Please solve No need for contour plot THanks Please solve in

Please solve. No need for contour plot. THanks

Please solve in u(x,t) = To(t) + Sum[Tn(t)*Cos[n*Pi*x/L] form if possible

For each non-homogeneous PDE. determine the specific solution. For each solution, generate a contour plot over the time interval [0.3]. (a) u_t = 1/36 u_xx + sin(t) cos(3/4x). 0

Solution

We consider the heat equation satisfying the initial conditions ( ut = kuxx, x [0,L], t > 0 u(x, 0) = (x), x [0,L] (7.1) We seek a solution u satisfying certain boundary conditions. The boundary conditions could be as follows: (a) Dirichlet u(0,t) = u(L,t) = 0. (b) Neumann ux(0,t) = ux(L,t). (c) Periodic u(L,t) = u(L,t) and ux(L,t) = ux(L,t). We look for solutions of the form u(x,t) = X(x)T(t) where X and T are function which have to be determined. Substituting u(x,t) = X(x)T(t) into the equation, we obtain X(x)T (t) = kX(x)T(t) from which, after dividing by kX(x)T(t), we get T (t) kT(t) = X(x) X(x) . The left side depends only on t whereas the right hand side depends only on x. Since they are equal, they must be equal to some constant . Thus T + kT = 0 (7.2) and X + X = 0. (7.3) 53 The general solution of the first equation is given T(t) = Bekt for an arbitrary constant B. The general solutions of the second equation are as follows. (1) If < 0, then X(x) = cosh x + sinh x. (2) If = 0, then X(x) = x + . (3) If > 0, then X(x) = cos x + sin x. (7.4) In addition, the function X which solves the second equation will satisfy boundary conditions depending on the boundary condition imposed on u. The problem ( X + X = 0 X satisfies boundary conditions (7.5) is called the eigenvalue problem, a nontrivial solution is called an eigenfunction associated with the eigenvalue .