Nonhomogeneous Boundary Value Problems and Problems in Higher Dimensions

This problem models heat propagation in a rod where the left end is kept at constant temperature T1, the right end is kept at temperature T2, the initial temperature is f(x) and at each point x, there is heat radiating at the rate F (x, t) at time t. Note that the PDE, the boundary conditions, and the initial condition, are nonhomogeneous. We can use the principle of superposition to decompose this problem into two subproblems The subproblems for v and w are