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Unformatted text preview: CAAM 336 Â· DIFFERENTIAL EQUATIONS Problem Set 6 Â· Solutions Posted Wednesday 29 September 2010. Due Wednesday 6 October 2010, 5pm. General advice: You may compute any integrals you encounter using symbolic mathematics tools such as WolframAlpha, Mathematica, or the Symbolic Math Toolbox in MATLAB. 1. [60 points: 13 points for (a), (b), (c), and (e); 8 points for (d)] Solve the following boundary value problems using the spectral method. For each problem, (i) write down the expansions of the right hand side functions as linear combinations of the eigenfunctions; (ii) write down the sum for the solution u obtained from the spectral method; and (iii) produce a plot showing the sum of the first twenty terms in the series for u . For parts (c)â€“(e) you may use the eigenvalues and eigenfunctions computed in Problem 1(d) of Problem Set 5, and the results of Section 5.2.3 of the text. (a)- d 2 u dx 2 ( x ) = e x , u (0) = 1, u (1) = 0. (b)- d 2 u dx 2 ( x )- 10 u ( x ) = e x , u (0) = 0, u (1) = 0. (c)- d 2 u dx 2 ( x ) = x + sin( Ï€x ), u (0) = du dx (1) = 0 . (d)- d 2 u dx 2 ( x ) = x + sin( Ï€x ), u (0) = du dx (1) = 1 . (e)- d 2 u dx 2 ( x ) = f ( x ), du dx (0) = u (1) = 0 , where f ( x ) = 1 , < x < 1 / 2; , 1 / 2 < x < 1 . (This f is not continuous; follow the usual procedure and see if you obtain a sensible answer.) Solution. (a) First we solve the problem with homogeneous Dirichlet boundary conditions using the spectral method, then we will add a correction to satisfy the inhomogeneous boundary conditions. For the operator with homogeneous Dirichlet conditions, we have eigenvalues Î» k = k 2 Ï€ 2 with associated (normalized) eigenfunctions u k ( x ) = âˆš 2sin( kÏ€x ). We shall call the solution to the problem with homogeneous Dirichlet conditions b u , which is given by the spectral method b u = âˆž X k =1 ( f,u k ) Î» k u k . To compute ( e x ,u k ), integrate twice by parts to obtain Z 1 e x âˆš 2sin( kÏ€x ) dx = âˆš 2 e x sin( kÏ€x ) 1- âˆš 2 kÏ€ Z 1 e x cos( kÏ€x ) dx =- âˆš 2 kÏ€ Z 1 e x cos( kÏ€x ) dx =- âˆš 2 kÏ€ e x cos( kÏ€x ) 1 + 1 kÏ€ Z 1 e x sin( kÏ€x ) dx = âˆš 2 kÏ€ ( 1- e (- 1) k )- âˆš 2 k 2 Ï€ 2 Z 1 e x sin( kÏ€x ) dx, from which we discern that âˆš 2 1 + 1 k 2 Ï€ 2 Z 1 e x sin( kÏ€x ) dx = âˆš 2 kÏ€ (1- (- 1) k e ) . We conclude that ( f,u k ) = Z 1 e x âˆš 2sin( kÏ€x ) dx = âˆš 2 kÏ€ 1 + k 2 Ï€ 2 (1- (- 1) k e ) . The spectral method thus gives b u = âˆž X k =1 ( f,u k ) Î» k u k = âˆž X k =1 2(1- (- 1) k e ) kÏ€ (1 + k 2 Ï€ 2 ) sin( kÏ€x ) . Now we need to add some function w to b u that will produce a u = b u + w that satisfies both the differential equation and the inhomogeneous boundary conditions. We want- d 2 u dx 2 =- d 2 b u dx 2- d 2 w dx 2 = e x , but since we already have- d 2 b u dx 2 = e x , we need- d 2 w dx 2 = 0 ....
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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