Handbook of Differential Equations

Handbook of Differential Equations

2nd Edition - January 28, 1992

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  • Author: Daniel Zwillinger
  • eBook ISBN: 9781483263960

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Handbook of Differential Equations, Second Edition is a handy reference to many popular techniques for solving and approximating differential equations, including numerical methods and exact and approximate analytical methods. Topics covered range from transformations and constant coefficient linear equations to Picard iteration, along with conformal mappings and inverse scattering. Comprised of 192 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.

Table of Contents

  • Preface


    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 Chaos in Dynamical Systems

    6 Classification of Partial Differential Equations

    7 Compatible Systems

    8 Conservation Laws

    9 Differential Resultants

    10 Existence and Uniqueness Theorems

    11 Fixed Point Existence Theorems

    12 Hamilton-Jacobi Theory

    13 Inverse Problems

    14 Limit Cycles

    15 Natural Boundary Conditions for a PDE

    16 Normal Forms: Near-Identity Transformations

    17 Self-Adjoint Eigenfunction Problems

    18 Stability Theorems

    19 Sturm-Liouville Theory

    20 Variational Equations

    21 Well-Posedness of Differential Equations

    22 Wronskians and Fundamental Solutions

    I.B Transformations

    23 Canonical Forms

    24 Canonical Transformations

    25 Darboux Transformation

    26 An Involutory Transformation

    27 Liouville Transformation - 1

    28 Liouville Transformation - 2

    29 Reduction of Linear ODEs to a First Order System

    30 Prüfer Transformation

    31 Modified Prüfer Transformation

    32 Transformations of Second Order Linear ODEs - 1

    33 Transformations of Second Order Linear ODEs - 2

    34 Transformation of an ODE to an Integral Equation

    35 Miscellaneous ODE Transformations

    36 Reduction of PDEs to a First Order System

    37 Transforming Partial Differential Equations

    38 Transformations of Partial Differential Equations

    II Exact Analytical Methods

    39 Introduction to Exact Analytical Methods

    40 Look Up Technique

    41 Look Up ODE Forms

    II.A Exact Methods for ODEs

    42 An N-th Order Equation

    43 Use of the Adjoint Equation

    44 Autonomous Equations

    45 Bernoulli Equation

    46 Clairaut's Equation

    47 Computer-Aided Solution

    48 Constant Coefficient Linear Equations

    49 Contact Transformation

    50 Delay Equations

    51 Dependent Variable Missing

    52 Differentiation Method

    53 Differential Equations with Discontinuities*

    54 Eigenfunction Expansions*

    55 Equidimensional-in-x Equations

    56 Equidimensional-in-y Equations

    57 Euler Equations

    58 Exact First Order Equations

    59 Exact Second Order Equations

    60 Exact N-th Order Equations

    61 Factoring Equations*

    62 Factoring Operators*

    63 Factorization Method

    64 Fokker-Planck Equation

    65 Fractional Differential Equations*

    66 Free Boundary Problems*

    67 Generating Functions*

    68 Green's Functions*

    69 Homogeneous Equations

    70 Method of Images*

    71 Integrable Combinations

    72 Integral Representations: Laplace's Method*

    73 Integral Transforms: Finite Intervals*

    74 Integral Transforms: Infinite Intervals*

    75 Integrating Factors*

    76 Interchanging Dependent and Independent Variables

    77 Lagrange's Equation

    78 Lie Groups: ODEs

    79 Operational Calculus*

    80 Pfaffian Differential Equations

    81 Reduction of Order

    82 Riccati Equation

    83 Matrix Riccati Equations

    84 Scale Invariant Equations

    85 Separable Equations

    86 Series Solution*

    87 Equations Solvable for x

    88 Equations Solvable for y

    89 Superposition*

    90 Method of Undetermined Coefficients*

    91 Variation of Parameters

    92 Vector Ordinary Differential Equations

    II.B Exact Methods for PDEs

    93 Bäcklund Transformations

    94 Method of Characteristics

    95 Characteristic Strip Equations

    96 Conformai Mappings

    97 Method of Descent

    98 Diagonalization of a Linear System of PDEs

    99 Duhamel's Principle

    100 Exact Equations

    101 Hodograph Transformation

    102 Inverse Scattering

    103 Jacobi's Method

    104 Legendre Transformation

    105 Lie Groups: PDEs

    106 Poisson Formula

    107 Riemann's Method

    108 Separation of Variables

    109 Similarity Methods

    110 Exact Solutions to the Wave Equation

    111 Wiener-Hopf Technique

    III Approximate Analytical Methods

    112 Introduction to Approximate Analysis

    113 Chaplygin's Method

    114 Collocation

    115 Dominant Balance

    116 Equation Splitting

    117 Floquet Theory

    118 Graphical Analysis: The Phase Plane

    119 Graphical Analysis: The Tangent Field

    120 Harmonic Balance

    121 Homogenization

    122 Integral Methods

    123 Interval Analysis

    124 Least Squares Method

    125 Lyapunov Functions

    126 Equivalent Linearization and Nonlinearization

    127 Maximum Principles

    128 McGarvey Iteration Technique

    129 Moment Equations: Closure

    130 Moment Equations: Itô Calculus

    131 Monge's Method

    132 Newton's Method

    133 Padé Approximants

    134 Perturbation Method: Method of Averaging

    135 Perturbation Method: Boundary Layer Method

    136 Perturbation Method: Functional Iteration

    137 Perturbation Method: Multiple Scales

    138 Perturbation Method: Regular Perturbation

    139 Perturbation Method: Strained Coordinates

    140 Picard Iteration

    141 Reversion Method

    142 Singular Solutions

    143 Soliton Type Solutions

    144 Stochastic Limit Theorems

    145 Taylor Series Solutions

    146 Variational Method: Eigenvalue Approximation

    147 Variational Method: Rayleigh-Ritz

    148 WKB Method

    IV.A Numerical Methods: Concepts

    149 Introduction to Numerical Methods

    150 Definition of Terms for Numerical Methods

    151 Available Software

    152 Finite Difference Methodology

    153 Finite Difference Formulas

    154 Excerpts from GAMS

    155 Grid Generation

    156 Richardson Extrapolation

    157 Stability: ODE Approximations

    158 Stability: Courant Criterion

    159 Stability: Von Neumann Test

    IV.B Numerical Methods for ODEs

    160 Analytic Continuation*

    161 Boundary Value Problems: Box Method

    162 Boundary Value Problems: Shooting Method*

    163 Continuation Method*

    164 Continued Fractions

    165 Cosine Method*

    166 Differential Algebraic Equations

    167 Eigenvalue/Eigenfunction Problems

    168 Euler's Forward Method

    169 Finite Element Method*

    170 Hybrid Computer Methods*

    171 Invariant Imbedding*

    172 Multigrid Methods*

    173 Parallel Computer Methods

    174 Predictor-Corrector Methods

    175 Runge-Kutta Methods

    176 Stiff Equations*

    177 Integrating Stochastic Equations

    178 Weighted Residual Methods*

    IV.C Numerical Methods for PDEs

    179 Boundary Element Method

    180 Differential Quadrature

    181 Domain Decomposition

    182 Elliptic Equations: Finite Differences

    183 Elliptic Equations: Monte Carlo Method

    184 Elliptic Equations: Relaxation

    185 Hyperbolic Equations: Method of Characteristics

    186 Hyperbolic Equations: Finite Differences

    187 Lattice Gas Dynamics

    188 Method of Lines

    189 Parabolic Equations: Explicit Method

    190 Parabolic Equations: Implicit Method

    191 Parabolic Equations: Monte Carlo Method

    192 Pseudo-Spectral Method

    Mathematical Nomenclature

    Differential Equation Index


Product details

  • No. of pages: 808
  • Language: English
  • Copyright: © Academic Press 1992
  • Published: January 28, 1992
  • Imprint: Academic Press
  • eBook ISBN: 9781483263960

About the Author

Daniel Zwillinger

Daniel Zwillinger

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.

Affiliations and Expertise

Rensselaer Polytechnic Institute, Troy, NY, USA

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