Partial Differential Equations: Finite Difference Methods

Abstract

There's no sense in being precise when you don't even know what you're talking about. John von Neumann (December 28, 1903–February 8, 1957) He was a great mathematician, physicist and computer scientist who had many research investigations to his credit. Mathematics is the science which draws necessary conclusions. Benjamin Peirce (April 4, 1809–October 6, 1880) He was a mathematician who had contributed to algebra, number theory, statistics and mechanics. Parabolic Equation (Heat Conduction or Diffusion Equation)Elliptic Equation (Laplace and Poisson Equations)Hyperbolic Equation (Wave Equation) Most of the problems posed by nature, and which are of interest to physicists and mathematicians are usually governed by a single or a system of differential equations. In general, a physical system involves more than one independent variable; in that case, our mathematical model contains partial differential equations (PDEs). PDEs play a vital role in the study of many branches of applied sciences and engineering; for example, fluid dynamics, heat transfer, elasticity, electromagnetic theory, optics, plasma physics, quantum mechanics, etc. In fact, the theories of modern physics, generally involve a mathematical model, as far as possible it is a set of PDEs. We first solve the mathematical model for solutions and then come to mathematical and physical interpretations of these solutions. So it is necessary to solve the mathematical model to study the physical system. Often, it is very difficult to solve these sets of PDEs explicitly for exact solutions. Consequently, numerical methods are applied to obtain approximate solutions of these equations. In fact, there is much current interest in obtaining numerical solutions of the PDEs. The finite difference method is a simple and most commonly used method to solve PDEs. In this method, various derivatives in the partial differential equation are replaced by their finite difference approximations, and the PDE is converted to a set of linear algebraic equations. This system of linear equations can be solved by any iterative procedure discussed in Chapter 5. Then, the solution of PDE is the solution of this system of linear equations. An important advantage of this method is that the most of the calculations can be carried out on the computer, and hence the solution is easy to obtain.