HW8 - May 29, 2007 MAE 105 Homework #8 Due: Tuesday,...

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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 xe 3 t /2 ,0 < x < π , t >0 ,( 1) with the boundary conditions u (0, t ) = 0, u ( , 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 < . (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’ sfor the unknown coefficients, A n ( t ).
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This note was uploaded on 05/08/2008 for the course MAE 105 taught by Professor Neiman-nassat during the Spring '07 term at UCSD.

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HW8 - May 29, 2007 MAE 105 Homework #8 Due: Tuesday,...

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