Download Computer Lab Assignment 4 - UCSB Chemical Engineering

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)}.