Notes on
Partial Dierential Equations
John K. Hunter
Department of Mathematics, University of California at Davis
Abstract. These are notes from a two-quarter class on PDEs that are heavily
based on t
viii
CHAPTER 7
Hyperbolic Equations
Hyperbolic PDEs arise in physical applications as models of waves, such as
acoustic, elastic, electromagnetic, or gravitational waves. The qualitative properties
of
124
CHAPTER 5
The Heat and Schrdinger Equations
o
The heat, or diusion, equation is
(5.1)
ut = u.
Section 4.A derives (5.1) as a model of heat ow.
Steady solutions of the heat equation satisfy Laplace
Appendix
May the Schwartz be with you!4
In this section, we summarize some results about Schwartz functions, tempered
distributions, and the Fourier transform. For complete proofs, see [19, 25].
5.A.
viii
6.A. VECTOR-VALUED FUNCTIONS
189
Appendix
In this appendix, we summarize some results about the integration and dierentiation of Banach-space valued functions of a single variable. In a rough sen
CHAPTER 4
Elliptic PDEs
One of the main advantages of extending the class of solutions of a PDE from
classical solutions with continuous derivatives to weak solutions with weak derivatives is that it
Appendix
4.A. Heat ow
As a simple physical application that leads to second order PDEs, we consider
the problem of nding the temperature distribution inside a body. Similar equations describe the dius
Appendix
In this appendix, we describe without proof some results from real analysis
which help to understand weak and distributional derivatives in the simplest context
of functions of a single varia
CHAPTER 1
Preliminaries
In this chapter, we collect various denitions and theorems for future use.
Proofs may be found in the references e.g. [3, 10, 19, 27, 32, 33].
1.1. Euclidean space
n
Let R be n
CHAPTER 2
Laplaces equation
There can be but one option as to the beauty and utility of this
analysis by Laplace; but the manner in which it has hitherto been
presented has seemed repulsive to the abl
Notes on
Partial Dierential Equations
John K. Hunter
Department of Mathematics, University of California at Davis
Abstract. These are notes from a two-quarter class on PDEs that are heavily
based on t