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Additional problem 1. Solve the following initial-boundary value problem for the heat equation ut = kuxx, x ∈ [0, L], t ∈ [0, ∞) u(0, t) = A, u(L, t) = B for t ∈ [0, ∞] u(x, 0) = f(x), for x ∈ [0, L], where A and B are constants, in general nonzero. Since we know how to solve problems with two homogeneous (i.e. zero) boundary conditions on a pair of opposite sides, we would be able to solve this problem if A = B = 0. One method to reduce the given problem to a problem of the type we like is to set u(x, t) = v(x, t) + l(x), where l(x) := αx + β is a linear function that - for an appropriate choice of the constants α and β - can be made to satisfy the boundary conditions l(0) = A, l(L) = B. Choose the constants α and β appropriately, show that the initial-boundary value problem for the function v(x, t) is vt = kvxx, x ∈ [0, L], t ∈ [0, ∞) v(0, t) = 0, v(L, t) = 0 for t ∈ [0, ∞] u(x, 0) = f(x) − l(x), for x ∈ [0, L], and solve this problem. Assume that the sine-Fourier expansion of the function f(x) is f(x) = X∞ n=1 fn sin nπx L . Be careful in formulating the problem for v(x, t); in particular, the initial condition for v(x, t) will differ from the one for u(x, t). Here are two facts that you will need (when you expand) the function l(x)). 1 = 4 π X n=1,3,5,... sin nπx L , x = 2L π X 1,3,5,... (−1)n+1 n sin nπL L . Show that as t → ∞, the function v(x, t) will tend to 0, so that the asymptotic behavior of the temperature u(x, t) is determined completely by the auxiliary function l(x) (recall that the constant k in the heat equation is strictly positive!). Sketch limt→∞ u(x, t) as a function of x.

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Heat Equation Problems
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