7.22, §3 Boundaries

Find u₀:

The x-boundary conditions are inhomogeneous:

Try finding a u₀ satisfying these boundary conditions:

Try a linear expression:

u₀ = A(t) + B(t) x

This can be solved to find

Identify the problem for the remainder:

Substitute u=u₀+v into the boundary conditions:

gives

Substitute u=u₀+v into the PDE :

where

Substitute u=u₀+v into the IC u(x,0)=f(x):

The problem for v is therefor: