# HW02 - APPM 4350/5350 Fourier Series and Boundary Value...

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APPM 4350/5350: Fourier Series and Boundary Value Problems Homework #2 Friday September 3, 2010 Due: Monday, September 20, 2010 In what follows, and in the future, all arbitrary functions are assumed to be piecewise smooth unless otherwise speciﬁed. 1. Consider: ∂T ∂t = κ 2 T ∂x 2 0 < x < L where κ > 0 is constant, T ( x = 0 ,t ) = T ( x = L,t ) = 0 Solve for T ( x,t ) with the following initial values: a) T ( x, 0) = T 0 sin Nπx L , b) T ( x, 0) = T 0 cos Nπx L where N is a non-negative integer, and T 0 is a constant. 2. Suppose: ∂T ∂t = κ 2 T ∂x 2 0 < x < L where κ > 0 is constant, ∂T ∂x ( x = 0 ,t ) = T ( x = L,t ) = 0 and T ( x, 0) = f ( x ). Find T ( x,t ) and the the equilibrium, temperature. Hint: integration or general formulae, which also will be discussed in class, can be used to show that the eigenfunctions are orthogonal. 3. Consider ∂T ∂t = κ 2 T ∂x 2 - αT 0 < x < L where κ,α > 0 are constant, ∂T ∂x ( x = 0 ,t ) = ∂T ∂x ( x = L,t ) = 0 and

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HW02 - APPM 4350/5350 Fourier Series and Boundary Value...

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