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Unformatted text preview: CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 12 Solutions 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: 15 points for (a); 10 points for (b); 5 points for (c); 20 points for (d)] On Problem Set 10, you solved the heat equation on a twodimensional 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 j,k (0) derived from the initial conditions u and v . (b) Write down the solution to the differential equation in part (a). (c) Use your solution to part (b) to write out a formula for the solution u ( x,y,t ). (d) Suppose the drum begins with zero velocity, v ( x,y ) = 0, and displacement u ( x,y ) = 200 xy (1 x )(1 y )( x 1 / 4)( y 1 / 4) = X j =1 X k =1 100(5 + 7( 1) j )(5 + 7( 1) k ) j 3 k 3 6 j,k ( x,y ) . Submit surface (or contour) plots of the solution at times t = 0 , . 5 , 1 . , 1 . 5 , 2 . 5, using j = 1 ,..., 10 and k = 1 ,..., 10 in the series. Solution. This question follows the same pattern as the first problem on this problem set. (a) Substitute the formula u ( x,y,t ) = X j =1 X k =1 a j,k ( t ) j,k ( x,y ) . into the two dimensional wave equation to obtain X j =1 X k =1 d 2 a j,k dt 2 ( t ) j,k ( x,y ) = X j =1 X k =1 a j,k ( t ) 2 x 2 + 2 y 2 j,k ( x,y ) , which implies X j =1 X k =1 d 2 a j,k dt 2 ( t ) j,k ( x,y ) = X j =1 X k =1 a j,k ( t ) j,k j,k ( 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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