Consider the one-dimensional heat equation for nonconstant thermal properties w...

Consider the one-dimensional heat equation for nonconstant thermal properties

with the initial condition u(x, 0) = f(x). [Hint: Suppose it is known that if u(x, t) = ??(x ) h(t), then

You may assume the eigenfuctions are known. Briefly discuss . Solve the initial value problem:

(a) with boundary conditions u(0, t) = 0 and u(L, t) = 0

*(b) with boundary conditions , Derive coefficients using orthogonality.]