A Precise Definition of Reduction of Partial Differential Equations

Journal of Mathematical Analysis and Applications

Volumn 238, Issue 1, 1999, Pages 101-123

  Zhdanov, R Z ^a   Tsyfra, I M ^a   Popovych, R O ^a  

^a INSTITUTE OF MATHEMATICS   (Ukraine)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0008875418     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jmaa.1999.6511     Document Type: Article

References (48)

1. Bluman G., Cole J. D. The general similarity solution of the heat equation. J. Math. Mech. 18:1969;1025-1042.
2. Olver P. J., Rosenau P. The construction of special solutions to partial differential equations. Phys. Lett. 114A:1986;107-112.
3. Olver P. J., Rosenau P. Group-invariant solutions of differential equations. SIAM J. Appl. Math. 47:1987;263-278.
4. Fushchych W. I., Tsyfra I. M. On a reduction and solutions of nonlinear wave equations with broken symmetry. J. Phys. A. 20:1987;L45-L48.
5. Fushchych W. I., Zhdanov R. Z. Symmetry and exact solutions of nonlinear spinor equations. Phys Rep. 172:1989;123-174.
6. Clarkson P., Kruskal M. D. New similarity solutions of the Boussinesq equation. J. Math. Phys. 30:1989;2201-2213.
7. Levi D., Winternitz P. Non-classical symmetry reduction: example of the Boussinesq equation. J. Phys. A. 22:1989;2915-2924.
8. Fushchych W. I. How to extend symmetry of differential equations? Symmetry and Solutions of Nonlinear Mathematical Physics Equations. 1987;Inst. of Math. Acad. of Sci. Ukraine, Kiev. p. 4-16.
9. Fushchych W. I. On symmetry and exact solutions of multi-dimensional nonlinear wave equations. Ukrain. Math. J. 39:1987;116-123.
10. Fushchych W. I., Nikitin A. G. Symmetries of Maxwell's Equations. 1987;Reidel, Dordrecht.
11. Fushchych W. I., Zhdanov R. Z. Symmetry and exact solutions of the nonlinear Dirac equation. Fizika Elementar. Chastits i Atom. Yadra. 19:1988;1154-1196.
12. Fushchych W. I., Zhdanov R. Z. On some new exact solutions of the nonlinear d'Alembert-Hamilton system. Phys. Lett. 141A:1989;113-115.
13. Fushchych W. I., Zhdanov R. Z. Non-Lie Ansätze and exact solutions of the nonlinear spinor equation. Ukrain. Math. J. 42:1990;958-962.
14. Fushchych W. I., Zhdanov R. Z. On non-Lie reduction of the nonlinear Dirac equation. J. Math. Phys. 32:1991;3488-3490.
15. Fushchych W. I., Zhdanov R. Z., Yehorchenko I. A. On the reduction of the nonlinear multi-dimensional wave equations and compatibility of the d'Alembert-Hamilton system. J. Math. Anal. Appl. 161:1991;352-360.
16. Zhdanov R. Z., Fushchych W. I. On non-Lie ansatzes and new exact solutions of the classical Yang-Mills equations. J. Nonlinear Math. Phys. 2:1995;172-181.
17. Zhdanov R. Z., Fushchych W. I. Conditional symmetry and new classical solutions of the Yang-Mills equations. J. Phys. A. 28:1995;6253-6263.
18. Zhdanov R. Z. On integration of nonlinear d'Alembert-eikonal system and conditional symmetry of nonlinear wave equations. J. Nonlinear Math. Phys. 4:1997;49-62.
19. Fushchych W. I., Zhdanov R. Z. Symmetries and Exact Solutions of Nonlinear Dirac Equations. 1997;Naukova Ukraina Publishing, Kyiv.
20. Fushchych W. I., Zhdanov R. Z. Conditional symmetry and reduction of partial differential equations. Ukrain Math. J. 44:1992;970-982.
21. Zhdanov R. Z. Conditional Lie-Bäcklund symmetry and reduction of evolution equations. J. Phys. A. 28:1995;3841-3850.
22. Olver P. J., Vorob'ev E. M. Nonclassical and conditional symmetries. Handbook of Lie Group Analysis. 1994;CRC Press, Boca Raton.
23. Popovych R. O. On reduction and Q-conditional symmetry. Proceedings of the Second International Conference 'Symmetry in Nonlinear Mathematical Physics' (Kyiv, July 7-13, 1997). 1997;Inst. of Math. Acad. of Sci. Ukraine, Kyiv. p. 437-443.
24. Popovych R. O., Korneva I. P. On Q-conditional symmetry of the linear n-dimensional heat equation. Symmetry and Analytic Methods in Mathematical Physics. 1998;Inst. of Math. Acad. of Sci. Ukraine, Kyiv. p. 201-211.
25. Popovych R. O. On a class of Q-conditional symmetries and solutions of evolutionary equations. Symmetry and Analytic Methods in Mathematical Physics. 1998;Inst. of Math. Acad. of Sci. Ukraine, Kyiv. p. 194-199.
26. Fushchych W. I. On symmetry and particular solutions of some multi-dimensional mathematical physics equations. Algebraic-Theoretical Methods in Mathematical Physics Problems. 1983;Inst. of Math. Acad. of Sci. Ukraine, Kiev. p. 4-23.
27. Fushchych W. I., Serov N. I. The symmetry and some exact solutions of nonlinear many-dimensional Liouville, d'Alembert and eikonal equations. J. Phys. A. 16:1983;3645-3656.
28. Olver P. J. Direct reduction and differential constraints. Proc. Roy. Soc. London Ser. A. 444:1994;509-523.
29. Fushchych W. I., Shtelen W. M., Serov M. I. Symmetry Analysis and Exact Solutions of Equations of Non-Linear Mathematical Physics. 1992;Kluwer Academic, Dordrecht.
30. Estevez P. G., Gordoa P. R. Nonclassical symmetries and the singular manifold method - theory and 6 examples. Stud. Appl. Math. 95:1995;517-557.
31. Zhdanov R. Z., Tsyfra I. M. Reduction of differential equations and conditional symmetry. Ukrain. Math. J. 48:1996;595-602.
32. Olver P. J. Applications of Lie Groups to Differential Equations. 1989;Springer-Verlag, New York.
33. Schutz B. F. Geometrical Methods of Mathematical Physics. 1982;Cambridge Univ. Press, London.
34. Vorob'ev E. M. Partial symmetries of system of differential equations. Soviet Math. Dokl. 33:1986;408-411.
35. Bateman H. Partial Differential Equations of Mathematical Physics. 1922;Cambridge Univ. Press, Cambridge.
36. Smirnov V. I., Sobolev S. L. New method of solution of the problem of elastic plane vibrations. Proc. Seismological Inst. Acad. Sci. USSR. 20:1932;37-40.
37. Smirnov V. I., Sobolev S. L. On application of the new method to study of elastic vibrations in the space with axial symmetry. Proc. Seismological Inst. Acad. Sci. USSR. 29:1933;43-51.
38. Collins C. B. Complex potential equations. I. A technique for solution. Proc. Cambridge Philos. Soc. 80:1976;165-171.
39. Cartan E. Oeuvres completes. 1995;Gauthier-Villars, Paris. p. 1431-1445.
40. Bateman H. Mathematical Analysis of Electrical and Optical Wave-Motion. 1955;Dover, New York.
41. Erugin N. P. On functionally-invariant solutions. Proc. Acad. Sci. USSR. 5:1944;385-386.
42. W. I. Fushchych, R. Z. Zhdanov, and, I. V. Revenko, Compatibility and solutions of the nonlinear d'Alembert and Hamilton equations, Preprint No. 90-39, Inst. of Math. Acad. of Sci. Ukraine, Kiev, 1990.
43. Fushchych W. I., Zhdanov R. Z., Revenko I. V. General solutions of the nonlinear wave and eikonal equations. Ukrain. Math. J. 43:1991;1471-1486.
44. Patera J., Sharp R. T., Winternitz P., Zassenhaus H. Subgroups of the Poincaré group and their invariants. J. Math Phys. 17:1976;977-985.
45. Grundland A., Harnad J., Winternitz P. Symmetry reduction for nonlinear relativistically invariant equations. J. Math. Phys. 25:1984;791-806.
46. Zhdanov R. Z., Panchak O. A. New conditional symmetries and exact solutions of the nonlinear wave equation. J. Phys. A. 31:1998;8727-8734.
47. Galaktionov V. A. On new exact blow-up solutions for nonlinear heat conduction equations with source and applications. Differential Integral Equations. 3:1990;863-874.
48. Nucci M. C. Iterating the nonclassical symmetries method. Physica D. 78:1994;124-134.