Differential Equations: Linear, Nonlinear, Ordinary, Partial

by A. C. King , J. Billingham , S. R. Otto

• ISBN13: 9780521816588
• ISBN10: 0521816580
• Format: Hardcover
• Copyright: 2003-06-30
• Publisher: Cambridge University Press

Summary

Finding and interpreting the solutions of differential equations is a central and essential part of applied mathematics. This book aims to enable the reader to develop the required skills needed for a thorough understanding of the subject. The authors focus on the business of constructing solutions analytically, and interpreting their meaning, using rigorous analysis where needed. MATLAB is used extensively to illustrate the material. There are many worked examples based on interesting and unusual real world problems. A large selection of exercises is provided, including several lengthier projects, some of which involve the use of MATLAB. The coverage is broad, ranging from basic second-order ODEs and PDEs, through to techniques for nonlinear differential equations, chaos, asymptotics and control theory. This broad coverage, the authors' clear presentation and the fact that the book has been thoroughly class-tested will increase its attraction to undergraduates at each stage of their studies.

Table of Contents

Preface	ix
Part One: Linear Equations	1	(200)
Variable Coefficient, Second Order, Linear, Ordinary Differential Equations	3	(28)
The Method of Reduction of Order	5	(2)
The Method of Variation of Parameters	7	(4)
Solution by Power Series: The Method of Frobenius	11	(20)
Legendre Functions	31	(27)
Definition of the Legendre Polynomials, Pn(x)	31	(4)
The Generating Function for Pn(x)	35	(3)
Differential and Recurrence Relations Between Legendre Polynomials	38	(1)
Rodrigues' Formula	39	(2)
Orthogonality of the Legendre Polynomials	41	(3)
Physical Applications of the Legendre Polynomials	44	(8)
The Associated Legendre Equation	52	(6)
Bessel Functions	58	(35)
The Gamma Function and the Pockhammer Symbol	58	(2)
Series Solutions of Bessel's Equation	60	(4)
The Generating Function for Jn(x), n an integer	64	(5)
Differential and Recurrence Relations Between Bessel Functions	69	(2)
Modified Bessel Functions	71	(1)
Orthogonality of the Bessel Functions	71	(6)
Inhomogeneous Terms in Bessel's Equation	77	(2)
Solutions Expressible as Bessel Functions	79	(1)
Physical Applications of the Bessel Functions	80	(13)
Boundary Value Problems, Green's Functions and Sturm--Liouville Theory	93	(30)
Inhomogeneous Linear Boundary Value Problems	96	(4)
The Solution of Boundary Value Problems by Eigenfunction Expansions	100	(7)
Sturm--Liouville Systems	107	(16)
Fourier Series and the Fourier Transform	123	(29)
General Fourier Series	127	(6)
The Fourier Transform	133	(8)
Green's Functions Revisited	141	(2)
Solution of Laplace's Equation Using Fourier Transforms	143	(2)
Generalization to Higher Dimensions	145	(7)
Laplace Transforms	152	(23)
Definition and Examples	152	(2)
Properties of the Laplace Transform	154	(3)
The Solution of Ordinary Differential Equations using Laplace Transforms	157	(5)
The Inversion Formula for Laplace Transforms	162	(13)
Classification, Properties and Complex Variable Methods for Second Order Partial Differential Equations	175	(26)
Classification and Properties of Linear, Second Order Partial Differential Equations in Two Independent Variables	175	(11)
Complex Variable Methods for Solving Laplace's Equation	186	(15)
Part Two: Nonlinear Equations and Advanced Techniques	201	(294)
Existence, Uniqueness, Continuity and Comparison of Solutions of Ordinary Differential Equations	203	(14)
Local Existence of Solutions	204	(6)
Uniqueness of Solutions	210	(1)
Dependence of the Solution of the Initial Conditions	211	(1)
Comparison Theorems	212	(5)
Nonlinear Ordinary Differential Equations: Phase Plane Methods	217	(39)
Introduction: The Simple Pendulum	217	(5)
First Order Autonomous Nonlinear Ordinary Differential Equations	222	(2)
Second Order Autonomous Nonlinear Ordinary Differential Equations	224	(25)
Third Order Autonomous Nonlinear Ordinary Differential Equations	249	(7)
Group Theoretical Methods	256	(18)
Lie Groups	257	(4)
Invariants Under Group Action	261	(1)
The Extended Group	262	(1)
Integration of a First Order Equation with a Known Group Invariant	263	(2)
Towards the Systematic Determination of Groups Under Which a First Order Equation is Invariant	265	(1)
Invariants for Second Order Differential Equations	266	(4)
Partial Differential Equations	270	(4)
Asymptotic Methods: Basic Ideas	274	(29)
Asymptotic Expansions	275	(5)
The Asymptotic Evaluation of Integrals	280	(23)
Asymptotic Methods: Differential Equations	303	(69)
An Instructive Analogy: Algebraic Equations	303	(3)
Ordinary Differential Equations	306	(45)
Partial Differential Equations	351	(21)
Stability, Instability and Bifurcations	372	(45)
Zero Eigenvalues and the Centre Manifold Theorem	372	(9)
Lyapunov's Theorems	381	(7)
Bifurcation Theory	388	(29)
Time-Optimal Control in the Phase Plane	417	(30)
Definitions	418	(1)
First Order Equations	418	(4)
Second Order Equations	422	(4)
Examples of Second Order Control Problems	426	(3)
Properties of the Controllable Set	429	(4)
The Controllability Matrix	433	(3)
The Time-Optimal Maximum Principle (TOMP)	436	(11)
An Introduction to Chaotic Systems	447	(48)
Three Simple Chaotic Systems	447	(5)
Mappings	452	(15)
The Poincare Return Map	467	(5)
Homoclinic Tangles	472	(12)
Quantifying Chaos: Lyapunov Exponents and the Lyapunov Spectrum	484	(11)
Appendix 1 Linear Algebra	495	(7)
Appendix 2 Continuity and Differentiability	502	(3)
Appendix 3 Power Series	505	(4)
Appendix 4 Sequences of Functions	509	(2)
Appendix 5 Ordinary Differential Equations	511	(6)
Appendix 6 Complex Variables	517	(9)
Appendix 7 A Short Introduction to MATLAB	526	(8)
Bibliography	534	(2)
Index	536