Table of Contents:
Handbook of Differential Equations
- Preface
- Introduction
- Introduction to the Electronic Version
- How to Use This Book
I.A Definitions and Concepts
- Definition of Terms
- Alternative Theorems
- Bifurcation Theory
- A Caveat for Partial Differential Equations
- Chaos in Dynamical Systems
- Classification of Partial Differential Equations
- Compatible Systems
- Conservation Laws
- Differential Resultants
- Existence and Uniqueness Theorems
- Fixed Point Existence Theorems
- Hamilton-Jacobi Theory
- Integrability of Systems
- Internet Resources
- Inverse Problems
- Limit Cycles
- Natural Boundary Conditions for a PDE
- Normal Forms: Near-Identity Transformations
- Random Differential Equations
- Self-Adjoint Eigenfunction Problems
- Stability Theorems
- Sturm-Liouville Theory
- Variational Equations
- Well Posed Differential Equations
- Wronskians and Fundamental Solutions
- Zeros of Solutions
I.B Transformations
- Canonical Forms
- Canonical Transformations
- Darboux Transformation
- An Involutory Transformation
- Liouville Transformation - 1
- Liouville Transformation - 2
- Reduction of Linear ODEs to a First Order System
- Prufer Transformation
- Modified Prufer Transformation
- Transformations of Second Order Linear ODEs - 1
- Transformations of Second Order Linear ODEs - 2
- Transformation of an ODE to an Integral Equation
- Miscellaneous ODE Transformations
- Reduction of PDEs to a First Order System
- Transforming Partial Differential Equations
- Transformations of Partial Differential Equations
II Exact Analytical Methods
- Introduction to Exact Analytical Methods
- Look-Up Technique
- Look-Up ODE Forms
II.A Exact Methods for ODEs
- An Nth Order Equation
- Use of the Adjoint Equation
- Autonomous Equations - Independent Variable Missing
- Bernoulli Equation
- Clairaut's Equation
- Computer-Aided Solution
- Constant Coefficient Linear Equations
- Contact Transformation
- Delay Equations
- Dependent Variable Missing
- Differentiation Method
- Differential Equations with Discontinuities
- Eigenfunction Expansions
- Equidimensional-in-x Equations
- Equidimensional-in-y Equations
- Euler Equations
- Exact First Order Equations
- Exact Second Order Equations
- Exact Nth Order Equations
- Factoring Equations
- Factoring Operators
- Factorization Method
- Fokker-Planck Equation
- Fractional Differential Equations
- Free Boundary Problems
- Generating Functions
- Green's Functions
- Homogeneous Equations
- Method of Images
- Integrable Combinations
- Integral Representation: Laplace's Method
- Integral Transforms: Finite Intervals
- Integral Transforms: Infinite Intervals
- Integrating Factors
- Interchanging Dependent and Independent Variables
- Lagrange's Equation
- Lie Groups: ODEs
- Operational Calculus
- Pfaffian Differential Equations
- Reduction of Order
- Riccati Equations
- Matrix Riccati Equations
- Scale Invariant Equations
- Separable Equations
- Series Solution
- Equations Solvable for x
- Equations Solvable for y
- Superposition
- Method of Undetermined Coefficients
- Variation of Parameters
- Vector Ordinary Differential Equations
II.B Exact Methods for PDEs
- Backlund Transformations
- Method of Characteristics
- Characteristic Strip Equations
- Conformal Mappings
- Method of Descent
- Diagonalization of a Linear System of PDEs
- Duhamel's Principle
- Exact Equations
- Hodograph Transformation
- Inverse Scattering
- Jacobi's Method
- Legendre Transformation
- Lie Groups: PDEs
- Poisson Formula
- Riemann's Method
- Separation of Variables
- Separable Equations: Stackel Matrix
- Similarity Methods
- Exact Solutions to the Wave Equation
- Wiener-Hopf Technique
III Approximate Analytical Methods
- Introduction to Approximate Analysis
- Chaplygin's Method
- Collocation
- Dominant Balance
- Equation Splitting
- Floquet Theory
- Graphical Analysis: The Phase Plane
- Graphical Analysis: The Tangent Field
- Harmonic Balance
- Homogenization
- Integral Methods
- Interval Analysis
- Least Squares Method
- Lyapunov Functions
- Equivalent Linearization and Nonlinearization
- Maximum Principles
- McGarvey Iteration Technique
- Moment Equations: Closure
- Moment Equations: Ito Calculus
- Monge's Method
- Newton's Method
- Pade Approximants
- Perturbation Method: Method of Averaging
- Perturbation Method: Boundary Layer Method
- Perturbation Method: Functional Iteration
- Perturbation Method: Multiple Scales
- Perturbation Method: Regular Perturbation
- Perturbation Method: Strained Coordinates
- Picard Iteration
- Reversion Method
- Singular Solutions
- Soliton-Type Solutions
- Stochastic Limit Theorems
- Taylor Series Solutions
- Variational Method: Eigenvalue Approximation
- Variational Method: Rayleigh-Ritz
- WKB Method
IV.A Numerical Methods: Concepts
- Introduction to Numerical Methods
- Definition of Terms for Numerical Methods
- Available Software
- Finite Difference Formulas
- Finite Difference Methodology
- Grid Generation
- Richardson Extrapolation
- Stability: ODE Approximations
- Stability: Courant Criterion
- Stability: Von Neumann Test
- Testing Differential Equation Routines
IV.B Numerical Methods for ODEs
- Analytic Continuation
- Boundary Value Problems: Box Method
- Boundary Value Problems: Shooting Method
- Continuation Method
- Continued Fractions
- Cosine Method
- Differential Algebraic Equations
- Eigenvalue/Eigenfunction Problems
- Euler's Forward Method
- Finite Element Method
- Hybrid Computer Methods
- Invariant Imbedding
- Multigrid Methods
- Parallel Computer Methods
- Predictor-Corrector Methods
- Runge-Kutta Methods
- Stiff Equations
- Integrating Stochastic Equations
- Symplectic Integration
- Use of Wavelets
- Weighted Residual Methods
IV.C Numerical Methods for PDEs
- Boundary Element Method
- Differential Quadrature
- Domain Decomposition
- Elliptic Equations: Finite Differences
- Elliptic Equations: Monte-Carlo Method
- Elliptic Equations: Relaxation
- Hyperbolic Equations: Method of Characteristics
- Hyperbolic Equations: Finite Differences
- Lattice Gas Dynamics
- Method of Lines
- Parabolic Equations: Explicit Method
- Parabolic Equations: Implicit Method
- Parabolic Equations: Monte-Carlo Method
- Pseudospectral Method
- Mathematical Nomenclature
- Index of Differential Equations
- Index