FINITE DIFFERENCE METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS

Abstract

Finite difference methods are well-known computational methods for solving differential equations that include approximating the derivatives using various difference schemes. During the last five decades, theoretical findings related to the precision, stability, and convergence of finite difference schemes (FDS) for differential equations have been discovered. In this paper, we look at various finite difference schemes for numerically solving partial differential equations in the context of heat transfer, wave equations, and the Laplace equation. The use of finite difference method (FDM) grids to solve partial difference equations (PDEs) is presented here. The solution of standard PDEs such as parabolic, hyperbolic, and elliptical form is the subject of this paper. The implicit, explicit, and Crank Nicolson schemes, as well as the Richardson Method, Du-Fort Method, and Frankel Method, are all used to solve PDEs. Depending on the form of differential equations, equilibrium, and convergence, there are many different forms and methods of FDS. The values at discrete grid points are obtained from numerical solutions of differential equations using finite difference. To obtain the numerical approximations to the time-dependent differential equations needed for computer simulations, explicit and implicit approaches are used. The numerical example results of the explicit and implicit schemes for the heat equation under unique initial and boundary conditions are implemented in this chapter. It also presents a common elliptic partial differential equation to determine the temperature at the inside nodes.