Math 126
Homework 2
(1) Modify the proof of the mean value formulas to show for n 3 that
1
1
1
n2 f dx,
gdS +
u(0) =
n 2
n(n 2)(n) B (0,r) |x|
r
B (0,r )
provided that
in BO (0, r)
on B(0, r)
u = f
u =g
(B O (x, r ) = cfw_y Rn | |x y | < r = open ball
Math 126
Homework 1
(1) Classify each of the partial dierential equations below as linear, semilinear, quasilinear
or fully nonlinear. What is the order of each PDE?
(a) u = f (u)
(b) utt + dut uxx = 0
(c) ut + uxxxx = 0
(d) ut + uux + uxxx = 0
(e) |Du| =
Math 126
Homework 3
(1) Let = (0, ) (0, ). Plot the solution to the following boundary-value problem for
Laplaces equation on a square,
u = 0
in
u(0, y ) = 0, u(a, y ) = y (y b)
0<y<b
u(x, 0) = 0, u(x, b) = 0
0 < x < a.
You are allowed to use the formula
Math 126
Homework 4
(1) Show that the Dirichlet problem
Lu = f x
u
= g x
with f C 2 (), g C 2 ( ) has at most one solution if c(x) 0.
(2) Show that the problem
2u 2u
+ 2 = u3
x2
y
in the domain D = cfw_(x, y ) : x2 + y 2 < 1 with u = 0 for x2 + y 2 = 1
Math 126
Homework 6
(1) Use separation of variables to construct the exact solution to the following heat problem
on
ut kuxx = 0 0 < x <
u(0, t) = 0, u(, t) = 0
u(x, 0) = x(x ) 0 < x <
For what values of k does the solution converge as t ? The solution
Math 126
Homework 5
(1) Consider the dierential equation
u (x) = f (x) for 0 < x < 1
u (0) =
u(1) = .
It is ideal to discretize the boundary conditions in a second order accurate manner so
that you do not lose the second order accuracy of the centered di
Math 126
Homework 7
(1) Let Rn be an open, bounded set with smooth boundary and T > 0. Prove there is
at most one smooth solution of this initial/boundary value problem for the heat equation
with Neumann boundary conditions:
ut u = f in T
u
[0, T ]
= 0