Differential and Integral Equations

387 Pages·2006·1.94 MB·English

by  Peter Collins

Differential and Integral Equations
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DIFFERENTIAL AND INTEGRAL EQUATIONS
Peter J. Collins
Senior Research Fellow, St Edmund Hall, Oxford
OXFORD
UNIVERSITY PRESS
OXFORD
UNIVERSITYPRESS
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Typeset by the author and Dr J.D. Lotay using LaTeX
Printed in Great Britain
on acid-free paper by
Antony Rowe Ltd., Chippenham, Wiltshire
ISBN 0-19-853382-9 978-0-19-853382-5
0-19-929789-4 978-0-19-929789-4 (Pbk)
Preface
It is a truism that differential and integral equations lie at the heart of mathematics,
being the inspiration of so many theoretical advances in analysis and applying to a wide
range of situations in the natural and social sciences.
Inafirst-yearcourseondifferentialequations, itisusualtogivetechniquesforsolving
some very particular equations. In this ‘second course’, the theoretical basis underlying
first-year work is explored and extended. Much of the discussion involves consideration
of the complementary topic of integral equations, which form an important subject of
study in their own right. Some basic results on integral equations of major significance
are included in later chapters.
The approach here is thoroughly classical, depending only on some results from real
variable theory (which are reviewed in Chapter 0) and from linear algebra (which are
summarised where needed), and some knowledge of the solution of elementary ordinary
differential equations (which is, in any case, treated in the Appendix). However, presen-
tation and proofs have been chosen with a view to useful generalisations in important
areas of functional analysis. I hope to satisfy, in both content and presentation, the (as
I see it) uncontradictory demands of the ‘pure’ and the ‘applied’ mathematician.
Applications of the material included in these notes are myriad. With the limited
space available here, a decision has been made to dwell on the understanding of basic
ideas and therefore only to include necessarily short discussion of examples from other
fields. The resulting gap may be filled by lectures and further reading.
Theworkherefirstappearedintheacademicyear1986–87,asacourseintheMethods
of Applied Mathematics, designed for mathematicians, physicists and economists in their
junior and senior undergraduate years, at the College of William and Mary in Virginia.
It has subsequently appeared, in more extended incarnations, as Oxford lecture notes
designed to cover the range of material most appropriate and most useful for modern
mathematicscoursesinEnglish-speakinguniversities. Theauthorwaspleasedtodiscover
that it has been found helpful not only to undergraduates who use mathematics, but also
to other non-mathematicians (for example, research chemists) as a handy and accessible
reference.
Thanks go to the many colleagues and students who, over the years, have taken the
trouble to discuss points in the text with me; in particular, Dominic Welsh and Alan Day
for their constructive criticism. Most recently, I have become indebted to Jason Lotay
for his invaluable help, especially in updating the typesetting and presentation of the
text, and to my patient editor at O.U.P.
Oxford, November 2005 P.J.C.
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Contents
Preface v
How to use this book xi
Prerequisites xiii
0 Some Preliminaries 1
1 Integral Equations and Picard’s Method 5
1.1 Integral equations and their relationship to differential equations . . . . 5
1.2 Picard’s method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Existence and Uniqueness 19
2.1 First-order differential equations in a single independent variable . . . . 20
2.2 Two simultaneous equations in a single variable . . . . . . . . . . . . . . 26
2.3 A second-order equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 The Homogeneous Linear Equation and Wronskians 33
3.1 Some linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Wronskians and the linear independence of solutions of the second-order
homogeneous linear equation . . . . . . . . . . . . . . . . . . . . . . . . 36
4 The Non-Homogeneous Linear Equation 41
4.1 The method of variation of parameters . . . . . . . . . . . . . . . . . . . 43
4.2 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
viii Contents
5 First-Order Partial Differential Equations 59
5.1 Characteristics and some geometrical considerations . . . . . . . . . . . 60
5.2 Solving characteristic equations . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 General solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.4 Fitting boundary conditions to general solutions . . . . . . . . . . . . . 70
5.5 Parametric solutions and domains of definition . . . . . . . . . . . . . . 75
5.6 A geometric interpretation of an analytic condition . . . . . . . . . . . . 83
6 Second-Order Partial Differential Equations 85
6.1 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2 Reduction to canonical form . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3 General solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.4 Problems involving boundary conditions . . . . . . . . . . . . . . . . . . 103
6.5 Appendix: technique in the use of the chain rule . . . . . . . . . . . . . 113
7 The Diffusion and Wave Equations and the Equation of Laplace 115
7.1 The equations to be considered . . . . . . . . . . . . . . . . . . . . . . . 116
7.2 One-dimensional heat conduction . . . . . . . . . . . . . . . . . . . . . . 119
7.3 Transverse waves in a finite string . . . . . . . . . . . . . . . . . . . . . . 123
7.4 Separated solutions of Laplace’s equation in polar co-ordinates and
Legendre’s equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.5 The Dirichlet problem and its solution for the disc . . . . . . . . . . . . 133
7.6 Radially symmetric solutions of the two-dimensional wave equation and
Bessel’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.7 Existence and uniqueness of solutions, well-posed problems . . . . . . . 138
7.8 Appendix: proof of the Mean Value Theorem for harmonic functions . . 144
8 The Fredholm Alternative 149
8.1 A simple case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.2 Some algebraic preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.3 The Fredholm Alternative Theorem . . . . . . . . . . . . . . . . . . . . . 155
8.4 A worked example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Contents ix
9 Hilbert–Schmidt Theory 165
9.1 Eigenvalues are real and eigenfunctions corresponding to distinct
eigenvalues are orthogonal . . . . . . . . . . . . . . . . . . . . . . . . . . 166
9.2 Orthonormal families of functions and Bessel’s inequality . . . . . . . . . 168
9.3 Some results about eigenvalues deducible from Bessel’s inequality . . . . 169
9.4 Description of the sets of all eigenvalues and all eigenfunctions. . . . . . 173
9.5 The Expansion Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
10 Iterative Methods and Neumann Series 181
10.1 An example of Picard’s method . . . . . . . . . . . . . . . . . . . . . . . 181
10.2 Powers of an integral operator . . . . . . . . . . . . . . . . . . . . . . . . 183
10.3 Iterated kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
10.4 Neumann series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
10.5 A remark on the convergence of iterative methods. . . . . . . . . . . . . 188
11 The Calculus of Variations 189
11.1 The fundamental problem . . . . . . . . . . . . . . . . . . . . . . . . . . 189
11.2 Some classical examples from mechanics and geometry . . . . . . . . . . 191
11.3 The derivation of Euler’s equation for the fundamental problem . . . . . 196
(cid:1)
11.4 The special case F = F(y,y ) . . . . . . . . . . . . . . . . . . . . . . . . 198
11.5 When F contains higher derivatives of y . . . . . . . . . . . . . . . . . . 201
11.6 When F contains more dependent functions . . . . . . . . . . . . . . . . 203
11.7 When F contains more independent variables . . . . . . . . . . . . . . . 208
11.8 Integral constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11.9 Non-integral constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
11.10 Varying boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 218
12 The Sturm–Liouville Equation 225
12.1 Some elementary results on eigenfunctions and eigenvalues . . . . . . . . 226
12.2 The Sturm–Liouville Theorem . . . . . . . . . . . . . . . . . . . . . . . . 229
12.3 Derivation from a variational principle . . . . . . . . . . . . . . . . . . . 233
12.4 Some singular equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
12.5 The Rayleigh–Ritz method. . . . . . . . . . . . . . . . . . . . . . . . . . 239