FINITE DIFFERENCE METHOD FOR SOLVING SECOND ORDER BOUNDARY VALUE PROBLEM

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FINITE DIFFERENCE METHOD FOR SOLVING SECOND ORDER BOUNDARY VALUE PROBLEM

CHAPTER ONE

INTRODUCTION

Background of the study

Finite difference method for boundary value problem for nonlinear elliptic equation with nonlocal conditions

Many physical phenomena have been formulated as a mathematical model with nonlocal boundary conditions. A short overview of these models is presented in many papers (see, for example, works [1, 2]). Particulary, many problems in thermoelasticity can be formulated as nonlocal problems (see [3,4,5] and the references therein). A separate class of such nonlocal models is boundary value problem for elliptic equation with nonlocal boundary conditions.

Numerical methods for boundary value problem of linear and nonlinear elliptic equations with various types of nonlocal conditions have been intensively investigated during past decades. Finite difference methods for linear elliptic equations with Bicadze–Samarski or multipoint nonlocal conditions were analyzed in works. In the papers, the main goal was the investigation of the existence and uniqueness of the solution of difference problem with integral conditions, as well as estimation of the error in certain norms.

Various iterative methods for the systems of linear difference equations, derived from elliptic equations with nonlocal conditions, and proofs of convergence of these methods can be found. The iterative methods are generalized for the systems of nonlinear difference equations with nonlocal conditions. Some other difference methods for elliptic equations with nonlocal conditions were described.

In most cases of elliptic equations with nonlocal conditions, the matrix of the discrete problem is characterized by the properties appropriate for M-matrices. This was used to prove the convergence of iterative methods for linear and nonlinear elliptic equations with nonlocal conditions.

Application of the M-matrices for the elliptic and parabolic differential equations with Dirichlet boundary conditions has been described by Varga (for some new research in this field, see the works.

In the present paper, the idea of application of M-matrices for theoretical investigation of difference methods with nonlocal conditions is further developed. Namely, M-matrices are used to prove the convergence of difference schemes. It is well known that the property of diagonal dominance of the matrix of discrete problem is necessary for applicability of the maximum principle. However, matrices with nonlocal conditions are not diagonally dominant. We can overcome this problem by applying the methodology of M-matrices.

We will use that in the spectrum of the matrix there are no eigenvalues with negative real part. This became apparent after investigating the structure of the spectrum of two-dimensional differential and appropriate difference operators with nonlocal conditions. As far as authors know, the idea of application of M-matrices for the convergence of the difference schemes in the case of nonlocal boundary conditions is applied for the first time. We note that in the case of nonlocal conditions, the matrix of system of finite difference equations is neither diagonally dominated nor symmetric.

Using the structure of the spectrum and properties of M-matrices, we succeed in proving the convergence of difference schemes for a wider class of nonlinear equations with nonlocal conditions than it was proved before. This is the main result of the research.

The structure of the paper is the following. The differential problem is formulated and its discrete approximation is provided. The difference problem for the error of the solution is investigated. The main properties of the matrix of difference problem are described, the case of matrix being as M-matrix is analyzed. The auxiliary lemmas on the evaluation of the solution of the system of the equations with an M-matrix are provided in Sect. 5. The majorant for the error is constructed. The results of numerical experiments are presented.

Statement of the problem

There may have been previous researches in this subject. This work gives further explanations and analysis in finite difference method for solving second order boundary value problem

Objectives of the study
To understand the impact of finite difference method on solving second order boundary value problem
To understand the relationship between finite difference method and solving second order boundary value problem

Research questions
What is the impact of finite difference method on solving second order boundary value problem
What is the relationship between finite difference method and solving second order boundary value problem