Homework Set # 4 Math 435 Summer SOLUTIONS
1. Solve the heat equation (i.e. - the diusion equation) 4uxx = ut on a rod of length 2 if
u(x, 0) = sin( x ) and u(0, t) = 0 = u(2, t).
2
Solution:
We are solving the heat equation on a nite interval (0, 2), wit
1. The solution is u(x, t) =
u(x, t) =
1
2
ex+ct + exct +
1
2c
x+ct
xct sin(s) dx
or
1
1 x+ct
e
+ exct +
( cos(x + ct) + cos(x ct)
2
2c
2. To solve the PDE uxx 3uxt 4utt = 0, you can rst factor the dierential operator to see
4
x
t
+
x t
u=0.
Now you can e
Math 435: Intro to Partial Dierential Equations
Instructor:
Oce:
Phone:
e-mail:
Oce Hours:
Summer 2013
Dr. Heather Finotti
233 Ayres Hall
865.974.4320
[email protected] (this is my preferred means of communication)
see website
Course Web Page:
http:/ww
Homework Set # 8 Math 435 SOLUTIONS Due date: 3/13/2013
1. (a) Find the Fourier sine series, the Fourier cosine series and the full Fourier series expansion
of ex on (0, 2) or (2, 2) as appropriate. (Note that once youve done the work for nding
the sine a
Homework Set # 6 Math 435 Summer
1. Solve uxx + uyy = 0 on the rectangle 0 < x < 2, 0 < y < 3, with boundary conditions
ux (0, y ) = 0
ux (2, y ) = 0
u(x, 0) = 0
u(x, 3) = 3x
Solution:
We can seperate variables to obtain u(x, y ) = X (x)Y (y ) where X = X
Homework Set # 6 Math 435 Summer
1. Solve uxx + uyy = 0 on the rectangle 0 < x < 2, 0 < y < 3, with boundary conditions
ux (0, y ) = 0
ux (2, y ) = 0
u(x, 0) = 0
u(x, 3) = 3x
2. Consider Laplaces equation on the disk x2 + y 2 < 9 with boundary condition u
Homework Set # 5 SOLUTIONS Math 435
1. Consider the Fourier sine series of each of the following functions. Do not compute the
coecients, but use the pointwise convergence theorem to discuss the convergence of each of
the series. Explain what you would ex
Homework Set # 5 Math 435
1. Consider the Fourier sine series of each of the following functions. Do not compute the
coecients, but use the pointwise convergence theorem to discuss the convergence of each of
the series. Explain what you would expect to se
Homework Set # 4 Math 435 Summer SOLUTIONS
1. Solve the heat equation (i.e. - the diusion equation) 4uxx = ut on a rod of length 2 if
u(x, 0) = sin( x ) and u(0, t) = 0 = u(2, t).
2
Solution:
We are solving the heat equation on a nite interval (0, 2), wit
Homework Set # 4 Math 435 Summer 2013
1. Solve the heat equation (i.e. - the diusion equation) 4uxx = ut on a rod of length 2 if
u(x, 0) = sin( x ) and u(0, t) = 0 = u(2, t). (no need to derive the solution here, just give it)
2
2. Solve the wave equation
1. The solution is u(x, t) =
u(x, t) =
1
2
ex+ct + exct +
1
2c
x+ct
xct sin(s) dx
or
1 x+ct
1
( cos(x + ct) + cos(x ct)
e
+ exct +
2
2c
2. To solve the PDE uxx 3uxt 4utt = 0, you can rst factor the dierential operator to see
4
x
t
+
x t
u=0.
Now you can e
Homework Set # 3 Math 435 Summer 2013
1. Solve the wave equation utt = c2 uxx on the whole real line with u(x, 0) = ex and ut (x, 0) =
sin(x).
2. Classify the PDE and then solve uxx 3uxt 4utt = 0, with u(x, 0) = (x) and ut (x, 0) = (x).
(Hint: Factor the
Homework Set # 2 Solutions Math 435 Summer 2013
1. Suppose that we have a uniform thin tube (approximable by one space dimension) of liquid
with some particles which are suspended in the liquid. If the liquid is owing through the
pipe at a constant rate c
Homework Set # 2 Math 435 Summer 2013
1. Suppose that we have a uniform thin tube (approximable by one space dimension) of liquid
with some particles which are suspended in the liquid. If the liquid is owing through the pipe
uniformly at a constant rate c
Homework Set # 1 SOLUTIONS Math 435 Spring 2013 Due date: 1/14/2013 1. Determine whether or not the following functions are solutions to the given PDE. (a) ux - 3uy = 0, u(x, y) = cos(y + 3x) (c) uxx + uyy = 0, u(x, y) = x2 + y 2
(b) ux - 3uy = 0, u(x, y)
Homework Set # 6 Math 435 Summer
1. Solve uxx + uyy = 0 on the rectangle 0 < x < 2, 0 < y < 3, with boundary conditions
ux (0, y) = 0
ux (2, y) = 0
u(x, 0) = 0
u(x, 3) = 3x
Solution:
We can seperate variables to obtain u(x, y) = X(x)Y (y) where X = X and