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     Handbook of Nonlinear Partial Differential Equations,  Second Edition    

A. D. Polyanin and V. F. Zaitsev

Handbook of Nonlinear Partial Differential Equations
Second Edition, Updated, Revised and Extended

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

Summary Preface Features Contents References Index (pdf 117K)

Contents (up to 3rd level)

See also full contents (up to 4th level): pdf 95.4K
  • 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)
  • 20.2. Adiabatic Gas Flow
  • 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

    Bibliography

    Index


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