PDE-1D - Introduction

Unsteady heat conduction is governed by a parabolic partial differential equation.

The equation for unsteady diffusion in one dimension is similar.

The temperature (or concentration) depend on two variables (time and spatial position), so that they are partial differential equations. The distinguishing characteristic of a parabolic equation is that it is evolutionary (like initial value problems ­ you start from somewhere), but has characteristics of two-point boundary value problems (the solution depends on both boundary conditions). In fact, usually for long times the parabolic equation becomes a two-point boundary value problem, since the time derivative becomes zero. A typical solution looks like

Indeed, the numerical methods are combinations of those used to solve boundary value problems (in x) with those used to solve initial value problems (in t).

Classification and other types of partial differential equations are described elsewhere.

Sometimes these problems can be solved with analytical methods, such as separation of variables or combination of variables. More usually, however, numerical methods are used. The subjects included are:
Separation of Variables
Combination of Variables
Numerical Methods - Overview
Finite Difference Methods in MATLAB
Orthogonal Collocation Methods
Orthogonal Collocation on Finite Elements
Finite Element Method
Method of Weighted Residuals
Spectral Methods
Errors
Stability
Comparison of Methods