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# On an effective solution of the Riemann problem for the second-order improperly elliptic equation in the rectangle

- Armenak O Babayan
^{1}and - Seyed Mohammadali Raeisian
^{2}Email author

**2013**:190

https://doi.org/10.1186/1687-1847-2013-190

© Babayan and Raeisian; licensee Springer 2013

**Received:**12 September 2012**Accepted:**31 May 2013**Published:**28 June 2013

## 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.

## Keywords

- improperly elliptic equation
- boundary value problem
- Riemann problem
- Bitzadze equation
- grid method

## 1 Introduction

*D*be a rectangle $D=\{(x,y):a<x<b,c<y<d\}$ in a complex plane with boundary $\mathrm{\Gamma}=\partial D$. We consider in

*D*the equation

*i.e.*, elliptic equation). Let the roots of (2) ${\lambda}_{k}$ with multiplicities ${m}_{k}$ satisfy the condition $\mathrm{\Im}{\lambda}_{k}>0$ and the roots ${\mu}_{j}$ with multiplicities ${l}_{j}$ satisfy the condition $\mathrm{\Im}{\mu}_{j}<0$. We suppose that ${\sum}_{k}{m}_{k}>{\sum}_{j}{l}_{k}$, so (1) is an improperly elliptic equation. It was shown in [1] that for the equation ${u}_{\overline{z}\overline{z}}=0$ (now known as a Bitzadze equation) the corresponding Dirichlet problem is not correct. It was shown later ([2–4]) that for arbitrary improperly elliptic equation (1) all of the classical boundary value problems are not correct (we say that the problem is correct if the corresponding homogeneous problem has a finite number of linearly independent solutions and the inhomogeneous problem is solvable if and only if the finite number of linearly independent conditions for the boundary functions are satisfied). Therefore another kind of boundary conditions must be introduced. In the works [5–8], different types of boundary conditions, whose number depends on the number of the roots of (2) with positive and negative imaginary parts, were introduced. In this paper we consider in

*D*the Bitzadze equation

*D*functions which satisfy Hölder condition in $D\cup \mathrm{\Gamma}$ with their first degree derivatives. We suppose that the function

*u*satisfies the Riemann-type boundary conditions

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

*D*. Therefore, problem (3)-(4) reduces to determination of the analytic functions Φ, Ψ by the condition (4). For this goal, we mention that if the function

*u*is a solution of problem (3)-(4), then the complex conjugate function $\overline{u}$ satisfies the equalities

*U*in the grid nodes:

*u*. From (6) we have

*D*; therefore, adding equalities (12) on the boundary of

*D*, we get the following problem with angled derivative for the Laplace equation:

*φ*- the real part of Φ. To determine the imaginary part of Φ, the function $\omega =\mathrm{\Im}\mathrm{\Phi}$, we have the similar boundary value problem

*D*; therefore, we have the Dirichlet problem for the Laplace equation to determine the real part of this function $\psi =\mathrm{\Re}\mathrm{\Psi}$:

Here a function ${\psi}_{0}$ is the already known function ${\psi}_{0}=U-\mathrm{\Re}(\overline{z}\mathrm{\Phi}(z))$. 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

*a*,

*b*,

*c*,

*d*are rational numbers. In this case we can divide the rectangle

*D*by $(N-1)(M-1)$ equidistant straight lines, parallel to coordinate axes and denote

*u*is a solution of problem (3), (4)) in the mesh points $({x}_{k},{y}_{j})$. First, we consider problem (9) to determine the function $U(x,y)=\mathrm{\Re}u(x,y)$. Passing to the discrete analogue of the Laplace operator

and ${\delta}_{h}$ is a standard grid analogue (forward divided difference) of the operator $\frac{\partial}{\partial \nu}$, ${f}_{h}$ and ${g}_{h}$ 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 $O({h}^{2})$ for a biharmonic equation and $O(h)$ for boundary conditions (see [13]). Therefore, from the stability of problem (18), we get the convergence of the grid function to $\{U({x}_{k},{y}_{m})\}$ ([9], p.30, Theorem 2.5). From the last two equations of (18), we get the values of the function ${U}_{h}$ in the points $({x}_{k},{y}_{m})$ for $k=0,1,N-1,N$; $m=0,1,\dots ,M$ and $k=0,1,\dots ,N$; $m=0,1,M-1,M$, and we find the values ${U}_{k}^{m}$ 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 [14].

*OX*axis. Values of ${\phi}_{k}^{m}$ inside ${\mathrm{\Gamma}}_{1h}$ and on the sides ${\mathrm{\Gamma}}_{1h}$ parallel to

*OY*axis will be found from the system of linear equations. The main matrix of this system may be reduced to the block tridiagonal form

in the grid points, here ${L}_{0}$, ${L}_{1}$, ${L}_{2}$ 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 ${L}_{3}$.

Summing up, we can say that the previous problem may be reduced to boundary value problems for properly elliptic equations.

## Declarations

## Authors’ Affiliations

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