# Table of Contents: Handbook of Differential Equations

• Preface
• Introduction
• Introduction to the Electronic Version
• 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. Integrability of Systems
14. Internet Resources
15. Inverse Problems
16. Limit Cycles
17. Natural Boundary Conditions for a PDE
18. Normal Forms: Near-Identity Transformations
19. Random Differential Equations
20. Self-Adjoint Eigenfunction Problems
21. Stability Theorems
22. Sturm-Liouville Theory
23. Variational Equations
24. Well Posed Differential Equations
25. Wronskians and Fundamental Solutions
26. Zeros of Solutions

## I.B Transformations

1. Canonical Forms
2. Canonical Transformations
3. Darboux Transformation
4. An Involutory Transformation
5. Liouville Transformation - 1
6. Liouville Transformation - 2
7. Reduction of Linear ODEs to a First Order System
8. Prufer Transformation
9. Modified Prufer Transformation
10. Transformations of Second Order Linear ODEs - 1
11. Transformations of Second Order Linear ODEs - 2
12. Transformation of an ODE to an Integral Equation
13. Miscellaneous ODE Transformations
14. Reduction of PDEs to a First Order System
15. Transforming Partial Differential Equations
16. Transformations of Partial Differential Equations

## II Exact Analytical Methods

1. Introduction to Exact Analytical Methods
2. Look-Up Technique
3. Look-Up ODE Forms

## II.A Exact Methods for ODEs

1. An Nth Order Equation
2. Use of the Adjoint Equation
3. Autonomous Equations - Independent Variable Missing
4. Bernoulli Equation
5. Clairaut's Equation
6. Computer-Aided Solution
7. Constant Coefficient Linear Equations
8. Contact Transformation
9. Delay Equations
10. Dependent Variable Missing
11. Differentiation Method
12. Differential Equations with Discontinuities
13. Eigenfunction Expansions
14. Equidimensional-in-x Equations
15. Equidimensional-in-y Equations
16. Euler Equations
17. Exact First Order Equations
18. Exact Second Order Equations
19. Exact Nth Order Equations
20. Factoring Equations
21. Factoring Operators
22. Factorization Method
23. Fokker-Planck Equation
24. Fractional Differential Equations
25. Free Boundary Problems
26. Generating Functions
27. Green's Functions
28. Homogeneous Equations
29. Method of Images
30. Integrable Combinations
31. Integral Representation: Laplace's Method
32. Integral Transforms: Finite Intervals
33. Integral Transforms: Infinite Intervals
34. Integrating Factors
35. Interchanging Dependent and Independent Variables
36. Lagrange's Equation
37. Lie Groups: ODEs
38. Operational Calculus
39. Pfaffian Differential Equations
40. Reduction of Order
41. Riccati Equations
42. Matrix Riccati Equations
43. Scale Invariant Equations
44. Separable Equations
45. Series Solution
46. Equations Solvable for x
47. Equations Solvable for y
48. Superposition
49. Method of Undetermined Coefficients
50. Variation of Parameters
51. Vector Ordinary Differential Equations

## II.B Exact Methods for PDEs

1. Backlund Transformations
2. Method of Characteristics
3. Characteristic Strip Equations
4. Conformal Mappings
5. Method of Descent
6. Diagonalization of a Linear System of PDEs
7. Duhamel's Principle
8. Exact Equations
9. Hodograph Transformation
10. Inverse Scattering
11. Jacobi's Method
12. Legendre Transformation
13. Lie Groups: PDEs
14. Poisson Formula
15. Riemann's Method
16. Separation of Variables
17. Separable Equations: Stackel Matrix
18. Similarity Methods
19. Exact Solutions to the Wave Equation
20. Wiener-Hopf Technique

## III Approximate Analytical Methods

1. Introduction to Approximate Analysis
2. Chaplygin's Method
3. Collocation
4. Dominant Balance
5. Equation Splitting
6. Floquet Theory
7. Graphical Analysis: The Phase Plane
8. Graphical Analysis: The Tangent Field
9. Harmonic Balance
10. Homogenization
11. Integral Methods
12. Interval Analysis
13. Least Squares Method
14. Lyapunov Functions
15. Equivalent Linearization and Nonlinearization
16. Maximum Principles
17. McGarvey Iteration Technique
18. Moment Equations: Closure
19. Moment Equations: Ito Calculus
20. Monge's Method
21. Newton's Method
22. Pade Approximants
23. Perturbation Method: Method of Averaging
24. Perturbation Method: Boundary Layer Method
25. Perturbation Method: Functional Iteration
26. Perturbation Method: Multiple Scales
27. Perturbation Method: Regular Perturbation
28. Perturbation Method: Strained Coordinates
29. Picard Iteration
30. Reversion Method
31. Singular Solutions
32. Soliton-Type Solutions
33. Stochastic Limit Theorems
34. Taylor Series Solutions
35. Variational Method: Eigenvalue Approximation
36. Variational Method: Rayleigh-Ritz
37. WKB Method

## IV.A Numerical Methods: Concepts

1. Introduction to Numerical Methods
2. Definition of Terms for Numerical Methods
3. Available Software
4. Finite Difference Formulas
5. Finite Difference Methodology
6. Grid Generation
7. Richardson Extrapolation
8. Stability: ODE Approximations
9. Stability: Courant Criterion
10. Stability: Von Neumann Test
11. Testing Differential Equation Routines

## IV.B Numerical Methods for ODEs

1. Analytic Continuation
2. Boundary Value Problems: Box Method
3. Boundary Value Problems: Shooting Method
4. Continuation Method
5. Continued Fractions
6. Cosine Method
7. Differential Algebraic Equations
8. Eigenvalue/Eigenfunction Problems
9. Euler's Forward Method
10. Finite Element Method
11. Hybrid Computer Methods
12. Invariant Imbedding
13. Multigrid Methods
14. Parallel Computer Methods
15. Predictor-Corrector Methods
16. Runge-Kutta Methods
17. Stiff Equations
18. Integrating Stochastic Equations
19. Symplectic Integration
20. Use of Wavelets
21. Weighted Residual Methods

## IV.C Numerical Methods for PDEs

1. Boundary Element Method
2. Differential Quadrature
3. Domain Decomposition
4. Elliptic Equations: Finite Differences
5. Elliptic Equations: Monte-Carlo Method
6. Elliptic Equations: Relaxation
7. Hyperbolic Equations: Method of Characteristics
8. Hyperbolic Equations: Finite Differences
9. Lattice Gas Dynamics
10. Method of Lines
11. Parabolic Equations: Explicit Method
12. Parabolic Equations: Implicit Method
13. Parabolic Equations: Monte-Carlo Method
14. Pseudospectral Method

• Mathematical Nomenclature
• Index of Differential Equations
• Index