# hw10 - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 10...

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CAAM 336 · DIFFERENTIAL EQUATIONS Problem Set 10 Posted Wednesday 3 November 2010. Due Wednesday 10 November 2010, 5pm. 1. [50 points: 8 points each for (a), (b), (d), (e); 4 points for (c); 14 points for (f)] This problem and the next study the heat equation in two dimensions. We begin with the steady-state problem. In place of the one dimensional equation, - u 00 = f , we now have - ( u xx ( x,y ) + u yy ( x,y )) = f ( x,y ) , 0 x 1 , 0 y 1 , with homogeneous Dirichlet boundary conditions u ( x, 0) = u ( x, 1) = u (0 ,y ) = u (1 ,y ) = 0 for all 0 x 1 and 0 y 1. The associated operator L is deﬁned as Lu = - ( u xx + u yy ) , acting on the space C 2 D [0 , 1] 2 consisting of twice continuously diﬀerentiable functions on [0 , 1] × [0 , 1] with homogeneous boundary conditions. We can solve the diﬀerential equation Lu = f using the spectral method just as we have seen in class before. This problem will walk you through the process; you may consult Section 8.2 of the text for hints.

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## This note was uploaded on 01/21/2012 for the course CAAM 330 taught by Professor Tompson during the Fall '09 term at UVA.

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hw10 - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 10...

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