On an effective solution of the Riemann problem for the second-order improperly elliptic equation in the rectangle
© Babayan and Raeisian; licensee Springer 2013
Received: 12 September 2012
Accepted: 31 May 2013
Published: 28 June 2013
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.
Keywordsimproperly elliptic equation boundary value problem Riemann problem Bitzadze equation grid method
reduces to ill-conditioned linear system with zeroes on the main diagonal, so we introduce another variant of the solution.
2 Description of the algorithm of solution
Here a function is the already known function . After determining the function ψ, we get the boundary value problem, analogous to (13), to determine the imaginary part of the function Ψ.
Thus, the main idea of this algorithm is to reduce the previous problem for an improperly elliptic equation to the boundary value problems for properly elliptic equations (in our case, to boundary value problems for biharmonic and Laplace equations). In the next section we describe the realization of this algorithm by the grid method.
3 Solution of problem (3), (4) by the grid method
and is a standard grid analogue (forward divided difference) of the operator , and are values of the functions f and g in boundary points of the grid. This problem approximates problem (9), and the rate of approximation is for a biharmonic equation and for boundary conditions (see ). Therefore, from the stability of problem (18), we get the convergence of the grid function to (, p.30, Theorem 2.5). From the last two equations of (18), we get the values of the function in the points for ; and ; , and we find the values in interior nodes from the linear system with a symmetric pentadiagonal matrix. Hence, we get the stability of (18) from the positive definiteness of the main matrix of this system, and an algorithm for the solution of this system may be found in .
in the grid points, here , , are the real constants. And at the final step, we must find the function Ψ. First, we solve the Dirichlet problem for the Laplace equation to determine the real part of Ψ, and then we determine the imaginary part of Ψ solving the problem analogous to (13). Here we get the last arbitrary constant .
Summing up, we can say that the previous problem may be reduced to boundary value problems for properly elliptic equations.
- Bitzadze AV: On a uniqueness of the Dirichlet problem for elliptic partial differential equations. Usp. Mat. Nauk 1948, 3(6(28)):211-212. (in Russian)Google Scholar
- Bitzadze AV: Boundary Value Problems for Elliptic Equations of Second Order. Nauka, Moscow; 1966. (in Russian), Engl. Transl. North-Holland, Amsterdam (1968)Google Scholar
- Tovmasyan NE: Non-Regular Differential Equations and Calculations of Electromagnetic Fields. World Scientific, Singapore; 1998.MATHView ArticleGoogle Scholar
- Wendland WL: Elliptic Systems in the Plane. Pitman, London; 1979.MATHGoogle Scholar
- Bikchantaev JA: The boundary value problem for the homogeneous elliptic equation with constant coefficients. Izv. Vysš. Učebn. Zaved., Mat. 1975, 6(157):56-61. (in Russian)Google Scholar
- Begehr H: Boundary value problems in complex analysis. I. Bol. Asoc. Mat. Venez. 2005, 12(1):65-85.MATHMathSciNetGoogle Scholar
- Begehr H: Boundary value problems in complex analysis. II. Bol. Asoc. Mat. Venez. 2005, 12(2):217-250.MATHMathSciNetGoogle Scholar
- Soldatov AP: The method of theory of functions for the boundary value problems on the plane 1. Smooth case. Izv. Akad. Nauk SSSR, Ser. Mat. 1991, 55(5):1070-1100. (in Russian)MATHGoogle Scholar
- Grossmann C, Roos H-G, Stynes M: Numerical Treatment of Partial Differential Equations. Springer, Berlin; 2007.MATHView ArticleGoogle Scholar
- Marchuk GI: Methods of Computational Mathematics. Nauka, Moscow; 1989. (in Russian)Google Scholar
- Wen GC: Approximate Methods and Numerical Analysis for Elliptic Complex Equations. Gordon and Breach, Amsterdam; 1999.MATHGoogle Scholar
- Samarskij AA: Theory of Difference Schemes. Dekker, New York; 2001.View ArticleGoogle Scholar
- Collatz L: The Numerical Treatment of Differential Equations. Springer, Berlin; 1966.Google Scholar
- Koulaei MH, Toutounian F: Factored sparse approximate inverse of block tridiagonal and block pentadiagonal matrices. Appl. Math. Comput. 2007, 184: 223-234. 10.1016/j.amc.2006.05.204MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.