1 Model Problem

A simple model problem needs to have a second derivative and a source term, both common in conservation laws. The simplest such model is as follows.

Suppose $u(x)$ is twice differentiable and satisfies

$$-u''+\kappa^2u=f(x),\ 0\le x\le 1,$$ (6)

with $u(0)=a$ and $u(1)=b$.

Although the differential equation can be solved analytically it needs to be treated as if a computed solution has to be found. There are a number of ways of discretising this model equation.