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On an effective solution of the Riemann problem for the second-order improperly elliptic equation in the rectangle
Abstract
In this paper we present the numerical method for the solution of the Riemann problem for the second-order improperly elliptic equation. First, we reduce this problem to boundary value problems for properly elliptic equations, and after that we solve these problems by the grid method.
MSC:
35G45, 35G15, 35J25, 35J57, 65N06, 65N20.
... In the last few decades a wide study on the Riemann-type boundary value problems for different classes of complex partial differential equations of arbitrary order has been done, see [2][3][4][5][6][7][8][9]. Important examples of such higher order differential equations are those yielding the well-known polyanalytic and polyharmonic functions. ...
... The only difference from the case of + is that in the latter case the functions ϕ and ψ may be found to be multi-valued, as a consequence of the presence of logarithmic terms. Therefore, the original problem may be reduced to the determination of the analytic functions ϕ and ψ such that F given by (5) satisfies the boundary conditions in (4). ...
... Therefore, ϕ(z) may also be determined by integration. Substituting (5) in the first boundary condition of (4), one obtains ...
In this paper we study a kind of Riemann problem for the Lamé–Navier system in the plane on a smooth as well as on a fractal closed contour. By using the Kolosov–Muskhelisvili formula, we reduce this problem to a pair of Riemann boundary value problems for analytic functions, and after that we get the necessary and sufficient conditions for the solvability of the problem and obtain explicit formulas for its solution.
Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G′/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd–Sokolov–Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics.
In this chapter we deal with the discretization of second-order initial-boundary value problems of parabolic and hyperbolic types. Unlike Chapter 2, the space discretization is handled by finite element methods. We shall begin with some basic properties that are used later to derive appropriate weak formulations of the problems considered. Then, moving on to numerical methods, various temporal discretizations are discussed for parabolic problems: one-step methods (Runge-Kutta), linear multi-step methods (BDF) and the discontinuous Galerkin method. An examination of similar topics for second-order hyperbolic problems follows. The initial value problems that are generated by semi-discretization methods are very stiff, which demands a careful choice of the time integration codes for such problems. Finally, at the end of the chapter some results on error control are given.
Contents Notation 1 Basics 1.1 Classification and Correctness 1.2 Fourier's Method, Integral Transforms 1.3 Maximum Principle, Fundamental Solution 2 Finite Difference Methods 2.1 Basic Concepts 2.2 Illustrative Examples 2.3 Transportation Problems and Conservation Laws 2.4 Elliptic Boundary Value Problems 2.5 Finite Volume Methods as Finite Difference Schemes 2.6 Parabolic Initial-Boundary Value Problems 2.7 Second-Order Hyperbolic Problems 3 Weak Solutions 3.1 Introduction 3.2 Adapted Function Spaces 3.3 VariationalEquationsand conformingApproximation 3.4 WeakeningV-ellipticity 3.5 NonlinearProblems 4 The Finite Element Method 4.1 A First Example 4.2 Finite-Element-Spaces 4.3 Practical Aspects of the Finite Element Method 4.4 Convergence of Conforming Methods 4.5 NonconformingFiniteElementMethods 4.6 Mixed Finite Elements 4.7 Error Estimators and adaptive FEM 4.8 The Discontinuous Galerkin Method 4.9 Further Aspects of the Finite Element Method 5 Finite Element Methods for Unsteady Problems 5.1 Parabolic Problems 5.2 Second-Order Hyperbolic Problems 6 Singularly Perturbed Boundary Value Problems 6.1 Two-Point Boundary Value Problems 6.2 Parabolic Problems, One-dimensional in Space 6.3 Convection-Diffusion Problems in Several Dimensions 7 Variational Inequalities, Optimal Control 7.1 Analytic Properties 7.2 Discretization of Variational Inequalities 7.3 Penalty Methods 7.4 Optimal Control of PDEs 8 Numerical Methods for Discretized Problems 8.1 Some Particular Properties of the Problems 8.2 Direct Methods 8.3 Classical Iterative Methods 8.4 The Conjugate Gradient Method 8.5Multigrid Methods 8.6 Domain Decomposition, Parallel Algorithms Bibliography: Textbooks and Monographs Bibliography: Original Papers Index
Preliminaries basic concepts of the theory of difference schemes homogeneous difference schemes difference schemes for elliptic equations different schemes for time-dependent equations with constant coefficients stability theory of differenceschemes symbols concluding remarks.
This paper is concerned with approaches to compute a factored sparse approximate inverse for block tridiagonal and block pentadiagonal matrices. Recurrence formulas are developed for computing sparse approximate inverse factors of these matrices using bordering technique. Resulting factored sparse approximate inverse is used as a preconditioner for the conjugate gradient method (PCG). As an application these formulas are simplified for computing the preconditioner for solving Lyapanuv matrix equations by PCG method. Numerical experiments on linear system, arising from discretization of partial differential equations are presented.
A systematic investigation of basic boundary value problems for complex partial differential equations of arbitrary order is started in these lectures restricted to model equations. In the first part [the author, Some boundary value problems for bi-bianalytic functions. In: Complex analysis, differential equations and related topics. Proceedings of the ISAAC conference on analysis, Yerevan, Armenia, 2002. Yerevan: National Academy of Sciences of Armenia, 233–252 (2004; Zbl 1063.30035)] the Schwarz, the Dirichlet, and the Neumann problems are treated for the inhomogeneous Cauchy-Riemann equation. The fundamental tools are the Gauss theorem and the Cauchy-Pompeiu representation. The principle of iterating these representation formulas is introduced which will enable treating higher order equations. Those are studied in a second part of these lectures. The whole course was delivered at the Simón Bolívar University in Caracas in May 2004. For Part II see Bol. Asoc. Mat. Venez. 12, No. 2, 217–250 (2005; Zbl 1116.30032).
Factored sparse approximate inverse of block tridiagonal and block pentadiagonal matrices doi:10.1186/1687-1847-2013-190 Cite this article as: Babayan and Raeisian: On an effective solution of the Riemann problem for the second-order improperly elliptic equation in the rectangle
- Koulaei
- Mh
- Toutounian
Koulaei, MH, Toutounian, F: Factored sparse approximate inverse of block tridiagonal and block pentadiagonal matrices. Appl. Math. Comput. 184, 223-234 (2007) doi:10.1186/1687-1847-2013-190 Cite this article as: Babayan and Raeisian: On an effective solution of the Riemann problem for the second-order improperly elliptic equation in the rectangle. Advances in Difference Equations 2013 2013:190.
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