URGENT: PDE: Please solve in detail question number 2.

1. Discuss the solution of the following problem showing all of the necessary details of your work at the bottom of this page and on another page if necessary). PDE: auxx = ut, 0 0: BC: ux(0, t) = 0; ux(L, t) = 0; IC: u(x,0) = f(x) (1) Using separation of variables, what is the ODE to solve for X(x) and what are the BC's for X(x)? Work? 3. (ii) What are the solutions Xn(x)? Work?? 3 (iii) What are eigenvalues and eigenfunctions for the X(x) equation? Show the work as to how they are obtained. (iv) What is the ODE to solve for T(t)? (v) What are the solutions Tn(t) and un(x, t)? 4 (vi) What is the solution u(x, t) to which you apply the initial condition u(x,) = f(x)? 2 (v) What is the final solution u(x, t) = 2. Now look at the same problem but change the BC's to read: u(0, t) = t; u(Tt, t) = e* ', u(x,0) = 0. Show the details of how you would solve this boundary value problem by letting u(x, t) = v(x, t) + S(x). What are the ODE and BC's to solve for S(x)? Discuss how you would try and solve for S(x) with the boundary conditions given. Write down an expression for the solution of the v(x, t) PDE problem assuming you had S(x), but DO NOT work out a solution. Write your final submitted work on this page and another one if necessary. 1. Discuss the solution of the following problem showing all of the necessary details of your work at the bottom of this page and on another page if necessary). PDE: auxx = ut, 0 0: BC: ux(0, t) = 0; ux(L, t) = 0; IC: u(x,0) = f(x) (1) Using separation of variables, what is the ODE to solve for X(x) and what are the BC's for X(x)? Work? 3. (ii) What are the solutions Xn(x)? Work?? 3 (iii) What are eigenvalues and eigenfunctions for the X(x) equation? Show the work as to how they are obtained. (iv) What is the ODE to solve for T(t)? (v) What are the solutions Tn(t) and un(x, t)? 4 (vi) What is the solution u(x, t) to which you apply the initial condition u(x,) = f(x)? 2 (v) What is the final solution u(x, t) = 2. Now look at the same problem but change the BC's to read: u(0, t) = t; u(Tt, t) = e* ', u(x,0) = 0. Show the details of how you would solve this boundary value problem by letting u(x, t) = v(x, t) + S(x). What are the ODE and BC's to solve for S(x)? Discuss how you would try and solve for S(x) with the boundary conditions given. Write down an expression for the solution of the v(x, t) PDE problem assuming you had S(x), but DO NOT work out a solution. Write your final submitted work on this page and another one if necessary