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Download Computer Lab Assignment 4 - UCSB Chemical Engineering
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CHEMICAL ENGINEERING 132A Professor Todd Squires Spring Quarter, 2007 Computer Lab Assignment 4 Laplace Transforms PRACTICE WITH LAPLACE TRANSFORMS New Commands: FullSimplify[%] A = {{1,2},{2,3}} (*two-by-two matrix*) MatrixForm[A] v = {4,5} (*Column Vector*) A.v (*Matrix Multiplication*) Inverse[A] (*Matrix Inversion*) Compute the following Laplace Transforms by hand, then use Mathematica to verify. First shifting theorem: compute the Laplace transform of a. e−2t t2 b. e−t sin t (1) (2) c. e−5t (3) a. 1 s2 +2s+1 1 1+(s−2)2 (4) Compute the inverse transform of b. (5) Second Shifting Theorem: Compute the Laplace Transform of a. H(t − 5) cos(t − 5) b. H(t − 2)t (6) (7) Compute the inverse Laplace Transform of a. b. e−s s(s−1) (8) −s e s+1 (9) Convolutions: Compute the Inverse Laplace Transform of F (s)G(s) directly, then by using the convolution theorem: 1) F (s) = 1/s2 , G(s) = 1/(s + 1) 2) F (s) = 1/(s + 1), G(s) = 1/(s + 2) (10) (11) (12) Extra credit bonus problem: Systems of equations: Solve the following set of equations dx dy dz +2 −3 dt dt dt dz dy − 3 + 2x dt dt dx dz + + 3z dt dt x(0) = 0, y(0) = 0, z(0) = 2 (13) = 0 (14) = 3 (15) = 0 (16) by 1) Laplace-transforming each equation by hand 2) Writing as a matrix equation A.x = b, where A is a threeby-three matrix, x = {X(s), Y (s), Z(s)} is a column vector, and b is a one-by-three column vector. Enter A and b into Mathematica. 3) Solve the equation by inverting the matrix: x = A−1 .b. That is, compute the inverse of A using mathematica, multply it by b. This gives the laplace-transformed solutions {X(s), Y (s), Z(s)}. 4) Invert the solutions to give the solutions {x(t), y(t), z(t)}.