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Unformatted text preview: ACM 95/100c Problem Set 2 Solutions Lei Zhang April 10, 2006 Each Problem is worth 10 points. Each part of a multipart problem is weighted equally. Problem 1 (Collaboration allowed) Consider the solution of the 1D heat equation φ t = a 2 φ xx < x < l with boundary conditions φ (0 ,t ) = φ ( l,t ) = 0 and initial conditions φ ( x, 0) = 1 Solve this problem using a series solution obtained from separation of variables. Choose various numerical values for a 2 , x , l and decide how many terms are required to represent the solution at (a) t = 0, (b) t = 1, Solution 1 By the result from class notes, a series solution of the heat equation which satisfies the boundary condition u (0 ,t ) = u ( l,t ) = 0 is u ( x,t ) = ∞ X n =1 u n sin( nπx/l )exp( a 2 n 2 π 2 t/l 2 ) (1) to satisfy the initial condition u ( x, 0) = 1, we need 1 = ∞ X n =1 u n sin( nπx/l ) (2) by orthogonality, u n = 2 l Z l sin( nπx l ) dx (3) = 2(1 ( 1) n ) nπ (4) i.e u ( x,t ) = ∞ X n =1 2(1 ( 1) n ) nπ sin( nπx/l )exp( a 2 n 2 π 2 t/l 2 ) (5) 1 take a = 1, l = 1, choose the first 10 terms of series solution, we have the following results at t = 0 and t = 1 in Figure (1). Note that exact solution is red, and truncated series solution is blue, 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 Solution at t = 0 Distance x u(x,0) exact solution series solution (a) t = 0 . 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 x 105 Solution at t = 1 Distance x u(x,1) exact solution series solution (b) t = 1 . Figure 1: a = 1, l = 1, 10 terms. choose the first 100 terms, we have the following results at t = 0 and t = 1 in Figure (2), we can clearly see that at t = 0 the series converges nonuniformly, i.e. Gibbs phenomena. However, at t = 1, the series converges uniformly, from Figure (3) we can see even 2 terms is already a very good approximation. 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 Solution at t = 0 Distance x u(x,0) exact solution series solution (a) t = 0 . 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 x 105 Solution at t = 1 Distance x u(x,1) exact solution series solution (b) t = 1 . Figure 2: a = 1, l = 1, 100 terms. 2 now take a = 1, and different l = 1 and l = 5, we just choose first 2 terms. Compare the result at t = 1 in Figure (3), we find that with smaller l , the series converges faster. 1 2 3 4 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Solution at t = 1 Distance x u(x,1) exact solution series solution (a) l = 5 . 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 x 105 Solution at t = 1 Distance x u(x,1) exact solution series solution (b) l = 1 . Figure 3: t = 1, a = 1, 2 terms. take different a = 0 . 1 and a = 1, with l = 1, we also choose first 2 terms. Compare the result at t = 1 in Figure (4), we find that with bigger a , the series converge faster....
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This note was uploaded on 01/08/2011 for the course ACM 95c taught by Professor Nilesa.pierce during the Spring '09 term at Caltech.
 Spring '09
 NilesA.Pierce

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