New to the Second Edition More than 1,000 pages with over 1,500 new first-, second-, third-, fourth-, and higher-order nonlinear equations with solutions Parabolic, hyperbolic, elliptic, and other systems of equations with solutions Some exact methods and transformations Symbolic and numerical methods for solving nonlinear PDEs with Maple™, Mathematica®, and MATLAB® Many new illustrative examples and tables A large list of references consisting of over 1,300 sources To accommodate different mathematical backgrounds, the authors avoid wherever possible the use of special terminology. They outline the methods in a schematic, simplified manner and arrange the material in increasing order of complexity.
Author(s): Andrei D. Polyanin, Valentin F. Zaitsev
Series: Handbooks of Mathematical Equations
Edition: 2
Publisher: Chapman and Hall/CRC
Year: 2011
Language: English
Pages: 1878
Tags: Математика;Дифференциальные уравнения;Дифференциальные уравнения в частных производных;
Title Page
......Page 4
Contents......Page 6
Untitled......Page 27
Preface to the new edition......Page 28
Preface to the first edition......Page 29
Authors......Page 32
Some notations and remarks......Page 34
Part I. Exact Solutions of Nonlinear Partial Differential Equations......Page 38
1. First-Order Quasilinear Equations......Page 40
2. First-Order Equations with Two Independent Variables Quadratic in Derivatives......Page 80
3. First-Order Nonlinear Equations with Two Independent Variables of General Form......Page 136
4. First-Order Nonlinear Equations with Three or More Independent Variables......Page 162
5. Second-Order Parabolic Equations with One Space Variable......Page 212
6. Second-Order Parabolic Equations with Two or More Space Variables......Page 404
7. Second-Order Hyperbolic Equations with One Space Variable......Page 470
8. Second-Order Hyperbolic Equations with Two or More Space Variables......Page 590
9. Second-Order Elliptic Equations with Two Space Variables......Page 678
10. Second-Order Elliptic Equations with Three or More Space Variables......Page 750
11. Second-Order Equations Involving Mixed Derivatives and Some Other Equations......Page 782
12. Second-Order Equations of General Form......Page 848
13. Third-Order Equations......Page 894
14. Fourth-Order Equations......Page 1014
15. Equations of Higher Orders......Page 1068
16. Systems of Two First-Order Partial Differential Equations......Page 1152
17. Systems of Two Parabolic Equations......Page 1170
18. Systems of Two Second-Order Klein–Gordon Type Hyperbolic Equations......Page 1210
19. Systems of Two Elliptic Equations......Page 1222
20. First-Order Hydrodynamic and Other Systems Involving Three or More Equations......Page 1234
21. Navier—Stokes and Related Equations......Page 1284
22. Systems of General Form......Page 1374
Part II. Exact Methods for Nonlinear Partial Differential Equations......Page 1390
23. Methods for Solving First-Order Quasilinear Equations......Page 1392
24. Methods for Solving First-Order Nonlinear Equations......Page 1410
25. Classification of Second-Order Nonlinear Equations......Page 1426
26. Transformations of Equations of Mathematical Physics......Page 1432
27. Traveling-WaveSolutions and Self-Similar Solutions......Page 1466
28. Elementary Theory of Using Invariants for Solving Equations......Page 1476
29. Method of Generalized Separation of Variables......Page 1496
30. Method of Functional Separation of Variables......Page 1524
31. Direct Method of Symmetry Reductions of Nonlinear Equations......Page 1540
32. Classical Method of Symmetry Reductions......Page 1550
33. Nonclassical Method of Symmetry Reductions......Page 1570
34. Method of Differential Constraints......Page 1576
35. Painlevé Test for Nonlinear Equations of Mathematical Physics......Page 1602
36. Methods of the Inverse Scattering Problem (Soliton Theory)......Page 1616
37. Conservation Laws......Page 1630
38. Nonlinear Systems of Partial Differential Equations......Page 1636
Part III. Symbolic and Numerical Solutions of Nonlinear PDEs with Maple, Mathematica, and MATLAB......Page 1660
39. Nonlinear Partial Differential Equations with Maple......Page 1662
40. Nonlinear Partial Differential Equations with Mathematica......Page 1724
41. Nonlinear Partial Differential Equations with MATLAB......Page 1772
Supplements. Painlevé Transcendents. Functional Equations......Page 1804
42. Painlevé Transcendents......Page 1806
43. Functional Equations......Page 1820
Bibliography......Page 1832
