4
Classication of Second-Order Equations
4.1
Types of Second-Order Equations
We now turn our attention to second-order equations
F (x, u, Du, D2 u) = 0.
In general, higher-order equations are more complicated to solve than rst-order equations.
Consequentl
5
Wave Equation in R
5.1
Derivation
Ref: Strauss, Section 1.3, Evans, Section 2.4
Consider a homogeneous string of length l and density = (x). Assume the string is
moving in the transverse direction, but not in the longitudinal direction. Let u(x, t) deno
6
6.1
Wave Equation on an Interval: Separation of Variables
Dirichlet Boundary Conditions
Ref: Strauss, Chapter 4
We now use the separation of variables technique to study the wave equation on a nite
interval. As mentioned above, this technique is much mo
Math 220A - Fall 2002 Homework 5 Solutions
1. Consider the initial-value problem for the hyperbolic equation - < x < , t > 0 utt + uxt - 20uxx = 0 u(x, 0) = (x) u (x, 0) = (x). t Use energy methods to show that the domain of dependence of the solution u a
7
Wave Equation in Higher Dimensions
We now consider the initial-value problem for the wave equation in n dimensions,
x Rn
utt c2 u = 0
u(x, 0) = (x)
ut (x, 0) = (x)
n
i=1
where u
7.1
(7.1)
ux i x i .
Method of Spherical Means
Ref: Evans, Sec. 2.4.1; St
2
First-Order Equations: Method of Characteristics
In this section, we describe a general technique for solving rst-order equations. We begin
with linear equations and work our way through the semilinear, quasilinear, and fully nonlinear cases. We start b