Advanced Numerical and Semi-Analytical Methods for Differential Equations

5Finite Difference Method

5.1 Introduction

Finite difference methods (FDM) are well‐known numerical methods to solve differential equations by approximating the derivatives using different difference schemes [1,2]. The finite difference approximations for derivatives are one of the simplest and oldest methods to solve differential equations [3]. Finite difference techniques in numerical applications began in the early 1950s as represented in Refs. [4,5], and their advancement was accelerated by the emergence of computers that offered a convenient framework for dealing with complex problems of science and technology. Theoretical results have been found during the last five decades related to accuracy, stability, and convergence of the finite difference schemes (FDS) for differential equations. Many science and engineering models involve nonlinear and nonhomogeneous differential equations, and solutions of these equations are sometimes beyond the reach by analytical methods. In such cases, FDM may be found to be practical, particularly for regular domains.

There are various types and ways of FDS [6,7] depending on the type of differential equations, stability, and convergence [8]. However, here only the basic idea of the titled method is discussed.

5.2 Finite Difference Schemes

In this section, the fundamental concept and FDS are addressed. The numerical solutions of differential equations based on finite difference provide us with the values at discrete grid points. Let us consider ...