SOLUTION TO PROBLEMS WEEK 5
1. Find the Fourier transform of the delta function (x x0 ).
Answer:
1
F ( (x x0 ) =
2
1
(x x0 )eix dx = eix0
2
2. Use the Fourier transform to solve the initial value problem:
ut = 2 uxx + f (x, t)
u(x, 0) = (x)
< x < , t
Assigned: Tuesday May 3, 2016
Due: Tuesday May 10, 2016
MATH 454: Homework 1
SPRING 2014
NOTE: For each homework assignment observe the following guidelines:
Include a cover page.
Always clearly label all plots (title, x-label, y-label, and legend).
Us
Assigned: Tuesday May 17, 2016
Due: Tuesday May 24, 2016
MATH 454: Homework 3
SPRING 2016
1. (a) Find the Fourier sine series for f (x) = 1 x defined on the interval 0 x 1.
(b) In MATLAB, plot the first 20 terms and the first 200 terms of the sine series
Assigned: Tuesday May 24, 2016
Due: Tuesday May 31, 2016
MATH 454: Homework 4
SPRING 2016
1. (a) Prove that Fs [f 00 (t)] = 2 f (0) 2 Fs [f (t)]
(b) Prove that Fc [f 00 (t)] = 2 f 0 (0) 2 Fc [f (t)]
2. Solve the following PDE using the Fourier cosine tran
SOLUTION TO EXAM 1
Write down your name. Show all your work. No work, no grade. Total 100 points. You
have 80 minutes to work on the problems.
1. (15pt) Classify the following equations in terms of their linearity, homogeneity, and
order. Let u = u(x, t).
TAKE HOME FINAL
DUE: 5PM, DECEMBER 14, 2010
Write your name on each page of your work. You are expected to work on the
problems independently. You may discuss with me about the problems and your
progress if you like.
Sign the following honor pledge on you
TAKE HOME FINAL-SOLUTION
1. (33 pt) We want to know how much time is needed to cook an egg. Assume
that the egg is a ball of radius a and is homogeneous. Its initial temperature is
T0 < 40 C , and it is cooked by immersion in boiling water at time t = 0 w
SOLUTION TO EXAM 2
Write down your name. Show all your work. No work, no grade. Total 100 points.
You have 80 minites to work on the problems.
1. (20pt) Solve
tux + 2ut = 2et
u(x, 0) = sin x.
(0.1)
Plot the characteristics.
Solution: let
(0.2)
dt
ds
dx
ds
SOLUTION TO PROBLEMS WEEK 9
1. Find the solution of the following initial-boundary value problem. What is your
physical interpretation of this problem? Does your solution agree with your intuition?
utt c2 uxx = 0,
0 < x < , t > 0
(0.1)
u(0, t) = sin t,
t>
SOLUTION TO PROBLEMS WEEK 8
1. Use the method of reections to solve the following Dirichlet problem for the heat
equation on the half-line.
ut = 2 uxx ,
0 < x < , t > 0
u(0, t) = 0,
t>0
u(x, 0) = f (x),
0<x<
(0.1)
Solution: In order to achieve the boundar
SOLUTION TO PROBLEMS WEEK 7
1. The midpoint of a piano string of tension T , density and length l is hit by a hammer
whose head diameter is 2a. A ee is sitting at a distance of l/4 from one end. (assume that
a < l/4.) How long does it take for the disturb
SOLUTION TO PROBLEMS WEEK 6
1. Verify the following formula for the Laplace transform:
L(f ) = sL(f ) f (0)
L(f ) = s2 L(f ) sf (0) f (0)
Answer: Using integration by parts, we have
L(f ) =
t=0
f (t)est dt = f (t)est
0
+s
0
est f (t) dt = sL(f ) f (0)
App
Assigned: Tuesday May 10, 2016
Due: Tuesday May 17, 2016
MATH 454: Homework 2
SPRING 2016
NOTE: For each homework assignment observe the following guidelines:
Always clearly label all plots (title, x-label, y-label, and legend).
Use the subplot command