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Math 257/316 Assignment 10 Due: Fri. Apr. 10 1. Find the eigenvalues and eigenfunctions of the following Sturm-Liouville problem on [0, 1] X 00 (x) + λX(x) = 0, X 0 (0) = 0, X(1) = 0, and use them to solve the following heat equation with mixed BCs and a source term: ut = uxx + cos(5πx/2)t, ux (0, t) = 0, 0 < x < 1, t > 0, u(1, t) = 0 u(x, 0) = cos(7πx/2) 2. Consider the problem of steady heat conduction in a semi-circular annulus ∆u = ∂ 2 u 1 ∂u 1 ∂2u + + = 0, ∂r2 r ∂r r2 ∂θ2 u(1, θ) = u(2, θ) = 0, 1 < r < 2, u(r, 0) = 0, 0 < θ < π, u(r, π) = 1. Separate variables, u(r, θ) = R(r)Θ(θ), and find the Euler equation to be solved for R(r). Show that it can also be converted to the Sturm-Liouville form 1 (rR0 )0 + λ R = 0, r R(1) = R(2) = 0. Find the eigenfunctions Rn and eigenvalues λn , and check the orthogonality condition Z 2 1 0 m 6= n Rn (r)Rm (r) dr = , C m=n 1 r where you should determine the constant C. Hence, find the solution u(r, θ) by eigenfunction expansion. Hint: remember to include the weighting function 1/r when calculating the generalized Fourier coefficients. 1