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Unformatted text preview: CAAM 336 · DIFFERENTIAL EQUATIONS Problem Set 10 · Solutions 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 steadystate 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 ) , ≤ x ≤ 1 , ≤ y ≤ 1 , with homogeneous Dirichlet boundary conditions u ( x, 0) = u ( x, 1) = u (0 ,y ) = u (1 ,y ) = 0 for all ≤ 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. (a) Show that L is symmetric, given the inner product ( v,w ) = Z 1 Z 1 v ( x,y ) w ( x,y ) dxdy. (b) Verify that the functions ψ j,k ( x,y ) = 2sin( jπx )sin( kπy ) are eigenfunctions of L for j,k = 1 , 2 ,.... (To do this, you simply need to show that Lψ j,k = λ j,k ψ j,k for some scalar λ j,k .) (c) What is the eigenvalue λ j,k associated with ψ j,k ? (d) Compute the inner product ( ψ j,k ,ψ j,k ) = k ψ j,k k 2 . (e) Let f ( x,y ) = x (1 y ). Compute the inner product ( f,ψ j,k ). (f) The solution to the diffusion equation is given by the spectral method, but now with a double sum to account for all the eigenvalues: u ( x,y ) = N X j =1 N X k =1 1 λ j,k ( f,ψ j,k ) ( ψ j,k ,ψ j,k ) ψ j,k ( x,y ) . In MATLAB plot the partial sum u 10 ( x,y ) = 10 X j =1 10 X k =1 1 λ j,k ( f,ψ j,k ) ( ψ j,k ,ψ j,k ) ψ j,k ( x,y ) . Hint for 3d plots: To plot ψ 1 , 1 ( x,y ) = 2sin( πx )sin( πy ), you could use x = linspace(0,1,40); y = linspace(0,1,40); [X,Y] = meshgrid(x,y); Psi11 = 2*sin(pi*X).*sin(pi*Y); surf(X,Y,Psi11) Solution. (a) To show that L is symmetric, we must show that ( Lu,v ) = ( u,Lv ) for all u,v ∈ C 2 D [0 , 1] 2 . We can establish this result by integrating by parts twice in each spatial dimension: ( Lu,v ) = Z 1 Z 1 ( u xx ( x,y ) + u yy ( x,y ) ) v ( x,y )d x d y = Z 1 Z 1 u xx ( x,y ) v ( x,y )d x d y Z 1 Z 1 u yy ( x,y ) v ( x,y )d y d x = Z 1 h u x ( x,y ) v ( x,y ) i 1 x =0 + h u ( x,y ) v x ( x,y ) i 1 x =0 Z 1 u ( x,y ) v xx ( x,y )d x d y + Z 1 h u y ( x,y...
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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.
 Fall '09
 Tompson

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