Hi,

I was working on my hwk, but would like to double check my answer. Could you please solve it?

PROBLEM 2: Heat flow with convection One-dimensional heat flow inside a pipe filled with a liquid flowing at speed Vo is modeled by the equation au at Vo ax (2) where k and Vo are positive constants. Both ends of the pipe (x = 0 and x = L) are fixed at zero temperature, and the initial temperature profile is u(x, 0) = f(x). (a) Write down the boundary conditions for u(x, t). (b) Applying separation of variables in the form u(x, t) = (x)G(t), show that o(x) and G(t) satisfy and da2 = -10 dG - Vodx dt = -AG, (3) where A is an unknown separation constant. What are the boundary conditions on d(x)? (c) Show that the equation for o(x) can be rewritten Vox d Voz do e e + -Ap = 0. dx (4) k (d) The eigenfunctions are given by VOX In (2) = e 2k sin L (5) for n = 1, 2,3.... Verify that these indeed satisfy the eigenvalue problem and boundary conditions. What are the corresponding eigenvalues In? (e) Prove the following orthogonality condition: On(2) Pm()e " da = 1/2 if n * m if n = m (6) (f) Using the principle of superposition and orthogonality, find the solution for u(x, t) in terms of the initial condition f (x)