Partial Differential Equations
Lecture Notes 2007 under construction
Vitali Liskevich
Department of Mathematics
Swansea University
Contents
1 Introduction 2
1.1 What is a PDE? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Change of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 First easy PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 First-order linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5.1 The constant coefficient equation–Transport equation . . . . . . . . . 6
1.5.2 Variable coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 First order nonlinear PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Initial and boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.8 Well-posedness of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.9 Classification of second-order PDEs . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Wave Equation 15
2.1 General solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Initial value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Reflection method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Non-homogeneous wave equation . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Bounded string. Fourier method . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.1 Homogeneous case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.2 Fourier series. Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5.3 Nonhomogeneous case . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.4 Energy and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Heat equation in 1D 32
3.1 Some qualitative properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Initial boundary value problem . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 Separation of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 Non-homogeneous case . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
0
CONTENTS CONTENTS
4 Harmonic functions 39
4.1 Divergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3 Maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 Harnack inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.5 Harmonicity revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.6 Fundamental solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.6.1 Green’s function of the ball . . . . . . . . . . . . . . . . . . . . . . . . 48
1 October 15, 2009
Chapter 1
Introduction
Recommended Texts:
1. Walter A. Strauss, Partial Differential Equations. Introduction. John Wiley & Sons,
1992.
2. Lawrence C. Evans, Partial Differential Equations. AMS 1998.
1.1 What is a PDE?
A partial differential equation (PDE) is an equation involving an unknown function of two or
more variables and certain of its partial derivatives. We can write a typical PDE, as follows.
Fix an integer k ≥ 1 and let U denote an open subset of R
n
.
An expression of the form
(1.1) F (D
k
u(x), D
k−1
u(x), ..., Du(x), u(x), x) = 0 (x ∈ U)
is called a kth-order partial differential equation, where
F : R
n
k
× R
n
k−1
× . . . R
n
× R × U
For example, the most general first order PDE in R
2
is of the form
F (x, y, u(x, y), u
x
(x, y), u
y
(x, y)) = F (x, y, u, u
x
, u
y
) = 0,
where we used the notation u
x
=
∂u
∂x
.
The most general second order in R
2
is of the form
F (x, y, u, u
x
, u
y
, u
xx
, u
xy
, u
yy
) = 0.
To solve the equation means to find a function u(x) for x in some region of R
n
so that
this equation is satisfied for all x in this region.
2
1.2. EXAMPLES CHAPTER 1. INTRODUCTION
PDEs have been studied in physics and mathematics since the times of Newton. Many
branches in physics are described using PDEs, and they are also fundamental in branches of
mathematics such as differential geometry and stochastic processes. Nowadays, PDEs also
find many applications in chemistry, biology, engineering and economics.
A PDE is called linear if F is linear in u and D
k
u for every k. Linearity means the
following. Write the equation in the form Lu = 0, where L is an operator acting as Lu =
F (D
k
u(x), D
k−1
u(x), ..., Du(x), u(x), x). The linearity of L means that for for the functions
u, v and a constant c we have that
L(u + v) = Lu + Lv, L(cu) = cLu.
The equation
Lu = 0
is called homogeneous linear equation.
The equation
Lu = f
is called inhomogeneous linear equation.
In this course we mainly consider second order partial differential equations with strong
motivations from physics: the wave equation, the heat equation and the Laplace equation.
Exercise 1 1. Which of the following operators L are linear
(a) Lu = u
x
+ xu
y
;
(b) Lu = u
x
+ uu
y
;
(c) Lu = u
x
+ u
2
y
;
(d) Lu = u
x
+ 5u
y
+ 6.
2. Show that the difference of two solutions of an inhomogeneous linear equation Lu = g
is a solution of the homogeneous equation Lu = 0.
1.2 Examples
There is no general theory known concerning the solvability of all partial differential equations.
Such a theory is extremely unlikely to exist, given the rich variety of physical, geometric, and
probabilistic phenomena which can be modeled by PDE. Instead, research focuses on various
particular partial differential equations that are important for applications within and outside
of mathematics, with the hope that insight from the origins of these PDE can give clues as
to their solutions.
Following are examples of PDEs with x ∈ R
n
and t ≥ 0.
a. Linear equations.
1. Laplace ’ s equation
∆u =
X
i=1
u
x
i
x
i
= 0.
3 October 15, 2009
