Non-linear reaction-diffusion systems with variable diffusivities: Lie symmetries, ansätze and exact solutions

Journal of Mathematical Analysis and Applications

Volumn 308, Issue 1, 2005, Pages 11-35

  Cherniha, Roman ^a   King, John R ^b  

^a INSTITUTE OF MATHEMATICS   (Ukraine)

^b UNIVERSITY OF NOTTINGHAM   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 19044368965     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jmaa.2004.10.034     Document Type: Article

References (30)

1. R. Cherniha J.R. King Lie symmetries of nonlinear multidimensional reaction-diffusion systems: I J. Phys. A: Math. Gen. 33 2000 267-282
2. R. Cherniha J.R. King Lie symmetries of nonlinear multidimensional reaction-diffusion systems: II J. Phys. A: Math. Gen. 36 2003 405-425
3. W. Zulehner W.F. Ames Group analysis of a semilinear vector diffusion equation Nonlinear Anal. 7 1983 945-969
4. A.M. Turing The chemical basis of morphogenesis Philos. Trans. Roy. Soc. London Ser. B 237 1952 37-72
5. J.D. Murray Mathematical Biology 1989 Springer Berlin
6. I.V. Knyazeva M.D. Popov A system of two diffusion equations in: N.H. Ibragimov (Eds.) CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1 1994 CRC Press Boca Raton, FL 171-176
7. R. Cherniha M. Serov Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection term European J. Appl. Math. 9 1998 527-542
8. G.W. Bluman J.D. Cole Similarity Methods for Differential Equations 1974 Springer Berlin
9. L.V. Ovsiannikov The Group Analysis of Differential Equations 1978 Nauka Moscow (in Russian)
10. P. Olver Applications of Lie Groups to Differential Equations 1986 Springer Berlin
11. W.I. Fushchych W.M. Shtelen M.I. Serov Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics 1993 Kluwer Academic Dordrecht
12. N.H. Ibragimov Lie Group Analysis of Differential Equations, vol. 1 1994 CRC Press Boca Raton, FL 133-136
13. V.A. Dorodnitsyn On invariant solutions of non-linear heat conduction with a source USSR Comput. Math. Math. Phys. 22 1982 115-122
14. J.R. King Exact multidimensional solutions to some nonlinear diffusion equations Quart. J. Mech. Appl. Math. 46 1993 419-436
15. P. Olver Direct reduction and differential constraints Proc. Roy. Soc. London Ser. A 46 1994 509-523
16. A.S. Fokas Q.M. Liu Nonlinear interaction of travelling waves of nonintegrable equations Phys. Rev. Lett. 72 1994 3293-3296
17. M.C. Nucci Iterating the nonclassical symmetries method Phys. D 78 1994 124-134
18. V.A. Galaktionov Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities Proc. Roy. Soc. Edinburgh Sect. A 125 1995 225-246
19. V.A. Galaktionov S.A. Posashkov Examples of nonsymmetric extinction and blow-up for quasilinear heat equations Differential Integral Equations 8 1995 87-103
20. M.C. Nucci Iterations of the non-classical symmetries method and conditional Lie-Backlund symmetries J. Phys. A: Math. Gen. 29 1996 8117-8122
21. R. Cherniha A constructive method for construction of new exact solutions of nonlinear evolution equations Rep. Math. Phys. 38 1996 301-312
22. G.W. Bluman J.D. Cole The general similarity solution of the heat equation J. Math. Mech. 18 1969 1025-1042
23. J.R. King Mathematical analysis of a model for substitutional diffusion Proc. Roy. Soc. London Ser. A 430 1990 377-404
24. R.M. Cherniha J.D. Fehribach New exact solutions for a free boundary system J. Phys. A: Math. Gen. 31 1998 3815-3829
25. R. Cherniha New non-Lie ansätze and exact solutions of nonlinear reaction-diffusion-convection equations J. Phys. A: Math. Gen. 31 1998 8179-8198
26. J.R. King Exact polynomial solutions to some nonlinear diffusion equations Phys. D 64 1993 35-65
27. W. Fushchych R. Zhdanov Antireduction and exact solutions of nonlinear heat equations J. Nonlinear Math. Phys. 1 1994 60-64
28. S.S. Titov Solutions of non-linear partial differential equations in the form of polynomials with respect to one of variables Chislennye Metody Mekhaniki Sploshnoy Sredy 8 1977 144-149
29. G.I. Barenblatt Self-Similarity and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics Cambridge Texts in Applied Mathematics 1996 Cambridge Univ. Press Cambridge
30. Q.M. Liu Exact solutions of nonlinear equations with quadratic nonlinearity J. Phys. A: Math. Gen. 34 2001 5083-5088