of first kind.
Yoshida, Hiroaki; Kobayashi, Takayuki; Hayashi, Hidemitsu; Kinjo, Tomoyuki; Washizu, Hitoshi; Fukuzawa, Kenji
2014-07-01
A boundary scheme in the lattice Boltzmann method (LBM) for the convection-diffusion equation, which correctly realizes the internal boundary condition at the interface between two phases with different transport properties, is presented. The difficulty in satisfying the continuity of flux at the interface in a transient analysis, which is inherent in the conventional LBM, is overcome by modifying the collision operator and the streaming process of the LBM. An asymptotic analysis of the scheme is carried out in order to clarify the role played by the adjustable parameters involved in the scheme. As a result, the internal boundary condition is shown to be satisfied with second-order accuracy with respect to the lattice interval, if we assign appropriate values to the adjustable parameters. In addition, two specific problems are numerically analyzed, and comparison with the analytical solutions of the problems numerically validates the proposed scheme.
Implicit Vector Integral Equations Associated with Discontinuous Operators
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Paolo Cubiotti
2014-01-01
Full Text Available Let I∶=[0,1]. We consider the vector integral equation h(u(t=ft,∫Ig(t,z,u(z,dz for a.e. t∈I, where f:I×J→R, g:I×I→ [0,+∞[, and h:X→R are given functions and X,J are suitable subsets of Rn. We prove an existence result for solutions u∈Ls(I, Rn, where the continuity of f with respect to the second variable is not assumed. More precisely, f is assumed to be a.e. equal (with respect to second variable to a function f*:I×J→R which is almost everywhere continuous, where the involved null-measure sets should have a suitable geometry. It is easily seen that such a function f can be discontinuous at each point x∈J. Our result, based on a very recent selection theorem, extends a previous result, valid for scalar case n=1.
Adaptive integral method with fast Gaussian gridding for solving combined field integral equations
Bakır, O.; Baǧ; Cı, H.; Michielssen, E.
Fast Gaussian gridding (FGG), a recently proposed nonuniform fast Fourier transform algorithm, is used to reduce the memory requirements of the adaptive integral method (AIM) for accelerating the method of moments-based solution of combined field integral equations pertinent to the analysis of scattering from three-dimensional perfect electrically conducting surfaces. Numerical results that demonstrate the efficiency and accuracy of the AIM-FGG hybrid in comparison to an AIM-accelerated solver, which uses moment matching to project surface sources onto an auxiliary grid, are presented.
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Mesgarani, H; Parmour, P [Department of Mathematics, Shahid Rajaee University, Lavizan, Tehran (Iran, Islamic Republic of); Aghazadeh, N [Department of Applied Mathematics, Azarbaijan University of Tarbiat Moallem, Tabriz 53751 71379 (Iran, Islamic Republic of)], E-mail: hmesgarani@sru.ac.ir, E-mail: pparmour@yahoo.com, E-mail: aghazadeh@azaruniv.edu
2010-02-15
In this paper, we apply Aitken extrapolation and epsilon algorithm as acceleration technique for the solution of a weakly singular nonlinear Volterra integral equation of the second kind. In this paper, based on Tao and Yong (2006 J. Math. Anal. Appl. 324 225-37.) the integral equation is solved by Navot's quadrature formula. Also, Tao and Yong (2006) for the first time applied Richardson extrapolation to accelerating convergence for the weakly singular nonlinear Volterra integral equations of the second kind. To our knowledge, this paper may be the first attempt to apply Aitken extrapolation and epsilon algorithm for the weakly singular nonlinear Volterra integral equations of the second kind.
Inviscid/Boundary-Layer Aeroheating Approach for Integrated Vehicle Design
Lee, Esther; Wurster, Kathryn E.
2017-01-01
A typical entry vehicle design depends on the synthesis of many essential subsystems, including thermal protection system (TPS), structures, payload, avionics, and propulsion, among others. The ability to incorporate aerothermodynamic considerations and TPS design into the early design phase is crucial, as both are closely coupled to the vehicle's aerodynamics, shape and mass. In the preliminary design stage, reasonably accurate results with rapid turn-representative entry envelope was explored. Initial results suggest that for Mach numbers ranging from 9-20, a few inviscid solutions could reasonably sup- port surface heating predictions at Mach numbers variation of +/-2, altitudes variation of +/-10 to 20 kft, and angle-of-attack variation of +/- 5. Agreement with Navier-Stokes solutions was generally found to be within 10-15% for Mach number and altitude, and 20% for angle of attack. A smaller angle-of-attack increment than the 5 deg around times for parametric studies and quickly evolving configurations are necessary to steer design decisions. This investigation considers the use of an unstructured 3D inviscid code in conjunction with an integral boundary-layer method; the former providing the flow field solution and the latter the surface heating. Sensitivity studies for Mach number, angle of attack, and altitude, examine the feasibility of using this approach to populate a representative entry flight envelope based on a limited set of inviscid solutions. Each inviscid solution is used to generate surface heating over the nearby trajectory space. A subset of a considered in this study is recommended. Results of the angle-of-attack sensitivity studies show that smaller increments may be needed for better heating predictions. The approach is well suited for application to conceptual multidisciplinary design and analysis studies where transient aeroheating environments are critical for vehicle TPS and thermal design. Concurrent prediction of aeroheating
ON ASYMTOTIC APPROXIMATIONS OF FIRST INTEGRALS FOR DIFFERENTIAL AND DIFFERENCE EQUATIONS
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W.T. van Horssen
2007-04-01
Full Text Available In this paper the concept of integrating factors for differential equations and the concept of invariance factors for difference equations to obtain first integrals or invariants will be presented. It will be shown that all integrating factors have to satisfya system of partial differential equations, and that all invariance factors have to satisfy a functional equation. In the period 1997-2001 a perturbation method based on integrating vectors was developed to approximate first integrals for systems of ordinary differential equations. This perturbation method will be reviewed shortly. Also in the paper the first results in the development of a perturbation method for difference equations based on invariance factors will be presented.
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Taohua Liu
2017-01-01
Full Text Available Fractional advection-dispersion equations, as generalizations of classical integer-order advection-dispersion equations, are used to model the transport of passive tracers carried by fluid flow in a porous medium. In this paper, we develop an implicit finite difference method for fractional advection-dispersion equations with fractional derivative boundary conditions. First-order consistency, solvability, unconditional stability, and first-order convergence of the method are proven. Then, we present a fast iterative method for the implicit finite difference scheme, which only requires storage of O(K and computational cost of O(KlogK. Traditionally, the Gaussian elimination method requires storage of O(K2 and computational cost of O(K3. Finally, the accuracy and efficiency of the method are checked with a numerical example.
A Time Marching Scheme for Solving Volume Integral Equations on Nonlinear Scatterers
Bagci, Hakan
2015-01-07
Transient electromagnetic field interactions on inhomogeneous penetrable scatterers can be analyzed by solving time domain volume integral equations (TDVIEs). TDVIEs are oftentimes solved using marchingon-in-time (MOT) schemes. Unlike finite difference and finite element schemes, MOT-TDVIE solvers require discretization of only the scatterers, do not call for artificial absorbing boundary conditions, and are more robust to numerical phase dispersion. On the other hand, their computational cost is high, they suffer from late-time instabilities, and their implicit nature makes incorporation of nonlinear constitutive relations more difficult. Development of plane-wave time-domain (PWTD) and FFT-based schemes has significantly reduced the computational cost of the MOT-TDVIE solvers. Additionally, latetime instability problem has been alleviated for all practical purposes with the development of accurate integration schemes and specially designed temporal basis functions. Addressing the third challenge is the topic of this presentation. I will talk about an explicit MOT scheme developed for solving the TDVIE on scatterers with nonlinear material properties. The proposed scheme separately discretizes the TDVIE and the nonlinear constitutive relation between electric field intensity and flux density. The unknown field intensity and flux density are expanded using half and full Schaubert-Wilton-Glisson (SWG) basis functions in space and polynomial temporal interpolators in time. The resulting coupled system of the discretized TDVIE and constitutive relation is integrated in time using an explicit P E(CE) m scheme to yield the unknown expansion coefficients. Explicitness of time marching allows for straightforward incorporation of the nonlinearity as a function evaluation on the right hand side of the coupled system of equations. Consequently, the resulting MOT scheme does not call for a Newton-like nonlinear solver. Numerical examples, which demonstrate the applicability
Signatures of chaos and non-integrability in two-dimensional gravity with dynamical boundary
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Fitkevich Maxim
2016-01-01
Full Text Available We propose a model of two-dimensional dilaton gravity with a boundary. In the bulk our model coincides with the classically integrable CGHS model; the dynamical boundary cuts of the CGHS strong-coupling region. As a result, classical dynamics in our model reminds that in the spherically-symmetric gravity: wave packets of matter fields either reflect from the boundary or form black holes. We find large integrable sector of multisoliton solutions in this model. At the same time, we argue that the model is globally non-integrable because solutions at the verge of black hole formation display chaotic properties.
Existence and Ulam stabilities for Hadamard fractional integral equations with random effects
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Said Abbas
2016-01-01
Full Text Available This article concerns the existence and Ulam stabilities for a class of random integral equations via Hadamard's fractional integral. Our main tools is a random fixed point theorem with stochastic domain.
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Yelda Aygar
2012-01-01
Full Text Available Let denote the operator generated in by , , , and the boundary condition , where , , , and , are complex sequences, , , and is an eigenparameter. In this paper we investigated the principal functions corresponding to the eigenvalues and the spectral singularities of .
Variational methods for boundary value problems for systems of elliptic equations
Lavrent'ev, M A
2012-01-01
Famous monograph by a distinguished mathematician presents an innovative approach to classical boundary value problems. The treatment employs the basic scheme first suggested by Hilbert and developed by Tonnelli. 1963 edition.
On series involving zeros of trascendental functions arising from Volterra integral equations
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Pietro Cerone
2001-05-01
Full Text Available Series arising from Volterra integral equations of the second kind are summed. The series involve inverse powers of roots of the characteristic equation. It is shown how previous similar series obtained from differential-difference equations are particular cases of the present development. A number of novel and interesting results are obtained. The techniques are demonstrated through illustrative examples.
Domoshnitsky, Alexander; Volinsky, Irina
2014-01-01
The impulsive delay differential equation is considered (Lx)(t) = x'(t) + ∑(i=1)(m) p(i)(t)x(t - τ(i) (t)) = f(t), t ∈ [a, b], x(t j) = β(j)x(t(j - 0)), j = 1,…, k, a = t0 Green's functions for this equation are obtained.
Chen, G.; Zheng, Q.; Coleman, M.; Weerakoon, S.
1983-01-01
This paper briefly reviews convergent finite difference schemes for hyperbolic initial boundary value problems and their applications to boundary control systems of hyperbolic type which arise in the modelling of vibrations. These difference schemes are combined with the primal and the dual approaches to compute the optimal control in the unconstrained case, as well as the case when the control is subject to inequality constraints. Some of the preliminary numerical results are also presented.
Mikhailov, SE
2006-01-01
Copyright @ 2006 Tech Science Press A quasi-static mixed boundary value problem of elastic damage mechanics for a continuously inhomogeneous body is considered. Using the two-operator Green-Betti formula and the fundamental solution of an auxiliary homogeneous linear elasticity with frozen initial, secant or tangent elastic coe±cients, a boundary-domain integro-differential formulation of the elasto-plastic problem with respect to the displacement rates and their gradients is derived. Usin...
Integration of equations of parabolic type by the method of nets
Saul'Yev, V K; Stark, M; Ulam, S
1964-01-01
International Series of Monographs in Pure and Applied Mathematics, Volume 54: Integration of Equations of Parabolic Type by the Method of Nets deals with solving parabolic partial differential equations using the method of nets. The first part of this volume focuses on the construction of net equations, with emphasis on the stability and accuracy of the approximating net equations. The method of nets or method of finite differences (used to define the corresponding numerical method in ordinary differential equations) is one of many different approximate methods of integration of partial diff
Integrable and nonintegrable cases of the Lax equations with a source
Mel'Nikov, V. K.
1994-06-01
The Korteweg-de Vries equation with a source given as a Fourier integral over eigenfunctions of the so-called generating operator is considered. It is shown that, depending on the choice of the basis of the eigenfunctions, we have the following three possibilities: (1) evolution equations for the scattering data are nonintegrable; (2) evolution equations for the scattering data are integrable but the solution of the Cauchy problem for the Korteweg-de Vries equation with a source at some t'> t o leaves the considered class of functions decreasing rapidly enough as x→±∞; (3) evolution equations for the scattering data are integrable and the solution of the Cauchy problem for the Korteweg-de vries equation with a source exists at all t> t o. All these possibilities are widespread and occur in other Lax equations with a source.
One out of many? Boundary negotiation and identity formation in postmerger integration
Drori, Israel; Wrzesniewski, Amy; Ellis, Shmuel
2013-01-01
This research investigates how boundaries are utilized during the postmerger integration process to influence the postmerger identity of the firm. We suggest that the boundaries that define the structures, practices, and values of firms prior to a merger become reinforced, contested, or revised in
Tang, Zhengming; Hong, Tao; Chen, Fangyuan; Zhu, Huacheng; Huang, Kama
2017-10-01
Microwave heating uniformity is mainly dependent on and affected by electric field. However, little study has paid attention to its stability characteristics in multimode cavity. In this paper, this problem is studied by the theory of Freedholm integral equation. Firstly, Helmholtz equation and the electric dyadic Green's function are used to derive the electric field integral equation. Then, the stability of electric field is demonstrated as the characteristics of solutions to Freedholm integral equation. Finally, the stability characteristics are obtained and verified by finite element calculation. This study not only can provide a comprehensive interpretation of electric field in multimode cavity but also help us make better use of microwave energy.
Shiryaeva, E V
2014-01-01
In paper [S.I. Senashov, A. Yakhno. 2012. SIGMA. Vol.8. 071] the variant of the hodograph method based on the conservation laws for two hyperbolic quasilinear equations of the first order is described. Using these results we propose a method which allows to reduce the Cauchy problem for the two quasilinear PDE's to the Cauchy problem for ODE's. The proposed method is actually some similar method of characteristics for a system of two hyperbolic quasilinear equations. The method can be used effectively in all cases, when the linear hyperbolic equation in partial derivatives of the second order with variable coefficients, resulting from the application of the hodograph method, has an explicit expression for the Riemann-Green function. One of the method's features is the possibility to construct a multi-valued solutions. In this paper we present examples of method application for solving the classical shallow water equations.
On the Analysis of Numerical Methods for Nonstandard Volterra Integral Equation
Mamba, H. S.; M. Khumalo
2014-01-01
We consider the numerical solutions of a class of nonlinear (nonstandard) Volterra integral equation. We prove the existence and uniqueness of the one point collocation solutions and the solution by the repeated trapezoidal rule for the nonlinear Volterra integral equation. We analyze the convergence of the collocation methods and the repeated trapezoidal rule. Numerical experiments are used to illustrate theoretical results.
Applications of normal S-iterative method to a nonlinear integral equation.
Gürsoy, Faik
2014-01-01
It has been shown that a normal S-iterative method converges to the solution of a mixed type Volterra-Fredholm functional nonlinear integral equation. Furthermore, a data dependence result for the solution of this integral equation has been proven.
On the Analysis of Numerical Methods for Nonstandard Volterra Integral Equation
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H. S. Mamba
2014-01-01
Full Text Available We consider the numerical solutions of a class of nonlinear (nonstandard Volterra integral equation. We prove the existence and uniqueness of the one point collocation solutions and the solution by the repeated trapezoidal rule for the nonlinear Volterra integral equation. We analyze the convergence of the collocation methods and the repeated trapezoidal rule. Numerical experiments are used to illustrate theoretical results.
Numerical solution of nonlinear Volterra-Fredholm integral equation by using Chebyshev polynomials
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R. Ezzati
2011-03-01
Full Text Available In this paper, we have used Chebyshev polynomials to solve linearand nonlinear Volterra-Fredholm integral equations, numerically.First we introduce these polynomials, then we use them to changethe Volterra-Fredholm integral equation to a linear or nonlinearsystem. Finally, the numerical examples illustrate the efficiencyof this method.
Jha, Abhinav K.; Zhu, Yansong; Wong, Dean F.; Rahmim, Arman
2017-03-01
Developing reconstruction methods for diffuse optical imaging requires accurate modeling of photon propagation, including boundary conditions arising due to refractive index mismatch as photons propagate from the tissue to air. For this purpose, we developed an analytical Neumann-series radiative transport equation (RTE)-based approach. Each Neumann series term models different scattering, absorption, and boundary-reflection events. The reflection is modeled using the Fresnel equation. We use this approach to design a gradient-descent-based analytical reconstruction algorithm for a three-dimensional (3D) setup of a diffuse optical imaging (DOI) system. The algorithm was implemented for a three-dimensional DOI system consisting of a laser source, cuboidal scattering medium (refractive index > 1), and a pixelated detector at one cuboid face. In simulation experiments, the refractive index of the scattering medium was varied to test the robustness of the reconstruction algorithm over a wide range of refractive index mismatches. The experiments were repeated over multiple noise realizations. Results showed that by using the proposed algorithm, the photon propagation was modeled more accurately. These results demonstrated the importance of modeling boundary conditions in the photon-propagation model.
Aguareles, M.
2014-06-01
In this paper we consider an oscillatory medium whose dynamics are modeled by the complex Ginzburg-Landau equation. In particular, we focus on n-armed spiral wave solutions of the complex Ginzburg-Landau equation in a disk of radius d with homogeneous Neumann boundary conditions. It is well-known that such solutions exist for small enough values of the twist parameter q and large enough values of d. We investigate the effect of boundaries on the rotational frequency of the spirals, which is an unknown of the problem uniquely determined by the parameters d and q. We show that there is a threshold in the parameter space where the effect of the boundary on the rotational frequency switches from being algebraic to exponentially weak. We use the method of matched asymptotic expansions to obtain explicit expressions for the asymptotic wavenumber as a function of the twist parameter and the domain size for small values of q. © 2014 Elsevier B.V. All rights reserved.
Monotonic solutions of functional integral and differential equations of fractional order
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Ahmed El-Sayed
2009-02-01
Full Text Available The existence of positive monotonic solutions, in the class of continuous functions, for some nonlinear quadratic integral equations have been studied by J. Banas. Here we are concerned with a singular quadratic functional integral equations. The existence of positive monotonic solutions $x \\in L_1[0,1]$ will be proved. The fractional order nonlinear functional differential equation will be given as a special case.
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Lakestani M.
2005-01-01
Full Text Available Compactly supported linear semiorthogonal B-spline wavelets together with their dual wavelets are developed to approximate the solutions of nonlinear Fredholm-Hammerstein integral equations. Properties of these wavelets are first presented; these properties are then utilized to reduce the computation of integral equations to some algebraic equations. The method is computationally attractive, and applications are demonstrated through an illustrative example.
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Quanquan Wang
2017-10-01
Full Text Available The marching-on-in-degree (MOD time-domain integral equation (TDIE solver for the transient electromagnetic scattering of the graphene is presented in this paper. Graphene’s dispersive surface impedance is approximated using rational function expressions of complex conjugate pole-residue pairs with the vector fitting (VF method. Enforcing the surface impedance boundary condition, TDIE is established and solved in the MOD scheme, where the temporal surface impedance is carefully convoluted with the current. Unconditionally stable transient solution in time domain can be ensured. Wide frequency band information is obtained after the Fourier transform of the time domain solution. Numerical results validate the proposed method.
Block-pulse functions approach to numerical solution of Abel’s integral equation
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Monireh Nosrati Sahlan
2015-12-01
Full Text Available This study aims to present a computational method for solving Abel’s integral equation of the second kind. The introduced method is based on the use of Block-pulse functions (BPFs via collocation method. Abel’s integral equations as singular Volterra integral equations are hard and heavy in computation, but because of the properties of BPFs, as is reported in examples, this method is more efficient and more accurate than some other methods for solving this class of integral equations. On the other hand, the benefit of this method is low cost of computing operations. The applied method transforms the singular integral equation into triangular linear algebraic system that can be solved easily. An error analysis is worked out and applications are demonstrated through illustrative examples.
An application of a semi-analytical method on linear fuzzy Volterra integral equations
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Tofigh Allahviranloo
2014-03-01
Full Text Available Recently, fuzzy integral equations have attracted some interest. In this paper, we focus on linear fuzzy Volterra integral equation of the second kind (FVIE-2 and propose a new method for numerical solving it. In fact, using parametric form of fuzzy numbers we convert a linear fuzzy Volterra integral equation of the second kind to a linear system of Volterra integral equations of the second kind in crisp case. We use variational iteration method (VIM and find the approximate solution of this system and hence obtain an approximation for fuzzy solution of the linear fuzzy Volterra integral equation of the second kind. Finally, using the proposed method, we give some illustrative examples.
Domoshnitsky, Alexander
2014-01-01
The impulsive delay differential equation is considered (Lx)(t) = x′(t) + ∑i=1 m p i(t)x(t − τ i(t)) = f(t), t ∈ [a, b], x(t j) = β j x(t j − 0), j = 1,…, k, a = t 0 Green's functions for this equation are obtained. PMID:24719584
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M. G. Crandall
1999-07-01
Full Text Available We study existence of continuous weak (viscosity solutions of Dirichlet and Cauchy-Dirichlet problems for fully nonlinear uniformly elliptic and parabolic equations. Two types of results are obtained in contexts where uniqueness of solutions fails or is unknown. For equations with merely measurable coefficients we prove solvability of the problem, while in the continuous case we construct maximal and minimal solutions. Necessary barriers on external cones are also constructed.
Dispersion estimates for spherical Schrödinger equations: the effect of boundary conditions
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Markus Holzleitner
2016-01-01
Full Text Available We investigate the dependence of the \\(L^1\\to L^{\\infty}\\ dispersive estimates for one-dimensional radial Schrödinger operators on boundary conditions at \\(0\\. In contrast to the case of additive perturbations, we show that the change of a boundary condition at zero results in the change of the dispersive decay estimates if the angular momentum is positive, \\(l\\in (0,1/2\\. However, for nonpositive angular momenta, \\(l\\in (-1/2,0]\\, the standard \\(O(|t|^{-1/2}\\ decay remains true for all self-adjoint realizations.
Topology and boundary shape optimization as an integrated design tool
Bendsoe, Martin Philip; Rodrigues, Helder Carrico
1990-01-01
The optimal topology of a two dimensional linear elastic body can be computed by regarding the body as a domain of the plane with a high density of material. Such an optimal topology can then be used as the basis for a shape optimization method that computes the optimal form of the boundary curves of the body. This results in an efficient and reliable design tool, which can be implemented via common FEM mesh generator and CAD type input-output facilities.
Integrating Observations of the Boundary Current Flow around Sri Lanka
2015-09-30
around Sri Lanka Uwe Send and Matthias Lankhorst Scripps Institution of Oceanography 9500 Gilman Drive, Mail Code 0230 La Jolla, CA 92093-0230...of Bengal. For this, the flow around Sri Lanka is critical since it exchanges salt and freshwater between the Bay of Bengal and the Arabian Sea...OBJECTIVES In-situ continuous observations of the boundary current flow around Sri Lanka will be collected over a period of several years. In order
De La Rosa Gomez, Alejandro; MacKay, Niall; Regelskis, Vidas
2017-04-01
We present a general method of folding an integrable spin chain, defined on a line, to obtain an integrable open spin chain, defined on a half-line. We illustrate our method through two fundamental models with sl2 Lie algebra symmetry: the Heisenberg XXX and the Inozemtsev hyperbolic spin chains. We obtain new long-range boundary Hamiltonians and demonstrate that they exhibit Yangian symmetries, thus ensuring integrability of the models we obtain. The method presented provides a ;bottom-up; approach for constructing integrable boundaries and can be applied to any spin chain model.
Zhou, Hua-Cheng; Guo, Bao-Zhu
2017-08-01
In this paper, we consider boundary output feedback stabilization for a multi-dimensional wave equation with boundary control matched unknown nonlinear internal uncertainty and external disturbance. A new unknown input type extended state observer is proposed to recover both state and total disturbance which consists of internal uncertainty and external disturbance. A key feature of the proposed observer in this paper is that we do not use the high-gain to estimate the disturbance. By the active disturbance rejection control (ADRC) strategy, the total disturbance is compensated (canceled) in the feedback loop, which together with a collocated stabilizing controller without uncertainty, leads to an output feedback stabilizing feedback control. It is shown that the resulting closed-loop system is well-posed and asymptotically stable under weak assumption on internal uncertainty and external disturbance. The numerical experiments are carried out to show the effectiveness of the proposed scheme.
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A.A. El-Deeb
2017-07-01
Full Text Available By some new analysis techniques, we generalize the results presented by Pachpatte in [Integral and Finite Difference Inequalities and Applications, volume 205, Elsevier, 2006] and and by S. D. Kendre in [Some nonlinear integral inequalities for Volterra–Fredholm integral equations, Adv. Inequal. App, 2014:Article 21, 2014.] to nonlinear retarded inequalities, and investigate also some new forms. Some applications are also presented in order to illustrate the usefulness of some of our results.
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Gampert, B.; Beyene, T. [Essen Univ. (Germany). Angewandte Mechanik
2000-07-01
the authors investigated heat transfer and wall shearing stress on a spinning thread from the boundary layer equations according to the similar solutions method for a wide range of technical parameters. Laminar incompressible flow with constant material characteristics is assumed, and the spinning thread is assumed to be a continuously moving circular cylinder. [German] In dieser Arbeit werden der Waermeuebergang und die Wandschubspannung am Spinnfaden aus den Grenzschichtgleichungen nach der Methode der aehnlichen Loesungen fuer einen weiten Bereich technischer Parameter bestimmt. Es wird eine laminare inkompressible Stroemung konstanter Stoffwerte vorausgesetzt und der Spinnfaden wird als kontinuierlich bewegter Kreiszylinder angenommen. (orig.)
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Schwartz, Peter; Barad, Michael; Colella, Phillip; Ligocki, Terry
2004-11-02
We present an algorithm for solving Poisson's equation and the heat equation on irregular domains in three dimensions. Our work uses the Cartesian grid embedded boundary algorithm for 2D problems of Johansen and Colella (1998, J. Comput. Phys. 147(2):60-85) and extends work of McCorquodale, Colella and Johansen (2001, J. Comput. Phys. 173(2):60-85). Our method is based on a finite-volume discretization of the operator, on the control volumes formed by intersecting the Cartesian grid cells with the domain, combined with a second-order accurate discretization of the fluxes. The resulting method provides uniformly second-order accurate solutions and gradients and is amenable to geometric multigrid solvers.
Third order difference equations with two rational integrals
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Sahadevan, R; Maheswari, C Uma [Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600 005, Tamil Nadu (India)], E-mail: ramajayamsaha@yahoo.co.in, E-mail: umagenie2004@hotmail.com
2009-10-30
A systematic investigation to derive three-dimensional analogs of two-dimensional Quispel, Roberts and Thompson (QRT) mappings is presented. The question of integrability of the obtained three-dimensional mappings with two independent integrals is also analyzed. It is also shown that there exist three-dimensional QRT maps with three n-dependent integrals.