3. Separation of Variables
3.0. Basics of the Method.
In this lecture we review the very basics of the method of separation of variables in 1D.
3.0.1. The method.
The idea is to write the solution as
u(x, t) =
Xn(x) Tn(t).
(3.1)
n
where Xn(x) Tn(t) solves
Math 436 Fall 2012 Homework 1 Solutions
Due Sept. 27 in Class
Note. All problem numbers refer to Updated version of lecture note.
Ex. 1.1. Derive in detail the equation for random walk in the following general case: The probability
of the particle at (x,
Math 436 Fall 2012 Homework 4 Solutions
Due Nov. 8 in Class
Note. All problem numbers refer to Updated version of lecture note.
Exercise 3.1. Consider the Telegraphers equation
uxx = utt + ut
(1)
(recall that > 0) over the interval x [0, L] subject to con
Math 436 Midterm Solution
Oct. 18, 2012 12:30pm 1:50pm Total 50 Pts.
NAME:
ID#:
Problem 1. (10 pts) Let c, be constants, > 0. Design a random walk model which leads to the equation
ut + c ux = uxx ,
(1)
then obtain Duhamels principle for the corresponding
Math 436 Fall 2012 Homework 3 Solutions
Due Oct. 25 in Class
Note. All problem numbers refer to Updated version of lecture note.
Ex. 2.28. d), e). Solve
ut + u2 + u = 0,
x
u(x, 0) = x
(1)
u(x, 0) = x2.
(2)
and
ut + u2 = 0,
x
Show that the solution of the
Math 436 Fall 2012 Homework 2
Due Oct. 11 in Class
Note. All problem numbers refer to Updated version of lecture note.
Ex. 2.2. Find the solution of the following Cauchy problems.
a) x ux + y u y = 2 x y,
with u = 2 on y = x2.
b) u ux u u y = u2 + (x + y)
Math 436 Fall 2012 Homework 5 Solutions
Due Nov. 22 in Class
Note. All problem numbers refer to Updated version of lecture note.
Ex. 3.11. Write the following equations into S-L form and discuss whether they are
regular or singular. Determine what is the
Math 436 Fall 2012 Homework 6 Solutions
Due Dec. 4 in Class
Note. All problem numbers refer to Updated version of lecture note.
Ex. 3.22. Construct a sequence fn(x)
f (x)2 dx 1/2 = 1 for all n.
R
0 for every x R, but
fn
=
Solution. Take
fn(x) =
1 x (n, n
2. Method of Characteristics
In this section we explore the method of characteristics when applied to linear and nonlinear equations of
order one and above.
2.1. Method of characteristics for rst order quasilinear equations.
2.1.1. Introduction to the met
4. Special Solutions and Stability
Most PDEs in science and engineering are nonlinear. They cannot be solved either explicitly or semiexplicitly (separation of variables, transform methods, Greens functions, etc.). For these equations one
important strate
1. From Random Movements to PDEs
1.1. Heat Equation.
1.1.1. 1D Random walk.
Consider the random walk of a particle along the real line. Let the rule of movement be: At each time
step of size , the particle jumps to left or right with distance h equally li