Fourier Series and Partial Differential Equations Lecture Notes

Chapter 1. Introduction 7
IVP: y ′′ = 6y 1
3, y(0) = 0, y ′ (0) = 0 has solutions y(x) = 0, y(x) = x3 (non-uniqueness);
BVP1: y ′′ +y = 0, y(0) = 1, y(2π) = 0 has no solution (non-existence);
BVP2: y ′′ + y = 0, y(0) = 0, y(2π) = 0 has infinitely many solutions, y(x) = csinx,
where c is an arbitrary constant (non-uniqueness).
1.2 Some preliminaries
We state, but do not prove, two preliminary results.
Theorem 1.1 (Leibniz’s Integral Rule) Let F(x,t) and ∂F/∂t be continuous in both x
and t in some region of the (x,t) plane including (t,x) ∈ [t0,t1] × [a(t),b(t)], and the
functions a(t) and b(t) and their partial derivatives be continuous for t ∈ [t0,t1]. Then
G(t) = d
dt
b(t)
a(t)
F(x,t)dx = b ′ (t)F(x,b(t))−a ′ b(t) ∂F(x,t)
(t)F(x,a(t)) + dx. (1.2)
a(t) ∂t
As a result, if a(t) and b(t) are constants with
then
Lemma 1.2 If f(x) is continuous then
Note that
1
h
b
G(t) = F(x,t)dx, (1.3)
a
dG
dt =
b ∂F(x,t)
dx. (1.4)
a ∂t
a+h
f(x)dx → f(a) as h → 0.
a
G(t+h)−G(t)
h
b
=
and the integrand tends to ∂F(x,t)/∂t as h → 0.
1.3 The equations we shall study
a
F(x,t+h)−F(x,t)
dx, (1.5)
h
It is proposed to study three linear second-order partial differential equations (PDEs) that
have applications throughout the physical sciences.
1.3.1 The wave equation
Here, we will look at finding y(x,t) such that
∂2y ∂t2 = c2∂2 y
∂x2, (1.6)
where, for example, y(x,t) is the transverse displacement of a stretched string at position
x and time t, and c is a positive constant—the wave speed.