Advanced Engineering Mathematics, 7th Edition

16.2 Heat Equation

INTRODUCTION

The basic idea in the following discussion is the same as in Section 16.1; we approximate a solution of a PDE—this time a parabolic PDE—by replacing the equation with a finite difference equation. But unlike the preceding section, we shall consider two finite-difference approximation methods for parabolic partial differential equations: one called an explicit method and the other an implicit method. For the sake of definiteness, we consider only the one-dimensional heat equation.

Difference Equation Replacement

To approximate the solution u(x, t) of the one-dimensional heat equation

(1)

we again replace the ...