 EqWorld The World of Mathematical Equations  ## Handbook of Nonlinear Partial Differential EquationsSecond Edition, Updated, Revised and Extended

Publisher: Chapman & Hall/CRC Press, Boca Raton-London-New York
Year of Publication: 2012
Number of Pages: 1912

## Contents (up to 3rd level)

• Authors
• Preface
• Some Remarks and Notation

### Part I. Exact Solutions of Nonlinear Partial Differential Equations

#### 1. First-Order Quasilinear Equations

• 1.1. Equations with Two Independent Variables Containing Arbitrary Parameters
• 1.2. Equations with Two Independent Variables Containing Arbitrary Functions
• 1.3. Other Quasilinear Equations

#### 2. First-Order Equations with Two Independent Variables Quadratic in Derivatives

• 2.1. Equations Containing Arbitrary Parameters
• 2.2. Equations Containing Arbitrary Functions

#### 3. First-Order Nonlinear Equations with Two Independent Variables of General Form

• 3.1. Nonlinear Equations Containing Arbitrary Parameters
• 3.2. Equations Containing Arbitrary Functions of Independent Variables
• 3.3. Equations Containing Arbitrary Functions of Derivatives

#### 4. First-Order Nonlinear Equations with Three or More Independent Variables

• 4.1. Nonlinear Equations with Three Variables Quadratic in Derivatives
• 4.2. Other Nonlinear Equations with Three Variables Containing Parameters
• 4.3. Nonlinear Equations with Three Variables Containing Arbitrary Functions
• 4.4. Nonlinear Equations with Four Independent Variables
• 4.5. Nonlinear Equations with Arbitrary Number of Variables Containing Arbitrary Parameters
• 4.6. Nonlinear Equations with Arbitrary Number of Variables Containing Arbitrary Functions

#### 5. Second-Order Parabolic Equations with One Space Variable

• 5.1. Equations with Power Law Nonlinearities
• 5.2. Equations with Exponential Nonlinearities
• 5.3. Equations with Hyperbolic Nonlinearities
• 5.4. Equations with Logarithmic Nonlinearities
• 5.5. Equations with Trigonometric Nonlinearities
• 5.6. Equations Involving Arbitrary Functions
• 5.7. Nonlinear Schrödinger Equations and Related Equations

#### 6. Second-Order Parabolic Equations with Two or More Space Variables

• 6.1. Equations with Two Space Variables Involving Power Law Nonlinearities
• 6.2. Equations with Two Space Variables Involving Exponential Nonlinearities
• 6.3. Other Equations with Two Space Variables Involving Arbitrary Parameters
• 6.4. Equations Involving Arbitrary Functions
• 6.5. Equations with Three or More Space Variables
• 6.6. Nonlinear Schrödinger Equations

#### 7. Second-Order Hyperbolic Equations with One Space Variable

• 7.1. Equations with Power Law Nonlinearities
• 7.2. Equations with Exponential Nonlinearities
• 7.3. Other Equations Involving Arbitrary Parameters
• 7.4. Equations Involving Arbitrary Functions
• 7.5. Equations of the Form

#### 8. Second-Order Hyperbolic Equations with Two or More Space Variables

• 8.1. Equations with Two Space Variables Involving Power Law Nonlinearities
• 8.2. Equations with Two Space Variables Involving Exponential Nonlinearities
• 8.3. Nonlinear Telegraph Equations with Two Space Variables
• 8.4. Equations with Two Space Variables Involving Arbitrary Functions
• 8.5. Equations with Three Space Variables Involving Arbitrary Parameters
• 8.6. Equations with Three or More Space Variables Involving Arbitrary Functions

#### 9. Second-Order Elliptic Equations with Two Space Variables

• 9.1. Equations with Power Law Nonlinearities
• 9.2. Equations with Exponential Nonlinearities
• 9.3. Equations Involving Other Nonlinearities
• 9.4. Equations Involving Arbitrary Functions

#### 10. Second-Order Elliptic Equations with Three or More Space Variables

• 10.1. Equations with Three Space Variables Involving Power Law Nonlinearities
• 10.2. Equations with Three Space Variables Involving Exponential Nonlinearities
• 10.3. Three-Dimensional Equations Involving Arbitrary Functions
• 10.4. Equations with n Independent Variables

#### 11. Second-Order Equations Involving Mixed Derivatives and Some Other Equations

• 11.1. Equations Linear in the Mixed Derivative
• 11.2. Equations Quadratic in the Highest Derivatives
• 11.3. Bellman-Type Equations and Related Equations

#### 12. Second-Order Equations of General Form

• 12.1. Equations Involving the First Derivative in t
• 12.2. Equations Involving Two or More Second Derivatives

#### 13. Third-Order Equations

• 13.1. Equations Involving the First Derivative in t
• 13.2. Equations Involving the Second Derivative in t
• 13.3. Hydrodynamic Boundary Layer Equations
• 13.4. Equations of Motion of Ideal Fluid (Euler Equations)
• 13.5. Other Third-Order Nonlinear Equations

#### 14. Fourth-Order Equations

• 14.1. Equations Involving the First Derivative in t
• 14.2. Equations Involving the Second Derivative in t
• 14.3. Equations Involving Mixed Derivatives

#### 15. Equations of Higher Orders

• 15.1. Equations Involving the First Derivative in t and Linear in the Highest Derivative
• 15.2. General Form Equations Involving the First Derivative in t
• 15.3. Equations Involving the Second Derivative in t
• 15.4. Other Equations

#### 16. Systems of Two First-Order Partial Differential Equations

• 16.1. Systems of the Form ux = F(u, w),  wt = G(u, w)
• 16.2. Other Systems of Two Equations

#### 17. Systems of Two Parabolic Equations

• 17.1. Systems of the Form ut = auxx + F(u, w),  wt = bwxx + G(u, w)
• 17.2. Systems of the Form ut = ax−n(xnux)x + F(u, w),  wt = bx−n(xnwx)x + G(u, w)
• 17.3. Other Systems of Two Parabolic Equations

#### 18. Systems of Two Second-Order Klein-Gordon Type Hyperbolic Equations

• 18.1. Systems of the Form utt = auxx + F(u, w),  wtt = bwxx + G(u, w)
• 18.2. Systems of the Form utt = ax−n(xnux)x + F(u, w),  wtt = bx−n(xnwx)x + G(u, w)

#### 19. Systems of Two Elliptic Equations

• 19.1. Systems of the Form uxx + uyy = F(u, w),  wxx + wyy = G(u, w)
• 19.2. Other Systems of Two Second-Order Elliptic Equations
• 19.3. Von Kármán Equations (Fourth-Order Elliptic Equations)

#### 20. First-Order Hydrodynamic and Other Systems Involving Three or More Equations

• 20.1. Equations of Motion of Ideal Fluid (Euler Equations)
• 20.3. Systems Describing Fluid Flows in the Atmosphere, Seas, and Oceans
• 20.4. Chromatography Equations
• 20.5. Other Hydrodynamic-Type Systems
• 20.6. Ideal Plasticity with the von Mises Yield Criterion

#### 21. Navier-Stokes and Related Equations

• 21.1. Navier-Stokes Equations
• 21.2. Solutions with One Nonzero Component of the Fluid Velocity
• 21.3. Solutions with Two Nonzero Components of the Fluid Velocity
• 21.4. Solutions with Three Nonzero Fluid Velocity Components Dependent on Two Space Variables
• 21.5. Solutions with Three Nonzero Fluid Velocity Components Dependent on Three Space Variables
• 21.6. Convective Fluid Motions
• 21.7. Boundary Layer Equations (Prandtl Equations)

#### 22. Systems of General Form

• 22.1. Nonlinear Systems of Two Equations Involving the First Derivatives with Respect to t
• 22.2. Nonlinear Systems of Two Equations Involving the Second Derivatives with Respect to t
• 22.3. Other Nonlinear Systems of Two Equations
• 22.4. Nonlinear Systems of Many Equations Involving the First Derivatives with Respect to t

### Part II. Exact Methods for Nonlinear Partial Differential Equations

#### 23. Methods for Solving First-Order Quasilinear Equations

• 23.1. Characteristic System. General Solution
• 23.2. Cauchy Problem. Existence and Uniqueness Theorem
• 23.3. Qualitative Features and Discontinuous Solutions of Quasilinear Equations
• 23.4. Quasilinear Equations of General Form

#### 24. Methods for Solving First-Order Nonlinear Equations

• 24.1. Solution Methods
• 24.2. Cauchy Problem. Existence and Uniqueness Theorem
• 24.3. Generalized Viscosity Solutions and Their Applications

#### 25. Classification of Second-Order Nonlinear Equations

• 25.1. Semilinear Equations in Two Independent Variables
• 25.2. Nonlinear Equations in Two Independent Variables

#### 26. Transformations of Equations of Mathematical Physics

• 26.1. Point Transformations: Overview and Examples
• 26.2. Hodograph Transformations (Special Point Transformations)
• 26.3. Contact Transformations. Legendre and Euler Transformations
• 26.4. Differential Substitutions. Von Mises Transformation
• 26.5. Bäcklund Transformations. RF Pairs
• 26.6. Some Other Transformations

#### 27. Traveling-Wave Solutions and Self-Similar Solutions

• 27.1. Preliminary Remarks
• 27.2. Traveling-Wave Solutions. Invariance of Equations under Translations
• 27.3. Self-Similar Solutions. Invariance of Equations under Scaling Transformations

#### 28. Elementary Theory of Using Invariants for Solving Equations

• 28.1. Introduction. Symmetries. General Scheme of Using Invariants for Solving Mathematical Equations
• 28.2. Algebraic Equations and Systems of Equations
• 28.3. Ordinary Differential Equations
• 28.4. Partial Differential Equations
• 28.5. General Conclusions and Remarks

#### 29. Method of Generalized Separation of Variables

• 29.1. Exact Solutions with Simple Separation of Variables
• 29.2. Structure of Generalized Separable Solutions
• 29.3. Simplified Scheme for Constructing Generalized Separable Solutions
• 29.4. Solution of Functional Differential Equations by Differentiation
• 29.5. Solution of Functional Differential Equations by Splitting
• 29.6. Titov-Galaktionov Method

#### 30. Method of Functional Separation of Variables

• 30.1. Structure of Functional Separable Solutions. Solution by Reduction to Equations with Quadratic Nonlinearities
• 30.2. Special Functional Separable Solutions. Generalized Traveling-Wave Solutions
• 30.3. Differentiation Method
• 30.4. Splitting Method. Solutions of Some Nonlinear Functional Equations and Their Applications

#### 31. Direct Method of Symmetry Reductions of Nonlinear Equations

• 31.1. Clarkson-Kruskal Direct Method
• 31.2. Some Modifications and Generalizations

#### 32. Classical Method of Symmetry Reductions

• 32.1. One-Parameter Transformations and Their Local Properties
• 32.2. Symmetries of Nonlinear Second-Order Equations. Invariance Condition
• 32.3. Using Symmetries of Equations for Finding Exact Solutions. Invariant Solutions
• 32.4. Some Generalizations. Higher-Order Equations
• 32.5. Symmetries of Systems of Equations of Mathematical Physics

#### 33. Nonclassical Method of Symmetry Reductions

• 33.1. General Description of the Method
• 33.2. Examples of Constructing Exact Solutions

#### 34. Method of Differential Constraints

• 34.1. Preliminary Remarks. Method of Differential Constraints for Ordinary Differential Equations
• 34.2. Description of the Method for Partial Differential Equations
• 34.3. First-Order Differential Constraints for PDEs
• 34.4. Second-Order Differential Constraints for PDEs. Some Generalized
• 34.5. Connection between the Method of Differential Constraints and Other Methods

#### 35. Painlevé Test for Nonlinear Equations of Mathematical Physics

• 35.1. Movable Singularities of Solutions of Ordinary Differential Equations
• 35.2. Solutions of Partial Differential Equations with a Movable Pole. Method Description
• 35.3. Performing the Painlevé Test and Truncated Expansions for Studying Some Nonlinear Equations

#### 36. Methods of the Inverse Scattering Problem (Soliton Theory)

• 36.1. Method Based on Using Lax Pairs
• 36.2. Method Based on a Compatibility Condition for Systems of Linear Equations
• 36.3. Method Based on Linear Integral Equations
• 36.4. Solution of the Cauchy Problem by the Inverse Scattering Problem Method

#### 37. Conservation Laws

• 37.1. Basic Definitions and Examples
• 37.2. Equations Admitting Variational Form. Noetherian Symmetries

#### 38. Nonlinear Systems of Partial Differential Equations

• 38.1. Overdetermined Systems of Two Equations
• 38.2. Pfaffian Equations and Their Solutions. Connection with Overdetermined Systems
• 38.3. Systems of First-Order Equations Describing Convective Mass Transfer with Volume Reaction
• 38.4. First-Order Hyperbolic Systems of Quasilinear Equations. Systems of Conservation Laws of Gas Dynamic Type
• 38.5. Systems of Second-Order Equations of Reaction-Diffusion Type

### Part III. Symbolic and Numerical Solutions of Nonlinear PDEs with Maple, Mathematica, and MATLAB

#### 39. Nonlinear Partial Differential Equations with Maple

• 39.1. Introduction
• 39.2. Brief Introduction to Maple
• 39.3. Analytical Solutions and Their Visualizations
• 39.4. Analytical Solutions of Nonlinear Systems
• 39.5. Constructing Exact Solutions Using Symbolic Computation. What Can Go Wrong
• 39.6. Some Errors That People Commonly Do When Constructing Exact Solutions with the Use of Symbolic Computations
• 39.7. Numerical Solutions and Their Visualizations
• 39.8. Analytical-Numerical Solutions

#### 40. Nonlinear Partial Differential Equations with Mathematica

• 40.1. Introduction
• 40.2. Brief Introduction to Mathematica
• 40.3. Analytical Solutions and Their Visualizations
• 40.4. Analytical Solutions of Nonlinear Systems
• 40.5. Numerical Solutions and Their Visualizations
• 40.6. Analytical-Numerical Solutions

#### 41. Nonlinear Partial Differential Equations with MATLAB

• 41.1. Introduction
• 41.2. Brief Introduction to MATLAB
• 41.3. Numerical Solutions via Predefined Functions
• 41.4. Solving Cauchy Problems. Method of Characteristics
• 41.5. Constructing Finite-Difference Approximations

### Supplements

#### 42. Painlevé Transcendents

• 42.1. Preliminary Remarks. Singular Points of Solutions
• 42.2. First Painlevé Transcendent
• 42.3. Second Painlevé Transcendent
• 42.4. Third Painlevé Transcendent
• 42.5. Fourth Painlevé Transcendent
• 42.6. Fifth Painlevé Transcendent
• 42.7. Sixth Painlevé Transcendent
• 42.8. Examples of Solutions to Nonlinear Equations in Terms of Painlevé Transcendents

#### 43. Functional Equations

• 43.1. Method of Differentiation in a Parameter
• 43.2. Method of Differentiation in Independent Variables
• 43.3. Method of Argument Elimination by Test Functions
• 43.4. Nonlinear Functional Equations Reducible to Bilinear Equations

#### Index

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