1. Consider the PDE Uy Upy = U (1) with Dirichlet boundary conditions on the interval [0, 7| and initial condition u(z,0) = #(). (a) Show that the substitution u(z,t) = e'v(x,t) transforms (1)) into a diffusion equation for v. What are the boundary conditions for v7 What initial condition does v satisfy? (b) Find the general solution of (1) with the given boundary and initial conditions as a Fourier series, u(z,t) = -, An(t)Xn(z) where you are also asked to determine, based on the boundary conditions, the appropriate set of eigenfunctions X,,. (c) Assuming (z,t) is continuously differentiable, show that the Fourier series solution converges for all > 0. (d) Consider the PDE that you have obtained in part (a) for v with the given boundary and initial conditions. Find the general solution of the PDE for v also as a Fourier series, v(z,t) = ), -, An(t)Xn(z) where X,, is a set of eigenfunctions you will determine based on the boundary conditions satisfied by v