sol8 - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 8...

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CAAM 336 · DIFFERENTIAL EQUATIONS Problem Set 8 · Solutions Posted Thursday 21 October 2010. Due Wednesday 27 October 2010, 5pm. 1. [50 points: 18 points for (a); 12 points for (b); 10 points each for (c) and (d)] Consider the following three matrices: (i) A = ± 0 1 1 0 ² (ii) A = ± - 50 49 49 - 50 ² (iii) A = ± 0 1 - 1 0 ² . (a) For each of the matrices (i)–(iii), compute the matrix exponential e t A . (You may use eig for the eigenvalues and eigenvectors, but please construct the matrix exponential ‘by hand’ (not with expm ). For diagonalizable A = VΛV - 1 , recall the formula e t A = V e t Λ V - 1 .) (b) Use your answers in part (a) to explain the behavior of x 0 ( t ) = Ax ( t ) as t → ∞ given that x (0) = [2 , 0] T (e.g., exponential growth, exponential decay, or neither) for each of the three matrices (i)–(iii). (c) For the matrix (ii), describe how large one can choose the time step Δ t so that the forward Euler method applied to x 0 ( t ) = Ax ( t ), x k +1 = x k + Δ t Ax k , will produce a solution that qualitatively matches the behavior of the true solution (i.e., the approximations x k should grow, decay, or remain of the same size as the true solution does). Answer the same question for the backward Euler method x k +1 = x k + Δ t Ax k +1 . (d) For the matrix in (iii), describe how the forward Euler method behaves for all Δ t as k → ∞ for x (0) = [1 , 1] T . Now describe how the backward Euler method must behave as k → ∞ for the same matrix and initial condition. Solution. [GRADERS: it is acceptable for students to use MATLAB to compute eigendecompositions, but they should not simply use the expm command. In particular, only give half credit if students computed e t A for a fixed value of t . The correct answer should depend on the variable t .] (a) We compute the matrix exponentials for each matrix in turn. (i) Note that det( λ I - A ) = λ 2 - 1 = ( λ + 1)( λ - 1) and hence the eigenvalues of A are λ 1 = - 1 and λ 2 = 1. The corresponding (normalized) orthogonal eigenvectors are u 1 = 2 2 ± 1 - 1 ² , u 2 = 2 2 ± 1 1 ² . As A is symmetric, if we set U = [ u 1 u 2 ] and Λ = diag( λ 1 2 ), we have A = UΛU * and e t A = U e t Λ U * = U ± e - t 0 0 e t ² U * . Multiplying this out gives e t A = 1 2 ± e t + e - t e t - e - t e t - e - t e t + e - t ² .
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(ii) If we denote the matrix in part (a) as A 1 , then we find that the A in part (b) can be written as A = - 50 I + 49 A 1 , from which it follows (a slight modification of problem 1 on the first exam) that A has eigenvalues λ 1 = - 99 and λ 2 = - 1 with the same eigenvectors as in part (a): u 1 = 2 2 ± 1 - 1 ² , u 2 = 2 2 ± 1 1 ² . Again A is symmetric, and we have that e tA = U e t Λ U * = 1 2 ± e - t + e - 99 t e - t - e - 99 t e - t - e - 99 t e - t + e - 99 t ² . (iii) [GRADERS: please be a bit more lenient with this problem, as this
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sol8 - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 8...

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