Question

12 2. Consider the heat equation where for simplicity we take c = 1. Thus au du ar2 at Suppose that a heat conducting rod of length a has the left end r = ( maintained at temperature ( while the right end at r = is insulated so that there is no heat flow. This gives us the boundary conditions au u(0,t) = 0, (7,0) = 0. Find the solution u(x, t) if the initial temperature distribution on the rod is given by u(x,0) = x2 – 271. Hint: After separating the variables with u(x, t) = F(2) G(t) and discarding the useless cases, one gets the simultaneous equations F"(x) + kF() = 0, G'(t) + kG(t) = 0 for positive numbers k. The acceptable values of k are determined by the fact that F(1) must satisfy the boundary conditions F(0) = 0 and F'(T) = 0. Show that the acceptable values of k are the values

Answer 1

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