Introduction to Partial Differential Equations: From Fourier Series to Boundary-Value Problems

By Arne Broman

This exceptionally well-written and well-organized text is the outgrowth of a course given every year for 45 years at the Chalmers University of Technology, Goteborg, Sweden. The object of the course was to give students a basic knowledge of Fourier analysis and certain of its applications. The text is self-contained with respect to such analysis; however, in areas where the author relies on results from branches of mathematics outside the scope of this book, references to widely used books are given.
Table of Contents:
Chapter 1. Fourier series
1.1 Basic concepts
1.2 Fourier series and Fourier coefficients
1.3 A minimizing property of the Fourier coefficients. The Riemann-Lebesgue theorem
1.4 Convergence of Fourier series
1.5 The Parseval formula
1.6 Determination of the sum of certain trigonometric series
Chapter 2. Orthogonal systems
2.1 Integration of complex-valued functions of a real variable
2.2 Orthogonal systems
2.3 Complete orthogonal systems
2.4 Integration of Fourier series
2.5 The Gram-Schmidt orthogonalization process
2.6 Sturm-Liouville problems
Chapter 3. Orthogonal polynomials
3.1 The Legendre polynomials
3.2 Legendre series
3.3 The Legendre differential equation. The generating function of the Legendre polynomials
3.4 The Tchebycheff polynomials
3.5 Tchebycheff series
3.6 The Hermite polynomials. The Laguerre polynomials
Chapter 4. Fourier transforms
4.1 Infinite interval of integration
4.2 The Fourier integral formula: a heuristic introduction
4.3 Auxiliary theorems
4.4 Proof of the Fourier integral formula. Fourier transforms
4.5 The convention theorem. The Parseval formula
Chapter 5. Laplace transforms
5.1 Definition of the Laplace transform. Domain. Analyticity
5.2 Inversion formula
5.3 Further properties of Laplace transforms. The convolution theorem
5.4 Applications to ordinary differential equations
Chapter 6. Bessel functions
6.1 The gamma function
6.2 The Bessel differential equation. Bessel functions
6.3 Some particular Bessel functions
6.4 Recursion formulas for the Bessel functions
6.5 Estimation of Bessel functions for large values of x. The zeros of the Bessel functions
6.6 Bessel series
6.7 The generating function of the Bessel functions of integral order
6.8 Neumann functions
Chapter 7. Partial differential equations of first order
7.1 Introduction
7.2 The differential equation of a family of surfaces
7.3 Homogeneous differential equations
7.4 Linear and quasilinear differential equations
Chapter 8. Partial differential equations of second order
8.1 Problems in physics leading to partial differential equations
8.2 Definitions
8.3 The wave equation
8.4 The heat equation
8.5 The Laplace equation
Answers to exercises; Bibliography; Conventions; Symbols; Index
Written on an advanced level, the book is aimed at advanced undergraduates and graduate students with a background in calculus, linear algebra, ordinary differential equations, and complex analysis. Over 260 carefully chosen exercises, with answers, encompass both routing and more challenging problems to help students test their grasp of the material.

• Language: English
• Publisher: Dover Publications
• Release date: Apr 27, 2012
• ISBN: 9780486153018

Book preview

INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS

from Fourier Series to Boundary-value Problems

ARNE BROMAN

PROFESSOR EMERITUS OF MATHEMATICS

CHALMERS UNIVERSITY OF TECHNOLOGY

GÖTEBORG, SWEDEN

DOVER PUBLICATIONS, INC., NEW YORK.

Copyright © 1970 by Addison-Wesley Publishing Company, Inc.

All rights reserved.

This Dover edition, first published in 1989, is an unabridged and corrected republication of the work originally published in 1970 by Addison-Wesley Publishing Company, Inc., Reading, Mass. Three sections have been simplified: Lemma 2.4.1, Lemma 4.4.13, and the hint for Exercise 115. Exercise 132 has been added, and Exercise 421 has been completely changed. The sections Conventions and Symbols have been moved from pp. vii–x to 176–179. The biography on page 184 is also new to this edition.

Library of Congress Cataloging-in-Publication Data

Broman, Arne, 1913–

Introduction to partial differential equations: from Fourier series to boundary-value problems / Arne Broman.

p. cm.

A corrected republication of the work originally published in 1970 by Addison-Wesley Publishing Company, Inc., Reading, Mass.—T.p. verso.

Includes bibliographical references.

eISBN 13: 978-0-486-15301-8

1. Differential equations, Partial. I. Title.

QA374.B7941989

515′.353—dc20

89-16923

CIP

Manufactured in the United States by Courier Corporation

66158X06

www.doverpublications.com

PREFACE

This text is an outgrowth of a course that has been given every year for some twenty-five years at Chalmers University of Technology, Göteborg, Sweden. The object of the course is to give the students some basic knowledge in Fourier Analysis and in certain of its applications.

The students taking the course have already taken basic courses in calculus, linear algebra, ordinary differential equations, and complex analysis. With such courses as prerequisites, this text is self-contained with respect to Fourier analysis and its applications. In some places where we rely on results from other branches of mathematics that may not have been presented in the basic courses mentioned, references to widely used books have been given.

The definitions and notations used are, as a rule, standard in the mathematical literature of today. In a few places where a definition or a notation is introduced for use in this text only, the word we is used in the statement.

The text is divided into eight chapters and subdivided into forty-four sections. At the end of each section exercises are given. These exercises are for the most part divided into two groups: first some routine exercises, and then some more challenging ones, the first of these being marked by an asterisk. The total number of exercises is two hundred and sixty-six.

Formulas of interest have been marked by numbers. Formulas that are not interesting in their own right but to which some reference is given are marked by letters. The first digit in the number of an exercise denotes the chapter in which it occurs.

In order to avoid too many repetitions, some conventions have been introduced, and are collected in an index.

The literature on Fourier analysis and its applications and on the other branches of mathematics that are touched upon in this text is quite extensive. The bibliography indicates a few of the most valuable books in these branches of mathematics.

It is possible for an instructor using this text to omit certain of the sections and starred exercises, and yet to offer a course that should be valuable to the students.

Many people have lectured or given tutorials on this course at Chalmers during the last two and a half decades. Many ideas in this text emanate from discussions I have had with these people and from lecture notes and collections of exercises that some of them have prepared.

Arne Broman

Göteborg, March 1970

CONTENTS

Chapter 1Fourier series

1.1Basic concepts

1.2Fourier series and Fourier coefficients

1.3A minimizing property of the Fourier coefficients. The Riemann–Lebesgue theorem

1.4Convergence of Fourier series

1.5The Parseval formula

1.6Determination of the sum of certain trigonometric series

Chapter 2Orthogonal systems

2.1Integration of complex-valued functions of a real variable

2.2Orthogonal systems

2.3Complete orthogonal systems

2.4Integration of Fourier series

2.5The Gram-Schmidt orthogonalization process

2.6Sturm-Liouville problems

Chapter 3Orthogonal polynomials

3.1The Legendre polynomials

3.2Legendre series

3.3The Legendre differential equation. The generating function of the Legendre polynomials

3.4The Tchebycheff polynomials

3.5Tchebycheff series

3.6The Hermite polynomials. The Laguerre polynomials

Chapter 4Fourier transforms

4.1Infinite interval of integration

4.2The Fourier integral formula: a heuristic introduction

4.3Auxiliary theorems

4.4Proof of the Fourier integral formula. Fourier transforms

4.5The convolution theorem. The Parseval formula

Chapter 5Laplace transforms

5.1Definition of the Laplace transform. Domain. Analyticity

5.2Inversion formula

5.3Further properties of Laplace transforms. The convolution theorem

5.4Applications to ordinary differential equations

Chapter 6Bessel functions

6.1The gamma function

6.2The Bessel differential equation. Bessel functions

6.3Some particular Bessel functions

6.4Recursion formulas for the Bessel functions

6.5Estimation of Bessel functions for large values of x. The zeros of the Bessel functions

6.6Bessel series

6.7The generating function of the Bessel functions of integral order

6.8Neumann functions

Chapter 7Partial differential equations of first order

7.1Introduction

7.2The differential equation of a family of surfaces

7.3Homogeneous differential equations

7.4Linear and quasilinear differential equations

Chapter 8Partial differential equations of second order

8.1Problems in physics leading to partial differential equations

8.2Definitions

8.3The wave equation

8.4The heat equation

8.5The Laplace equation

Answers to exercises

Bibliography

Conventions

Symbols

Index

CHAPTER 1

FOURIER SERIES

1.1 BASIC CONCEPTS

This section (1.1) is devoted to some definitions, notations and conventions concerning intervals, functions and integrals. In what follows we shall adhere to these conventions, etc., unless the context clearly implies some other convention.

means belongs to (is a member of).

1.1.1 Definitions

The set of all real numbers will be denoted by R. Suppose that a and b belong to R and that a < b. The sets of all x R such that a < x < b, a ≤ x ≤ b, a < x ≤ b or a ≤ x < b will be denoted by

respectively. The sets (1) are called open interval, closed interval and half-open intervals respectively. The terms interval and finite interval denote any among the sets (1). The number b – a is called the length of each of the intervals (1).

In many contexts it is immaterial which among the intervals (1) is considered. In such situations we shall simply write "the interval (a, b)", and in general it is left to the reader to see that our arguments apply equally to any of the intervals (1).

The sets of all x R such that a < x, a ≤ x, x < a, x ≤ a, or x is any real number are denoted by

respectively. The sets (2) are called infinite intervals. The second and fourth among them are half-open infinite intervals. The remaining three are open infinite intervals.

1.1.2 Definitions

A real-valued function of a real variable is a set of ordered pairs of real numbers (x, y) such that no two pairs have the same first component. A function f, in this chapter, is a real-valued function of a real variable. In this text the set of its first components x, the domain of f, is a finite or infinite interval, or such an interval with at most a finite number of x-values missing from each of its finite sub-intervals. The set of its second components y is the range of f. The domain and the range of f are denoted by Df and Rf respectively. A function f is bounded if Rf is a subset of some finite interval. A function that is not bounded is called unbounded. If x Df, the number y in the corresponding pair (x, y) is often denoted by f(x), and the number y or f(x) is called the value of the function f at the point x. We shall, however, often let f(x) or y = f(x) denote the function f; it will always be clear from the context which of the two meanings of f(x) is used. If a function consists of all those pairs (x, y) that satisfy a certain equation, we shall often use this equation to define the function; e.g. the phrase "the function y = sin x" denotes the function {(x, y); x R and y = sin x}. Further, we shall often let expressions such as "the function y = sin x, 0 < x < 2π" define a function; in this example the function {(x, y); 0 < x < 2π and y = sin x}.

Many authors of mathematical texts define a function as a rule that associates a unique number y with each number x belonging to a certain set. These authors use the term graph (of a function) where we use the term function.

means implies (has as a consequence).

Suppose that S is a nonempty set of real numbers. Also suppose that S has an upper bound c, i.e. c is a real number such that y S y ≤ c. Then S has a least upper bound (see [13], Theorem 1.36).* This least upper bound is called the supremum of S, and is denoted by sup S or supy S y. Analogously, if S has a lower bound, then S has a greatest lower bound, called the infimum of S, and denoted by inf S or infy S y.

1.1.3 Definitions

Suppose that f is a bounded function whose domain includes the interval (a, b) with the possible exception of finitely many points (Fig. 1.1). Let D be a subdivision of the interval (a, b) i.e. a strictly increasing finite sequence of numbers xv, where v = 0, 1, . . . , n, such that x0 = a and xn = b. (To index finite sequences, Greek letters will be used.) Let the numbers mv and Mv be defined by the equalities

Figure 1.1

where Iv denotes the set {x; xv−1 < x < xv and x Df}. Consider the sums s(D) and S(D) defined by the equations

(the lower sum and the upper sum respectively of f (the lower and the upper integral respectively of f on the interval) be defined by the equalities

where the supremum and the infimum are taken over the set of all subdivisions of (a, b. If

the function is said to be integrable on (a, b), and the common value of the two members of (3) is called the integral of f on (a, b) and is denoted by

It is seen that, m these definitions of integrable and integral, the interval (a, b) can be replaced by any among the last three intervals in (1). It is also seen that a function that is continuous on a closed interval [a, b], is integrable on [a, b].

We want to extend these definitions to certain unbounded functions. To this end, suppose that the function f satisfies the conditions in the comments applying to Fig. 1.1 with the word bounded deleted. First suppose that the range of f contains no negative numbers. Let m be a positive integer, and let fm be the function defined by

(The curve y = f(x) is truncated by the line y = m.) Suppose that fm(x) is integrable on (a, b) for every m and that the limit

exists, i.e. that (5) denotes a real number. Then f is said to be integrable on (a, b) and the number (5) is called its integral on (a, b) and is denoted by (4).

Then suppose that the range of f contains negative numbers. Let the functions f + and f − (the positive part and the negative part of f) be defined by

Suppose that the functions f + and – f − are both integrable on (a, b). Then f is said to be integrable on (a, b), and by its integral on (a, b) denoted by (4), is understood

We sum up our definitions in the following way. A function f is said to be integrable on the interval (a, b) if it satisfies the conditions given in the context for any of the formulas (3), (5) and (6). The number given by (3), (5) or (6) is called the integral of f on (a, b); it is denoted by (4).

Every function occurring in this text as an integrand on a finite interval is integrable in the above sense (Riemann integrable) on the interval of integration, and the operations that we perform on integrals can be proved to be legitimate. For a definition of a concept of integral (due to Lebesgue) which applies to a considerably wider class of functions than we study in this text, see [13], Chapter 10.

EXERCISE

*101.Decide for each of the following functions if it is integrable in the sense above on the interval (0, 1)

1.2 FOURIER SERIES AND FOURIER COEFFICIENTS

1.2.1 Definitions

Let a function f(x) and a number p ≠ 0 be given. If, for every x in the domain of f(x), the numbers x + p and x − p belong to the domain and f(x + p) = f(x), then the function is said to be periodic with the period p.

If p is a period of f(x), then every nonzero multiple of p is a period. If two functions f(x) and g(x) have the period p, then their sum and product have the same period.

1.2.2 Definitions

The functions cos x and sin x both have period 2π. It follows that the functions cos vx and sin vx, v being a nonnegative integer, have period 2π. Further, every trigonometric polynomial, i.e. every function of the form

or, using the sum notation,

the av and bv being given constants, has the period 2πin the first term is introduced for convenience; see the comment on formula (6a) below.) The integer n, occurring in (1), is called the order of the trigonometric polynomial.

Suppose that we want to study a given function g(x) of period p. We can then standardize the period to 2π by studying the function f(x) =