Handbook of Nonlinear Partial Differential Equations, Second Edition (original) (raw)

| | | | A. D. Polyanin and V. F. Zaitsev 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 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 Schr�dinger 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 Schr�dinger 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 K�rm�n 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

21.1. 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. B�cklund 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. Painlev� 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 Painlev� 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. Painlev� Transcendents

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