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Unformatted text preview: CAAM 336 Â· DIFFERENTIAL EQUATIONS Problem Set 12 Posted Friday 26 November 2010. Due Friday 3 December 2010, 5pm. This problem set counts for 100 points, plus a 20 point bonus. 1. [50 points] On Problem Set 10, you solved the heat equation on a two-dimensional square domain. Now we will investigate the wave equation on the same domain, a model of a vibrating membrane stretched over a square frameâ€”that is, a square drum: u tt ( x,y,t ) = u xx ( x,y,t ) + u yy ( x,y,t ) , with 0 â‰¤ x â‰¤ 1, and 0 â‰¤ y â‰¤ 1, and t â‰¥ 0. Take homogeneous Dirichlet boundary conditions u ( x, ,t ) = u ( x, 1 ,t ) = u (0 ,y,t ) = u (1 ,y,t ) = 0 for all x and y such that 0 â‰¤ x â‰¤ 1 and 0 â‰¤ y â‰¤ 1 and all t â‰¥ 0, and consider the initial conditions u ( x,y, 0) = u ( x,y ) = âˆž X j =1 âˆž X k =1 a j,k (0) Ïˆ j,k ( x,y ) , u t ( x,y, 0) = v ( x,y ) = âˆž X j =1 âˆž X k =1 b j,k (0) Ïˆ j,k ( x,y ) . Here Ïˆ j,k ( x,y ) = 2sin( jÏ€x )sin( kÏ€y ), for j,k â‰¥ 1, are the eigenfunctions of the operator Lu =- ( u xx + u yy ) , with homogeneous Dirichlet boundary conditions given in Problem Set 10. You may use without proof that these eigenfunctions are orthogonal, and use the eigenvalues Î» j,k = ( j 2 + k 2 ) Ï€ 2 computed for Problem Set 10. (a) We wish to write the solution to the wave equation in the form u ( x,y,t ) = âˆž X j =1 âˆž X k =1 a j,k ( t ) Ïˆ j,k ( x,y ) . Show that the coefficients a j,k ( t ) obey the ordinary differential equation a 00 j,k ( t ) =- Î» j,k a j,k ( t ) with initial conditions a j,k (0) , a j,k (0) = b...
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- Fall '09
- Vibrating string, Partial differential equation, wave equation, Dirichlet boundary conditions