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HW8 - MAE 105 Homework#8 Due Tuesday Name NOTE To receive...

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May 29, 2007 MAE 105 Homework #8 Due: Tuesday, 06/05/07 Name:_________________________ NOTE: To receive full credit, you must write down all steps neatly and draw neat and complete sketches. PROBLEM 1: Consider the nonhomogeneous heat equation u t = 2 u x 2 + x cos x e 3 t /2 , 0 < x < π , t > 0, (1) with the boundary conditions u (0, t ) = 0, u ( π , t ) = 0, (2) and the initial condition u ( x , 0) = x sin x . (3) (a) (1 Point) Find the Fourier coefficients in the following Fourier series of x and cos x : x sin( x ) = n = 1 Σ a n sin nx , (4) x cos x = n = 1 Σ b n sin nx , (5) for 0 < x < π . (b) (0.5 Point) Express the solution of the original PDE (1), i.e., u ( x , t ), as an infinite series of sin nx , with time- dependent coefficients, say, A n ( t ). [Write down this series.] (c) (1 Point) Substitute the infinite series expressions for u ( x , t ) and x cos x into the nonhomogeneous PDE (3), col- lect coefficients of sin nx , use the orthogonality of these eigenfunctions, and find the necessary ODE’s for the unknown coefficients, A n ( t ).
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