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Unformatted text preview: CAAM 336 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 framethat is, a square drum: u tt ( x,y,t ) = u xx ( x,y,t ) + u yy ( x,y,t ) , with 0 x 1, and 0 y 1, and t 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 x 1 and 0 y 1 and all t 0, and consider the initial conditions u ( x,y, 0) = u ( x,y ) = X j =1 X k =1 a j,k (0) j,k ( x,y ) , u t ( x,y, 0) = v ( x,y ) = X j =1 X k =1 b j,k (0) j,k ( x,y ) . Here j,k ( x,y ) = 2sin( jx )sin( ky ), for j,k 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 j,k = ( j 2 + k 2 ) 2 computed for Problem Set 10. (a) We wish to write the solution to the wave equation in the form u ( x,y,t ) = X j =1 X k =1 a j,k ( t ) j,k ( x,y ) . Show that the coefficients a j,k ( t ) obey the ordinary differential equation a 00 j,k ( t ) =- j,k a j,k ( t ) with initial conditions a j,k (0) , a j,k (0) = b...
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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