Consider the nonhomogeneous heat equation (with a steady heat source): Solve th...

Consider the nonhomogeneous heat equation (with a steady heat source):

Solve this equation with the initial condition

u(x, 0) = f(x)

and the boundary conditions

u(0, t) = 0 and u(L, t) = 0.

Assume that a continuous solution exists (with continuous derivatives). [Hints: Expand the solution as a Fourier sine series (i.e., use the method of eigenfunctions expansion). Expand g(x) as a Fourier sine series. Solve for the Fourier sine series of the solution. Justify all differentiations with respect to x.]