Handbook of Differential Equations

Handbook of Differential Equations is a handy reference to many popular techniques for solving and approximating differential equations, including exact analytical methods, approximate analytical methods, and numerical methods. Topics covered range from transformations and constant coefficient linear equations to finite and infinite intervals, along with conformal mappings and the perturbation method.

Comprised of 180 chapters, this book begins with an introduction to transformations as well as general ideas about differential equations and how they are solved, together with the techniques needed to determine if a partial differential equation is well-posed or what the "natural" boundary conditions are. Subsequent sections focus on exact and approximate analytical solution techniques for differential equations, along with numerical methods for ordinary and partial differential equations.

This monograph is intended for students taking courses in differential equations at either the undergraduate or graduate level, and should also be useful for practicing engineers or scientists who solve differential equations on an occasional basis.

Autorentext

Dr. Daniel Zwillinger is a Senior Principal Systems Engineer for the Raytheon Company. He was a systems requirements "book boss for the Cobra Judy Replacement (CJR) ship and was a requirements and test lead for tracking on the Ungraded Early Warning Radars (UEWR). He has improved the Zumwalt destroyer's software accreditation process and he was test lead on an Active Electronically Scanned Array (AESA) radar. Dan is a subject matter expert (SME) in Design for Six Sigma (DFSS) and is a DFSS SME in Test Optimization, Critical Chain Program Management, and Voice of the Customer. He is currently leading a project creating Trust in Autonomous Systems. At Raytheon, he twice won the President's award for best Six Sigma project of the year: on converting planning packages to work packages for the Patriot missile, and for revising Raytheon's timecard system. He has managed the Six Sigma white belt training program. Prior to Raytheon, Dan worked at Sandia Labs, JPL, Exxon, MITRE, IDA, BBN, and The Mathworks (where he developed an early version of their Statistics Toolbox).

For ten years, Zwillinger was owner and president of Aztec Corporation. As a small business, Aztec won several Small Business Innovation Research (SBIR) contracts. The company also created several software packages for publishing companies. Prior to Aztec, Zwillinger was a college professor at Rensselaer Polytechnic Institute in the department of mathematics.

Dan has written several books on mathematics on the topics of differential equations, integration, statistics, and general mathematics. He is editor-in-chief of the Chemical Rubber Company's (CRC's) "Standard Mathematical Tables and Formulae , and is on the editorial board for CRC's "Handbook of Chemistry and Physics . Zwillinger holds a bachelor's degree in mathematics from the Massachusetts Institute of Technology (MIT). He earned his doctorate in applied mathematics from the California Institute of Technology (Caltech). Zwillinger is a certified Raytheon Six Sigma Expert and an ASQ certified Six Sigma Black Belt. He also holds a pilot's license.

Inhalt

Preface
Introduction
How to Use This Book
I.A Definitions and Concepts
1 Definition of Terms
2 Alternative Theorems
3 Bifurcation Theory
4 A Caveat for Partial Differential Equations
5 Classification of Partial Differential Equations
6 Compatible Systems
7 Conservation Laws
8 Differential Resultants
9 Fixed Point Existence Theorems
10 Hamilton-Jacobi Theory
11 Limit Cycles
12 Natural Boundary Conditions for a PDE
13 Self-Adjoint Eigenfunction Problems
14 Sturm-Liouville Theory
15 Variational Equations
16 Well-Posedness of Differential Equations
17 Wronskians and Fundamental Solutions
I.B Transformations
18 Canonical Forms
19 Canonical Transformations
20 Darboux Transformation
21 An Involutory Transformation
22 Liouville Transformation - 1
23 Liouville Transformation - 2
24 Reduction of Linear ODEs to a First Order System
25 Transformations of Second Order Linear ODEs - 1
26 Transformations of Second Order Linear ODEs - 2
27 Transformation of an ODE to an Integral Equation
28 Miscellaneous ODE Transformations
29 Reduction of PDEs to a First Order System
30 Transforming Partial DifFerential Equations
31 Transformations of Partial DifFerential Equations
II Exact Analytical Methods
32 Introduction to Exact Analytical Methods
33 Look Up Technique
II.A Exact Methods for ODEs
34 An N-th Order Equation
35 Use of the Adjoint Equation*
36 Autonomous Equations
37 Bernoulli Equation
38 Clairaut's Equation
39 Computer-Aided Solution
40 Constant Coefficient Linear Equations
41 Contact Transformation
42 Delay Equations
43 Dependent Variable Missing
44 Differentiation Method
45 Differential Equations with Discontinuities*
46 Eigenfunction Expansions*
47 Equidimensional-in-x Equations
48 Equidimensional-in-y Equations
49 Euler Equations
50 Exact First Order Equations
51 Exact Second Order Equations
52 Exact N-th Order Equations
53 Factoring Equations*
54 Factoring Operators*
55 Factorization Method
56 Fokker-Planck Equation
57 Fractional Differential Equations*
58 Free Boundary Problems*
59 Generating Functions*
60 Green's Functions*
61 Homogeneous Equations
62 Method of Images*
63 Integrable Combinations
64 Integral Representations: Laplace's Method*
65 Integral Transforms: Finite Intervals*
66 Integral Transforms: Infinite Intervals*
67 Integrating Factors*
68 Interchanging Dependent and Independent Variables
69 Lagrange's Equation
70 Lie Groups: ODEs
71 Operational Calculus*
72 PfafSan Differential Equations
73 Prüfer Substitution
74 Reduction of Order
75 Riccati Equation - 1
76 Riccati Equation - 2
77 Matrix Riccati Equations
78 Scale Invariant Equations
79 Separable Equations
80 Series Solution*
81 Equations Solvable for x
82 Equations Solvable for y
83 Superposition*
84 Method of Undetermined Coefficients*
85 Variation of Parameters
86 Vector Ordinary Differential Equations
II.B Exact Methods for PDEs
87 Bäcklund Transformations
88 Method of Characteristics
89 Characteristic Strip Equations
90 Conformai Mappings
91 Method of Descent
92 Diagonalization of a Linear System of PDEs
93 Duhamel's Principle
94 Hodograph Transformation
95 Inverse Scattering
96 Jacobi's Method
97 Legendre Transformation
98 Lie Groups: PDEs
99 Poisson Formula
100 Riemann's Method
101 Separation of Variables
102 Similarity Methods
103 Exact Solutions to the Wave Equation
104 Wiener-Hopf Technique
III Approximate Analytical Methods
105 Introduction to Approximate Analysis
106 Chaplygin's Method
107 Collocation
108 Dominant Balance
109 Equation Splitting
110 Equivalent Linearization
111 Equivalent Nonlinearization
112 Floquet Theory
113 Graphical Analysis: The Phase Plane
114 Graphical Analysis: The Tangent Field
115 Harmonic Balance
116 Homogenization
117 Integral Methods
118 Interval Analysis
119 Least Squares Method
120 Liapunov Functions
121 Maximum Principles
122 McGarvey Iteration Technique
123 Moment Equations: Closure
124 Moment Equations: Itô Calculus
125 Monge's Method
126 Newton's Method
127 Padé Approximants
128 Perturbation Method: Method of Averaging
129 Perturbation Method: Boundary Layer Method
130 Perturbation Method: Functional Iteration
131 Perturbation Method: Multiple Scales
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