92.445/545 Partial Dierential Equations
Spring 2013
Homework Assignment # 4 Solutions
1. (Pinchover and Rubinstein problem 3.1) Find the canonical form of the following pde. Be sure
to show the change of coordinates that reduces the pde to canonical form.
92.445/545 Partial Dierential Equations
Midterm Exam
Due March 18
Spring 2013
PLEASE SHOW ALL WORK! You will not receive full credit if you do not show your
work.
You must do your own work. Please do not consult anyone other than me regarding
this exam. V
92.445/545 Partial Dierential Equations
Spring 2013
Homework Assignment # 6 Solutions
1. Solve the following BVP for the Laplace equation.
u = 0
on 0 < x < , 0 < y <
u(x, 0) = sin(2x)
0<x<
u(x, ) = 0
0<x<
u(0, y ) = sin(y )
0<y<
u(, y ) = 0
0<y<
As state
92.445/545 Partial Dierential Equations
Spring 2013
Optional Homework Assignment # 7 Solutions
1. a) Find the Fourier series for the sawtooth function given by f (x) = x on the interval x .
The Fourier series for the function f on the interval L x L is gi
92.445/545 Partial Dierential Equations
Midterm Exam Solutions
Spring 2013
Problem 1. (10 points)
Is the function given by u(x, y ) = x2 + y 2 a solution of the pde yux xuy = 0? Why or why not?
u(x, y ) = x2 + y 2 ux = 2x and uy = 2y.
Therefore, yux xuy =
92.445/545 Partial Dierential Equations
Spring 2013
Homework Assignment # 5 Solutions
PLEASE SHOW ALL WORK! You will not receive full credit if you do not show your
work. You may work together, but everyone must turn in his/her own homework set.
1. Solve
92.445/545 Partial Dierential Equations
Spring 2013
Final Exam Solutions
Problem 1. (10 points) Consider the pde yuxx + xuyy + ux uy = x2 + y 2
a) In what region(s) of the xy plane is this pde elliptic?
Recall that the second order linear pde in 2 indepen
92.445/545 Partial Dierential Equations Spring 2013
Homework Assignment # 3 Solutions
1. Solve the following Cauchy problem on a semi-innite domain.
utt 4uxx = 0
on 0 < x < , t > 0
x
u(x, 0) = xe
ut (x, 0) = 0
u(0, t) = 0
As we discussed in class, the sol
92.445/545 Partial Dierential Equations
Spring 2013
Final Exam
Due May 13 - No Extensions
PLEASE SHOW ALL WORK! You will not receive full credit if you do not show your
work. Citing Mathematica or an online or hard copy reference (except for evaluating
in
92.445/545 Partial Dierential Equations Spring 2013
Homework Assignment # 1 Solutions
1. Pinchover and Rubinstein problem 1.1 Show that each of the following equations has a
solution of the form u(x, y ) = f (ax + by ) for a proper choice of constants a,
92.445/545 Partial Dierential Equations Spring 2013
Homework Assignment # 2 Solutions
1. Consider the pde ut + 4u3 + 4u ux = 0, and consider a weak solution in the form of a shock
wave: u =
u if x < (t)
u+ if x > (t)
Find the speed of propagation (t) of t
92.445/545 Partial Dierential Equations
Spring 2013
Homework Assignment # 8 Solutions
PLEASE SHOW ALL WORK! You will not receive full credit if you do not show your
work. You may work together, but everyone must turn in his/her own homework set.
1. Use th