Question: Problem: Given the following initial value problem for the heat equation on [0, 1] - 2u, 0 0, u(0, t) = 0, u(1, f) =0,

Problem: Given the following initial value problem for the heat equation on [0, 1] - 2u, 00, u(0, t) = 0, u(1, f) =0, t>0, u(r. 0) = 7 sin (3mr) a) If u(x, t) is the solution to the problem above, find an initial boundary value problem satisfied by w(r, t) = e"'u(x, t). b) Solve the problem found in part a) for w(x, t). c) Find the solution u(r, () to the original problem. d) Find the time 7, such that u(x, ()

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