sol10 - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 10...

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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 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. (a) Show that L is symmetric, given the inner product ( v, w ) = Z 1 0 Z 1 0 v ( x, y ) w ( x, y ) dxdy. (b) Verify that the functions ψ j,k ( x, y ) = 2 sin( jπx ) sin( kπy ) are eigenfunctions of L for j, k = 1 , 2 , . . . . (To do this, you simply need to show that 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 ) = 2 sin( 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)
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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 0 Z 1 0 ( u xx ( x, y ) + u yy ( x, y ) ) v ( x, y ) d x d y = - Z 1 0 Z 1 0 u xx ( x, y ) v ( x, y ) d x d y - Z 1 0 Z 1 0 u yy ( x, y ) v ( x, y ) d y d x = Z 1 0 - 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 0 u ( x, y ) v xx ( x, y ) d x d y + Z 1 0 - h u y ( x, y ) v ( x, y ) i 1 y =0 + h u ( x, y ) v y ( x, y ) i 1 y =0 - Z 1 0 u ( x, y ) v yy ( x, y ) d y d x = - Z 1 0 Z 1 0 u ( x, y ) v xx ( x, y ) d x d y - Z 1 0 Z 1 0 u ( x, y ) v yy ( x, y ) d y d x = - Z 1 0 Z 1 0 u ( x, y ) ( v xx ( x, y ) + v yy ( x, y ) ) d x d y = ( u, Lv ) .
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