M534H
INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
UMASS AMHERST, 2015
INSTRUCTOR: NESTOR GUILLEN
1. Lecture I
1.1. Notation. Let us set up quickly some notation for the semester (more notation to be be added later).
N, Z, R will denote respectively the
2. Lecture II
Harmonic functions
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
A quick Review of Calculus Identities
Harmonic functions, Laplace and Poissons equation.
Examples
Boundary Value Problems. Uniqueness (Energy method)
The mean value proper
Math 534H
Homework V
(Due Tuesday, April 14th)
(1) Find the solution to the initial value problems
a)
@t u + 5@x u =
2u
u(x, 0) = (1
b)
x2 ) +
@t u + @x u = cos(2x)
u(x, 0) = 1 + sin(2x)
c)
@t u
@x u = g(x, t)
u(x, 0) = 1 + sin(2x)
In the last problem, g(
Math 534H
Homework III
(Due Thursday, February 26th)
(1) Consider the square = cfw_(x1 , x2 ) | 0 x1 1, 0 x2 1 and for every pair of numbers
n, m 2 Z, the functions
u(x1 , x2 ) = sin(2nx1 ) sin(2mx2 ).
Compute u(x1 , x2 ). Then, say whether there is a nit
Math 534H
Homework IV
(Due Thursday, April 2nd)
(1) Find a explicit expression for the solution to
@t u = @xx u + 10u
u(x, 0) = cos(x) + cos(3x)
in R R+ ,
Hint: Note that for every n, @xx (cos(nx) + 10 cos(nx) = (10
data is a sum of cosines.
n2 )u, use th
Math 534H
Homework I
(Due Tuesday, January 27th)
(1) Use separation of variables to solve the initial value problem:
u0 (t) = u2 (t)
u(0) = 1
(2) Use separation of variables to solve the initial value problem:
u0 (t)
3u(t) = 0
u(0) = 1
(3) Use separation