Handbook of Differential Equations (Errata added)

870 Pages·1998·4.47 MB·English

by  Daniel Zwillinger

Handbook of Difierential Equations
3rd edition
Daniel Zwillinger
Academic Press, 1997
Contents
Preface
Introduction
Introduction to the Electronic Version
How to Use This Book
I.A Deflnitions and Concepts
1 Deflnition of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Alternative Theorems . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Bifurcation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 A Caveat for Partial Difierential Equations . . . . . . . . . . . . 27
5 Chaos in Dynamical Systems . . . . . . . . . . . . . . . . . . . . 29
6 Classiflcation of Partial Difierential Equations . . . . . . . . . . . 36
7 Compatible Systems . . . . . . . . . . . . . . . . . . . . . . . . . 43
8 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . 47
9 Difierential Resultants . . . . . . . . . . . . . . . . . . . . . . . . 50
10 Existence and Uniqueness Theorems . . . . . . . . . . . . . . . . 53
11 Fixed Point Existence Theorems . . . . . . . . . . . . . . . . . . 58
12 Hamilton-Jacobi Theory . . . . . . . . . . . . . . . . . . . . . . . 61
13 Integrability of Systems . . . . . . . . . . . . . . . . . . . . . . . 65
14 Internet Resources . . . . . . . . . . . . . . . . . . . . . . . . . . 71
15 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
16 Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
17 Natural Boundary Conditions for a PDE . . . . . . . . . . . . . . 83
18 Normal Forms: Near-Identity Transformations . . . . . . . . . . 86
19 Random Difierential Equations . . . . . . . . . . . . . . . . . . . 91
20 Self-Adjoint Eigenfunction Problems . . . . . . . . . . . . . . . . 95
21 Stability Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 101
22 Sturm-Liouville Theory . . . . . . . . . . . . . . . . . . . . . . . 103
23 Variational Equations . . . . . . . . . . . . . . . . . . . . . . . . 109
24 Well Posed Difierential Equations . . . . . . . . . . . . . . . . . . 115
25 Wronskians and Fundamental Solutions . . . . . . . . . . . . . . 119
26 Zeros of Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . 123
I.B Transformations
27 Canonical Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
28 Canonical Transformations . . . . . . . . . . . . . . . . . . . . . 132
29 Darboux Transformation . . . . . . . . . . . . . . . . . . . . . . . 135
30 An Involutory Transformation . . . . . . . . . . . . . . . . . . . . 139
31 Liouville Transformation - 1 . . . . . . . . . . . . . . . . . . . . . 141
32 Liouville Transformation - 2 . . . . . . . . . . . . . . . . . . . . . 144
33 Reduction of Linear ODEs to a First Order System . . . . . . . . 146
34 Prufer Transformation . . . . . . . . . . . . . . . . . . . . . . . . 148
35 Modifled Prufer Transformation . . . . . . . . . . . . . . . . . . . 150
36 Transformations of Second Order Linear ODEs - 1 . . . . . . . . 152
37 Transformations of Second Order Linear ODEs - 2 . . . . . . . . 157
38 Transformation of an ODE to an Integral Equation . . . . . . . . 159
39 Miscellaneous ODE Transformations . . . . . . . . . . . . . . . . 162
40 Reduction of PDEs to a First Order System . . . . . . . . . . . . 166
41 Transforming Partial Difierential Equations . . . . . . . . . . . . 168
42 Transformations of Partial Difierential Equations . . . . . . . . . 173
II Exact Analytical Methods
43 Introduction to Exact Analytical Methods . . . . . . . . . . . . . 178
44 Look-Up Technique . . . . . . . . . . . . . . . . . . . . . . . . . . 179
45 Look-Up ODE Forms. . . . . . . . . . . . . . . . . . . . . . . . . 219
II.A Exact Methods for ODEs
46 An Nth Order Equation . . . . . . . . . . . . . . . . . . . . . . . 224
47 Use of the Adjoint Equation . . . . . . . . . . . . . . . . . . . . . 226
48 Autonomous Equations - Independent Variable Missing . . . . . 230
49 Bernoulli Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 235
50 Clairaut’s Equation. . . . . . . . . . . . . . . . . . . . . . . . . . 237
51 Computer-Aided Solution . . . . . . . . . . . . . . . . . . . . . . 240
52 Constant Coe–cient Linear Equations . . . . . . . . . . . . . . . 247
53 Contact Transformation . . . . . . . . . . . . . . . . . . . . . . . 249
54 Delay Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
55 Dependent Variable Missing . . . . . . . . . . . . . . . . . . . . . 260
56 Difierentiation Method . . . . . . . . . . . . . . . . . . . . . . . . 262
57 Difierential Equations with Discontinuities . . . . . . . . . . . . . 264
58 Eigenfunction Expansions . . . . . . . . . . . . . . . . . . . . . . 268
59 Equidimensional-in-x Equations . . . . . . . . . . . . . . . . . . . 275
60 Equidimensional-in-y Equations . . . . . . . . . . . . . . . . . . . 278
61 Euler Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
62 Exact First Order Equations . . . . . . . . . . . . . . . . . . . . 284
63 Exact Second Order Equations . . . . . . . . . . . . . . . . . . . 287
64 Exact Nth Order Equations . . . . . . . . . . . . . . . . . . . . . 290
65 Factoring Equations . . . . . . . . . . . . . . . . . . . . . . . . . 292
66 Factoring Operators . . . . . . . . . . . . . . . . . . . . . . . . . 294
67 Factorization Method . . . . . . . . . . . . . . . . . . . . . . . . 300
68 Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . . . 303
69 Fractional Difierential Equations . . . . . . . . . . . . . . . . . . 308
70 Free Boundary Problems . . . . . . . . . . . . . . . . . . . . . . . 311
71 Generating Functions. . . . . . . . . . . . . . . . . . . . . . . . . 315
72 Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 318
73 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . 327
74 Method of Images . . . . . . . . . . . . . . . . . . . . . . . . . . 330
75 Integrable Combinations . . . . . . . . . . . . . . . . . . . . . . . 334
76 Integral Representation: Laplace’s Method. . . . . . . . . . . . . 336
77 Integral Transforms: Finite Intervals . . . . . . . . . . . . . . . . 342
78 Integral Transforms: Inflnite Intervals . . . . . . . . . . . . . . . 347
79 Integrating Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 356
80 Interchanging Dependent and Independent Variables . . . . . . . 360
81 Lagrange’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . 363
82 Lie Groups: ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . 366
83 Operational Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 379
84 Pfa–an Difierential Equations. . . . . . . . . . . . . . . . . . . . 384
85 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . 389
86 Riccati Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
87 Matrix Riccati Equations . . . . . . . . . . . . . . . . . . . . . . 395
88 Scale Invariant Equations . . . . . . . . . . . . . . . . . . . . . . 398
89 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . 401
90 Series Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
91 Equations Solvable for x . . . . . . . . . . . . . . . . . . . . . . . 409
92 Equations Solvable for y . . . . . . . . . . . . . . . . . . . . . . . 411
93 Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
94 Method of Undetermined Coe–cients. . . . . . . . . . . . . . . . 415
95 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . 418
96 Vector Ordinary Difierential Equations . . . . . . . . . . . . . . . 421
II.B Exact Methods for PDEs
97 Backlund Transformations . . . . . . . . . . . . . . . . . . . . . . 428
98 Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . 432
99 Characteristic Strip Equations . . . . . . . . . . . . . . . . . . . 438
100 Conformal Mappings . . . . . . . . . . . . . . . . . . . . . . . . . 441
101 Method of Descent . . . . . . . . . . . . . . . . . . . . . . . . . . 446
102 Diagonalization of a Linear System of PDEs . . . . . . . . . . . . 449
103 Duhamel’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . 451
104 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
105 Hodograph Transformation . . . . . . . . . . . . . . . . . . . . . 456
106 Inverse Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . 460
107 Jacobi’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
108 Legendre Transformation . . . . . . . . . . . . . . . . . . . . . . 467
109 Lie Groups: PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . 471
110 Poisson Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
111 Riemann’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . 481
112 Separation of Variables. . . . . . . . . . . . . . . . . . . . . . . . 487
113 Separable Equations: Stackel Matrix . . . . . . . . . . . . . . . . 494
114 Similarity Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 497
115 Exact Solutions to the Wave Equation . . . . . . . . . . . . . . . 501
116 Wiener-Hopf Technique . . . . . . . . . . . . . . . . . . . . . . . 505
III Approximate Analytical Methods
117 Introduction to Approximate Analysis . . . . . . . . . . . . . . . 510
118 Chaplygin’s Method . . . . . . . . . . . . . . . . . . . . . . . . . 511
119 Collocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514
120 Dominant Balance . . . . . . . . . . . . . . . . . . . . . . . . . . 517
121 Equation Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . 520
122 Floquet Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
123 Graphical Analysis: The Phase Plane . . . . . . . . . . . . . . . 526
124 Graphical Analysis: The Tangent Field. . . . . . . . . . . . . . . 532
125 Harmonic Balance . . . . . . . . . . . . . . . . . . . . . . . . . . 535
126 Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
127 Integral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 542
128 Interval Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
129 Least Squares Method . . . . . . . . . . . . . . . . . . . . . . . . 549
130 Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . . . . 551
131 Equivalent Linearization and Nonlinearization . . . . . . . . . . . 555
132 Maximum Principles . . . . . . . . . . . . . . . . . . . . . . . . . 560
133 McGarvey Iteration Technique . . . . . . . . . . . . . . . . . . . 566
134 Moment Equations: Closure . . . . . . . . . . . . . . . . . . . . . 568
135 Moment Equations: Ito Calculus . . . . . . . . . . . . . . . . . . 572
136 Monge’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
137 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 578
138 Pade Approximants . . . . . . . . . . . . . . . . . . . . . . . . . 582
139 Perturbation Method: Method of Averaging . . . . . . . . . . . . 586
140 Perturbation Method: Boundary Layer Method . . . . . . . . . . 590
141 Perturbation Method: Functional Iteration . . . . . . . . . . . . 598
142 Perturbation Method: Multiple Scales . . . . . . . . . . . . . . . 605
143 Perturbation Method: Regular Perturbation . . . . . . . . . . . . 610
144 Perturbation Method: Strained Coordinates . . . . . . . . . . . . 614
145 Picard Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
146 Reversion Method . . . . . . . . . . . . . . . . . . . . . . . . . . 621
147 Singular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 623
148 Soliton-Type Solutions . . . . . . . . . . . . . . . . . . . . . . . . 626
149 Stochastic Limit Theorems . . . . . . . . . . . . . . . . . . . . . 629
150 Taylor Series Solutions . . . . . . . . . . . . . . . . . . . . . . . . 632
151 Variational Method: Eigenvalue Approximation . . . . . . . . . . 635
152 Variational Method: Rayleigh-Ritz . . . . . . . . . . . . . . . . . 638
153 WKB Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
IV.A Numerical Methods: Concepts
154 Introduction to Numerical Methods. . . . . . . . . . . . . . . . . 648
155 Deflnition of Terms for Numerical Methods . . . . . . . . . . . . 651
156 Available Software . . . . . . . . . . . . . . . . . . . . . . . . . . 654
157 Finite Difierence Formulas . . . . . . . . . . . . . . . . . . . . . . 661
158 Finite Difierence Methodology. . . . . . . . . . . . . . . . . . . . 670
159 Grid Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
160 Richardson Extrapolation . . . . . . . . . . . . . . . . . . . . . . 679
161 Stability: ODE Approximations . . . . . . . . . . . . . . . . . . . 683
162 Stability: Courant Criterion . . . . . . . . . . . . . . . . . . . . . 688
163 Stability: Von Neumann Test . . . . . . . . . . . . . . . . . . . . 692
164 Testing Difierential Equation Routines . . . . . . . . . . . . . . . 694
IV.B Numerical Methods for ODEs
165 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . 698
166 Boundary Value Problems: Box Method . . . . . . . . . . . . . . 701
167 Boundary Value Problems: Shooting Method . . . . . . . . . . . 706
168 Continuation Method . . . . . . . . . . . . . . . . . . . . . . . . 710
169 Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . 713
170 Cosine Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716
171 Difierential Algebraic Equations. . . . . . . . . . . . . . . . . . . 720
172 Eigenvalue/Eigenfunction Problems. . . . . . . . . . . . . . . . . 726
173 Euler’s Forward Method . . . . . . . . . . . . . . . . . . . . . . . 730
174 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . 734
175 Hybrid Computer Methods . . . . . . . . . . . . . . . . . . . . . 744
176 Invariant Imbedding . . . . . . . . . . . . . . . . . . . . . . . . . 747
177 Multigrid Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 752
178 Parallel Computer Methods . . . . . . . . . . . . . . . . . . . . . 755
179 Predictor-Corrector Methods . . . . . . . . . . . . . . . . . . . . 759
180 Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . 763
181 Stifi Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770
182 Integrating Stochastic Equations . . . . . . . . . . . . . . . . . . 775
183 Symplectic Integration . . . . . . . . . . . . . . . . . . . . . . . . 780
184 Use of Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784
185 Weighted Residual Methods . . . . . . . . . . . . . . . . . . . . . 786
IV.C Numerical Methods for PDEs
186 Boundary Element Method . . . . . . . . . . . . . . . . . . . . . 792
187 Difierential Quadrature . . . . . . . . . . . . . . . . . . . . . . . 796
188 Domain Decomposition . . . . . . . . . . . . . . . . . . . . . . . 800
189 Elliptic Equations: Finite Difierences . . . . . . . . . . . . . . . . 805
190 Elliptic Equations: Monte-Carlo Method . . . . . . . . . . . . . . 810
191 Elliptic Equations: Relaxation . . . . . . . . . . . . . . . . . . . 814
192 Hyperbolic Equations: Method of Characteristics . . . . . . . . . 818
193 Hyperbolic Equations: Finite Difierences . . . . . . . . . . . . . . 824
194 Lattice Gas Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 828
195 Method of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 831
196 Parabolic Equations: Explicit Method . . . . . . . . . . . . . . . 835
197 Parabolic Equations: Implicit Method . . . . . . . . . . . . . . . 839
198 Parabolic Equations: Monte-Carlo Method . . . . . . . . . . . . 844
199 Pseudospectral Method . . . . . . . . . . . . . . . . . . . . . . . 851
Mathematical Nomenclature
Errata
Preface
WhenIwasagraduatestudentinappliedmathematicsattheCaliforniaInstitute
of Technology, we solved many di(cid:11)erential equations (both ordinary di(cid:11)erential
equations and partial di(cid:11)erential equations). Given a di(cid:11)erential equation to
solve, I would thinkof all thetechniquesI knewthat might solvethat equation.
Eventually, the number of techniques I knew became so large that I began to
forget some. Then, I would have to consult books on di(cid:11)erential equations to
familiarize myself with a techniquethat I remembered only vaguely. This was a
slowprocessandoftenunrewarding;Imightspendtwentyminutesreadingabout
a technique only to realize that it did not apply to the equation I was trying to
solve.
Eventually, I created a list of the di(cid:11)erent techniques that I knew. Each
techniquehadabriefdescriptionofhowthemethodwasusedandtowhattypes
of equations it applied. As I learned more techniques, they were added to the
list. This book is a direct result of that list.
At Caltech we were taught the usefulness of approximate analytic solutions
and the necessity of being able to solve di(cid:11)erential equations numerically when
exactorapproximatesolutiontechniquescouldnotbefound. Hence,approximate
analyticalsolutiontechniquesandnumericalsolutiontechniqueswerealsoadded
to thelist.
Given a di(cid:11)erential equation to analyze, most people spend only a small
amount of time using analytical tools and then use a computer to see what
the solution \looks like." Because this procedure is so prevalent, this edition
includes an expanded section on numerical methods. New sections on sympletic
integration (see page 780) and theuse of wavelets (see page 784) also havebeen
added.
Inwritingthisbook,Ihaveassumedthatthereaderisfamiliarwithdi(cid:11)eren-
tial equations and their solutions. The object of this book is not to teach novel
techniquesbut to provide a handy reference to many popular techniques. All of
thetechniquesincludedareelementaryintheusualmathematicalsense;because
thisbookisdesignedtobefunctionalitdoesnotincludemanyabstractmethods
of limited applicability. This handbook has been designed to serve as both a
reference book and as a complement to a text on di(cid:11)erential equations. Each
techniquedescribed is accompanied by several references; these allow each topic
to bestudied in more detail.
Itishopedthatthisbookwillbeusedbystudentstakingcoursesindi(cid:11)erential
equations (at either the undergraduate or the graduate level). It will introduce
the student to more techniques than they usually see in a di(cid:11)erential equations
xv
xvi Preface
classandwillillustratemanydi(cid:11)erenttypesoftechniques. Furthermore,itshould
actasaconcisereferenceforthetechniquesthatastudenthaslearned. Thisbook
shouldalsobeusefulforthepracticingengineerorscientistwhosolvesdi(cid:11)erential
equations on an occasional basis.
A feature of this book is that it has sections dealing with stochastic di(cid:11)er-
ential equations and delay di(cid:11)erential equations as well as ordinary di(cid:11)erential
equationsandpartialdi(cid:11)erentialequations. Stochasticdi(cid:11)erentialequationsand
delaydi(cid:11)erentialequationsareoftenstudiedonlyinadvancedtextsandcourses;
yet, the techniques used to analyze these equations are easy to understand and
easy to apply.
Had this book been available when I was a graduate student, it would have
saved me much time. It has saved me time in solving problems that arose from
my own work in industry (the Jet Propulsion Laboratory, Sandia Laboratories,
EXXONResearch and Engineering, The MITRE Corporation, BBN).
Parts of the text have been utilized in di(cid:11)erential equations classes at the
Rensselaer Polytechnic Institute. Students’ comments have been used to clarify
thetext. Unfortunately,theremaystillbesomeerrorsinthetext;Iwouldgreatly
appreciate receiving notice of any such errors.
Many people have been kind enough to send in suggestions for additional
material to add and corrections of existing material. There are too many to
name them individually, but Alain Moussiaux stands out for all of the checking
hehas performed. Thank you all!
This book is dedicated tomy wife, Janet Taylor.
Boston, Mass. 1997 Daniel Zwillinger
[email protected]
CD-ROM Handbook of Di(cid:11)erential Equations (cid:13)cAcademic Press 1997
Introduction
Thisbookisacompilationofthemostimportantandwidelyapplicablemethods
for solving and approximating di(cid:11)erential equations. As a reference book, it
providesconvenient access tothese methods and contains examples of theiruse.
The book is divided into four parts. The (cid:12)rst part is a collection of trans-
formations and general ideas about di(cid:11)erential equations. This section of the
book describes thetechniquesneededtodeterminewhethera partialdi(cid:11)erential
equation is well posed, what the \natural" boundary conditions are, and many
other things. At the beginning of this section is a list of de(cid:12)nitions for many of
theterms that describe di(cid:11)erential equations and their solutions.
The second part of the book is a collection of exact analytical solution
techniques for di(cid:11)erential equations. The techniques are listed (nearly) alpha-
betically. First is a collection of techniques for ordinary di(cid:11)erential equations,
thenacollectionoftechniquesforpartialdi(cid:11)erentialequations. Thosetechniques
that can be used for both ordinary di(cid:11)erential equations and partial di(cid:11)erential
equations havea star ((cid:3)) nextto themethod name. For nearly every technique,
thefollowing are given:
(cid:15) thetypesof equations to which themethod is applicable
(cid:15) theidea behind themethod
(cid:15) theprocedurefor carrying out themethod
(cid:15) at least one simple example of themethod
(cid:15) any cautions that should be exercised
(cid:15) notes for more advanced users
(cid:15) references to the literature for more discussion or more examples
The material for each method has deliberately been kept short to simplify
use. Proofs havebeen intentionally omitted.
Itishopedthat,byworkingthroughthesimpleexample(s)given,themethod
willbeunderstood. Enoughinsightshouldbegainedfromworkingtheexample(s)
to apply the method to other equations. Further references are given for each
methodsothattheprinciplemaybestudiedinmoredetailorsomoreexamples
may be seen. Note that not all of the references listed at the end of a method
may be referred to in thetext.
Theauthorhasfoundthatcomputerlanguagesthatperformsymbolicmanip-
ulations(e.g.,Macsyma,Maple,andMathematica)areveryusefulforperforming
the calculations necessary to analyze di(cid:11)erential equations. Hence, there is
a section comparing the capabilities of these languages and, for some exact
analytical techniques,examples of theiruse are given.
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