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## Initial-boundary Value Problem (initial-boundary + value_problem)
## Selected Abstracts## A Riemann solver and upwind methods for a two-phase flow model in non-conservative form INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2006C. E. CastroAbstract We present a theoretical solution for the Riemann problem for the five-equation two-phase non-conservative model of Saurel and Abgrall. This solution is then utilized in the construction of upwind non-conservative methods to solve the general initial-boundary value problem for the two-phase flow model in non-conservative form. The basic upwind scheme constructed is the non-conservative analogue of the Godunov first-order upwind method. Second-order methods in space and time are then constructed via the MUSCL and ADER approaches. The methods are systematically assessed via a series of test problems with theoretical solutions. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## A boundary value problem for the spherically symmetric motion of a pressureless gas with a temperature-dependent viscosity MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2009Bernard DucometAbstract We consider an initial-boundary value problem for the equations of spherically symmetric motion of a pressureless gas with temperature-dependent viscosity µ(,) and conductivity ,(,). We prove that this problem admits a unique weak solution, assuming Belov's functional relation between µ(,) and ,(,) and we give the behaviour of the solution for large times. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## On an initial-boundary value problem for a wide-angle parabolic equation in a waveguide with a variable bottom MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2009V. A. DougalisAbstract We consider the third-order Claerbout-type wide-angle parabolic equation (PE) of underwater acoustics in a cylindrically symmetric medium consisting of water over a soft bottom B of range-dependent topography. There is strong indication that the initial-boundary value problem for this equation with just a homogeneous Dirichlet boundary condition posed on B may not be well-posed, for example when B is downsloping. We impose, in addition to the above, another homogeneous, second-order boundary condition, derived by assuming that the standard (narrow-angle) PE holds on B, and establish a priori H2 estimates for the solution of the resulting initial-boundary value problem for any bottom topography. After a change of the depth variable that makes B horizontal, we discretize the transformed problem by a second-order accurate finite difference scheme and show, in the case of upsloping and downsloping wedge-type domains, that the new model gives stable and accurate results. We also present an alternative set of boundary conditions that make the problem exactly energy conserving; one of these conditions may be viewed as a generalization of the Abrahamsson,Kreiss boundary condition in the wide-angle case. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## On the asymptotic stability of steady solutions of the Navier,Stokes equations in unbounded domains MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2007Francesca CrispoAbstract We consider the problem of the asymptotic behaviour in the L2 -norm of solutions of the Navier,Stokes equations. We consider perturbations to the rest state and to stationary motions. In both cases we study the initial-boundary value problem in unbounded domains with non-compact boundary. In particular, we deal with domains with varying and possibly divergent exits to infinity and aperture domains. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Global existence of weak-type solutions for models of monotone type in the theory of inelastic deformations MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2002Krzysztof CheThis article introduces the notion of weak-type solutions for systems of equations from the theory of inelastic deformations, assuming that the considered model is of monotone type (for the definition see [Lecture Notes in Mathematics, 1998, vol. 1682]). For the boundary data associated with the initial-boundary value problem and satisfying the safe-load condition the existence of global in time weak-type solutions is proved assuming that the monotone model is rate-independent or of gradient type. Moreover, for models possessing an additional regularity property (see Section 5) the existence of global solutions in the sense of measures, defined by Temam in Archives for Rational Mechanics and Analysis, 95: 137, is obtained, too. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Bernstein Ritz-Galerkin method for solving an initial-boundary value problem that combines Neumann and integral condition for the wave equation NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2010S.A. YousefiAbstract In this article, the Ritz-Galerkin method in Bernstein polynomial basis is implemented to give an approximate solution of a hyperbolic partial differential equation with an integral condition. We will deal here with a type of nonlocal boundary value problem, that is, the solution of a hyperbolic partial differential equation with a nonlocal boundary specification. The nonlocal conditions arise mainly when the data on the boundary cannot be measured directly. The properties of Bernstein polynomial and Ritz-Galerkin method are first presented, then Ritz-Galerkin method is used to reduce the given hyperbolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique presented in this article. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 [source] ## On the numerical solution of hyperbolic PDEs with variable space operator NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2009Allaberen AshyralyevAbstract The first and second order of accuracy in time and second order of accuracy in the space variables difference schemes for the numerical solution of the initial-boundary value problem for the multidimensional hyperbolic equation with dependent coefficients are considered. Stability estimates for the solution of these difference schemes and for the first and second order difference derivatives are obtained. Numerical methods are proposed for solving the one-dimensional hyperbolic partial differential equation. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009 [source] ## Numerical solution to a linearized KdV equation on unbounded domain NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2008Chunxiong ZhengAbstract Exact absorbing boundary conditions for a linearized KdV equation are derived in this paper. Applying these boundary conditions at artificial boundary points yields an initial-boundary value problem defined only on a finite interval. A dual-Petrov-Galerkin scheme is proposed for numerical approximation. Fast evaluation method is developed to deal with convolutions involved in the exact absorbing boundary conditions. In the end, some numerical tests are presented to demonstrate the effectiveness and efficiency of the proposed method.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008 [source] ## On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2005Mehdi DehghanAbstract Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second-order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one-dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 [source] ## A dual-reciprocity boundary element solution of a generalized nonlinear Schrödinger equation NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2004Whye-Teong AngAbstract A time-stepping dual-reciprocity boundary element method is presented for the numerical solution of an initial-boundary value problem governed by a generalized non-linear Schrödinger equation. To test the method, two specific problems with known exact solutions are solved. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20, 2004. [source] ## On the choice of initial conditions of difference schemes for parabolic equations PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2007Givi BerikelashviliWe study finite difference schemes to approximate the first initial-boundary value problem for linear second order parabolic equations and obtain some convergence rate estimates. When difference schemes are constructed for such problems, in the process of obtaining convergence rate estimates compatible with smoothness of the solution, various authors assume that the solution of the problem can be extended to the exterior of the domain of integration, preserving the Sobolev class. Our investigations show that this restriction can be removed if, instead of using the exact initial condition, we use certain approximations of the initial conditions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## On the initial-boundary value problem of the incompressible viscoelastic fluid system COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 4 2008Fanghua LinIn this paper, we shall establish the local well-posedness of the initial-boundary value problem of the viscoelastic fluid system of the Oldroyd model. We shall also prove that the local solutions can be extended globally and that the global solutions decay exponentially fast as time goes to infinity provided that the initial data are sufficiently close to the equilibrium state. © 2007 Wiley Periodicals, Inc. [source] ## Continuous dependence on the geometry and on the initial time for a class of parabolic problems I MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2007L. E. PayneAbstract In this paper, we investigate the continuous dependence on the geometry and the initial time for solutions u(x, t) of a class of nonlinear parabolic initial-boundary value problems. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Continuous dependence on the geometry and on the initial time for a class of parabolic problems II MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2007G. A. PhilippinAbstract Extending the investigations initiated in an earlier paper, the authors deal in this paper with the solutions of another class of initial-boundary value problems for which continuous dependence inequalities on the geometry and the initial time are established. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## A Petrov-Galerkin method with quadrature for semi-linear second-order hyperbolic problems NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2006B. BialeckiAbstract We propose and analyze a Crank,Nicolson quadrature Petrov,Galerkin (CNQPG) -spline method for solving semi-linear second-order hyperbolic initial-boundary value problems. We prove second-order convergence in time and optimal order H2 norm convergence in space for the CNQPG scheme that requires only linear algebraic solvers. We demonstrate numerically optimal order Hk, k = 0,1,2, norm convergence of the scheme for some test problems with smooth and nonsmooth nonlinearities. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 [source] |