Math 678. Homework 5 Solutions.
#1
Consider a subsolution of the heat equation vt v 0 in UT .
(a) The proof follows the argument given in Theorem 3, p.53-54, with the exception being that (r) 0, from where it follows that
v (x, t)
1
4rn
v (y, s)
E (x,t;r
Math 678. Homework 3 Solutions.
#1
We need to derive the formula for the solution of the IVP
ut u + cu = f. in Rn (0, )
u = g, on Rn cfw_t = 0
Some observations: (1) the solution to the non-homogeneous problem can be
obtained from the solution to the homo
Math 678. Fall 2011.
Homework #2.
Due Wednesday 09/28/11 in class.
Solutions should represent individual work, with all necessary details. Only facts discussed
in class or given in the main textbook can be used without proof (except the facts known
from c
Math 678. Homework 2 Solutions.
#3, p.85
Following the proof of the mean value property, denote (r) =
u(y )dS (y )
B (0,r )
u(y )dS = f (y )dS .
and use the same argument to show that (r) =
B (0,r )
r
Then x some > 0 and notice that (r) ( ) =
as
0. Sinc
Math 678. Homework 1 Solutions.
#2, p.12
Think about circles denoting the partial derivative, with m consecutive circles
representing an m-th order derivative in the corresponding variable. Dividers
are placed between variables, and can be separated by m
Math 678. Fall 2011.
Homework #1.
Due Wednesday 09/14/11 in class.
Solutions should represent individual work, with all necessary details. Only facts discussed
in class or given in the main textbook can be used without proof. Only selected problems
will b
Math 678. Fall 2011.
Homework #3.
Due Wednesday 10/12/11 in class.
Solutions should represent individual work, with all necessary details. Only facts discussed
in class or given in the main textbook can be used without proof (except the facts known
from c
Math 678. Homework 4 Solutions.
#1
(a) In order to get the general solution of uxy = 0, use integration in one of
the variables, e.g. in y rst. This gives ux = f (x) and u(x, y ) = f (x)dx + G(y ).
Denote F (x) = f (x)dx to get the conclusion: general sol
Math 678. Fall 2011.
Homework #6.
Due Wednesday 11/30/11 in class.
Solutions should represent individual work, with all necessary details. Only facts discussed
in class or given in the main textbook can be used without proof (except the facts known
from c
Math 678. Fall 2011.
Homework #5.
Due Wednesday 11/16/11 in class.
Solutions should represent individual work, with all necessary details. Only facts discussed
in class or given in the main textbook can be used without proof (except the facts known
from c
Math 678. Fall 2011.
Homework #4.
Due Wednesday 11/02/11 in class.
Solutions should represent individual work, with all necessary details. Only facts discussed
in class or given in the main textbook can be used without proof (except the facts known
from c
Math 678. Fall 2011. Final Exam.
Due Wednesday 12/14/11 by 5pm
Solutions should represent individual work, with all necessary details. Only facts discussed
in class or given in the main textbook can be used without proof (except the facts known
from calcu