Crank-Nicolson Method

Background for Parabolic Equations

Heat Equation

As an example of parabolic partial differential equations, we consider the one-dimensional heat equation


for0 < x < aand0 < t < b.

with the initial condition


fort = 0and
.

and the boundary conditions


forx = 0and
,


forx = aand
.

The heat equation models the temperature in an insulated rod with ends held at constant temperatures
and
and the initial temperature distribution along the rod beingf(x).Although analytic solutions to the heat equation can be obtained with Fourier series, we use the problem as a prototype of a parabolic equation for numerical solution.

Program (Forward-Difference method for the heat equation)

To approximate the solution of the heat equation
over the rectangle
with
,for
.and
,for
.