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Unformatted text preview: Math Methods Solutions for Assignment 11 Nov. 17, 2010 Fall 2010 Partial Differential Equations Due Dec. 1, 2010 1. Consider a thin half pipe of unit radius laying on the ground. It is heated by radiation from above. We take the initial temperature of the pipe and the temperature of the ground to be zero. We model this problem with a heat equation with a source term. u t = u xx + A sin x ; u (0 ,t ) = u ( ,t ) = 0 , u ( x, 0) = 0 We search for the solution as u = sin( x ) T ( t ). This satisfies the required boundary conditions. For T ( t ) we have sin x =- sin( x ) T + A sin x, T (0) = 0 T + T = A, T (0) = 0 T = A + ce- t T = A (1- e- t ) And the solution is u = A sin( x )(1- e- t ) 2. Obtain Poissons formula to solve the Dirichlet problem for the circular region 0 r < R, < 2 . That is, determine a solution ( r, ) to Laplaces equation: 2 = 0 in polar coordinates given ( R, ). Show that ( r, ) = 1 2 Z 2 ( R, ) R 2- r 2 R 2 + r 2- 2 Rr cos( - ) d We expand the solution in a Fourier series = 1 2 a ( r ) + X n =1 a n ( r )cos( n ) + X n =1 b n ( r )sin( n ). We substitute the series into the Laplaces equation 1 r r r r + 1 r 2 2 2 = 0, and obtain the differential equations for the coefficients a 00 + 1 r a = 0 , a 00 n + 1 r a n- 1 r 2 n 2 a n = 0...
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