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ps2sol - ACM 95/100c Problem Set 2 Solutions Lei Zhang Each...

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ACM 95/100c Problem Set 2 Solutions Lei Zhang April 10, 2006 Each Problem is worth 10 points. Each part of a multi-part problem is weighted equally. Problem 1 (Collaboration allowed) Consider the solution of the 1-D heat equation φ t = a 2 φ xx 0 < 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 0 sin( nπx l ) dx (3) = 2(1 - ( - 1) n ) (4) i.e u ( x, t ) = X n =1 2(1 - ( - 1) n ) sin( nπx/l ) exp( - a 2 n 2 π 2 t/l 2 ) (5) 1
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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 0.2 0.4 0.6 0.8 1 0 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 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 x 10 -5 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 0.2 0.4 0.6 0.8 1 0 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 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 x 10 -5 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
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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. 0 1 2 3 4 5 0 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 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 x 10 -5 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. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Solution at t = 1 Distance x u(x,1) exact solution series solution (a) a = 0 . 1 . 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 x 10 -5 Solution at t = 1 Distance x u(x,1) exact solution series solution (b) a = 1 . Figure 4: t = 1, l = 1, 2 terms Note: the convergence is in the limit of your computer’s floating point precision, since as l gets smaller or a gets bigger, the exponential decay term will be underflow very soon.
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