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Handbook of Nonlinear Partial Differential Equations, Second Edition > Contents




Handbook of Nonlinear Partial Differential Equations Second Edition, Updated, Revised and Extended
Publisher: Chapman & Hall/CRC Press, Boca RatonLondonNew York
Year of Publication: 2012
Number of Pages: 1912

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. FirstOrder 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. FirstOrder Equations with Two Independent Variables Quadratic in Derivatives
 2.1. Equations Containing Arbitrary Parameters
 2.2. Equations Containing Arbitrary Functions
3. FirstOrder 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. FirstOrder 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. SecondOrder 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. SecondOrder 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. SecondOrder 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. SecondOrder 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. SecondOrder 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. SecondOrder 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. ThreeDimensional Equations Involving Arbitrary Functions
 10.4. Equations with n Independent Variables
11. SecondOrder 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. BellmanType Equations and Related Equations
12. SecondOrder Equations of General Form
 12.1. Equations Involving the First Derivative in t
 12.2. Equations Involving Two or More Second Derivatives
13. ThirdOrder 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 ThirdOrder Nonlinear Equations
14. FourthOrder 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 FirstOrder Partial Differential Equations
 16.1. Systems of the Form u_{x} = F(u, w), w_{t} = G(u, w)
 16.2. Other Systems of Two Equations
17. Systems of Two Parabolic Equations
 17.1. Systems of the Form u_{t} = au_{xx} + F(u, w), w_{t} = bw_{xx} + G(u, w)
 17.2. Systems of the Form u_{t} = ax^{−n}(x^{n}u_{x})_{x} + F(u, w), w_{t} = bx^{−n}(x^{n}w_{x})_{x} + G(u, w)
 17.3. Other Systems of Two Parabolic Equations
18. Systems of Two SecondOrder KleinGordon Type Hyperbolic Equations
 18.1. Systems of the Form u_{tt} = au_{xx} + F(u, w), w_{tt} = bw_{xx} + G(u, w)
 18.2. Systems of the Form u_{tt} = ax^{−n}(x^{n}u_{x})_{x} + F(u, w), w_{tt} = bx^{−n}(x^{n}w_{x})_{x} + G(u, w)
19. Systems of Two Elliptic Equations
 19.1. Systems of the Form u_{xx} + u_{yy} = F(u, w), w_{xx} + w_{yy} = G(u, w)
 19.2. Other Systems of Two SecondOrder Elliptic Equations
 19.3. Von Kármán Equations (FourthOrder Elliptic Equations)
20. FirstOrder 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 HydrodynamicType Systems
 20.6. Ideal Plasticity with the von Mises Yield Criterion
21. NavierStokes and Related Equations
 21.1. NavierStokes 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 FirstOrder 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 FirstOrder 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 SecondOrder 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. TravelingWave Solutions and SelfSimilar Solutions
 27.1. Preliminary Remarks
 27.2. TravelingWave Solutions. Invariance of Equations under Translations
 27.3. SelfSimilar 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. TitovGalaktionov 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 TravelingWave 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. ClarksonKruskal Direct Method
 31.2. Some Modifications and Generalizations
32. Classical Method of Symmetry Reductions
 32.1. OneParameter Transformations and Their Local Properties
 32.2. Symmetries of Nonlinear SecondOrder Equations. Invariance Condition
 32.3. Using Symmetries of Equations for Finding Exact Solutions. Invariant Solutions
 32.4. Some Generalizations. HigherOrder 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. FirstOrder Differential Constraints for PDEs
 34.4. SecondOrder 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 FirstOrder Equations Describing Convective Mass Transfer with Volume Reaction
 38.4. FirstOrder Hyperbolic Systems of Quasilinear Equations. Systems of Conservation Laws of Gas Dynamic Type
 38.5. Systems of SecondOrder Equations of ReactionDiffusion 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. AnalyticalNumerical 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. AnalyticalNumerical 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 FiniteDifference 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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