Suppose that we want to test a computer program that implements non-homogeneous Dirichlet conditions of the type To this end, we want to compare the numerical solution with an exact solution of the...

Suppose that we want to test a computer program that implements non-homogeneous Dirichlet conditions of the type

To this end, we want to compare the numerical solution with an exact solution of the problem. Show that the function

is a solution of the diffusion equation, where a, b, and c are constants. The nice feature of this solution is that it obeys certain time-dependent non-homogeneous boundary values by choosing certain values of the constants a, b, and c. Use the freedom in choosing a, b, and c to find an exact solution of a diffusion problem with a non-homogeneous Dirichlet condition at x D 0 and a non-homogeneous Neumann condition at x = 1.