New exact solutions of one nonlinear equation in mathematical biology and their properties

Ukrainian Mathematical Journal

Volumn 53, Issue 10, 2001, Pages 1712-1727

  Cherniha, R M ^a  

^a INSTITUTE OF MATHEMATICS   (Ukraine)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 10844253282     PISSN: 00415995     EISSN: 15739376     Source Type: Journal    
DOI: 10.1023/A:1015252112419     Document Type: Article

References (25)

1. W. F. Ames, Nonlinear Partial Differential Equations in Engineering, Academic Press, New York (1972).
2. R. Arts, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, I, II, Clarendon Press, Oxford (1975).
3. J. D. Murray, Nonlinear Differential Equation Models in Biology, Clarendon Press, Oxford (1977).
4. R. M. Cherniha, "Symmetry and exact solutions of heat-and-mass transfer equations in Tokamak plasma," Dop. Nats. Akad. Nauk Ukr., No. 4, 17-21 (1995).
5. R. M. Cherniha, "A constructive method for construction of new exact solutions of nonlinear evolution equations," Rep. Math. Phys., 38,301-312(1996).
6. R. M. Cherniha, "Application of one constructive method for the construction of non-Lie solutions of nonlinear evolution equations," Ukr. Math. Zh., 49, 910-925 (1997).
7. R. A. Fisher, "The wave of advance of advantageous genes," Ann. Eugenics, 7, 353-369 (1937).
8. J. D. Murray, Mathematical Biology, Springer, Berlin, 1989.
9. N. Shigesada, K. Kawasaki and E. Teramoto, "Spatial segregation of interacting species," J. Theoret. Biol., 79, 83-99 (1979).
10. E. Conway and J. Smoller, "Diffusion and predator-prey interaction," SIAM J. Appl.Math., 33, 673-686 (1977).
11. P. N. Brown, "Decay to uniform states in ecological interactions," SIAM J. Appl.Math., 38, 22-37 (1980).
12. Y. Lou and W. M. Ni, "Diffusion, self-diffusion, and cross-diffusion," J. Diff. Eqs.,131, 79-131 (1996).
13. L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978).
14. W. Fushchich, W. Shtelen, and M. Serov, Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics, Kluwer, Dordrecht (1993).
15. V. I. Fushchych, M. I. Serov, and V. I. Chopyk, "Conditional invariance and nonlinear heat equations," Dop. Akad. Nauk Ukr. RSR, Ser. A., No. 9, 17-22 (1988).
16. V.A. Dutka, "On the efficiency of application of one version of the finite-element method to the solution of nonlinear transient heat-conduction problems," Inzh.-Fiz. Zh., Termofiz., 70, 284-289 (1997).
17. V. A. Dorodnitsyn, "On invariant solutions of a nonlinear heat-conduction equation with a source," USSR Comput. Math. Math. Phys., 22, 115-122(1982).
18. R. Cherniha, "New non-Lie ansalze and exact solutions of nonlinear reaction-diffusion-convection equations," J. Phys. A: Math.Gen., 31, 8179-8198 (1998).
19. R. M. Cherniha and M. I. Serov, "Symmetries, ansalze, and exact solutions of nonlinear second-order evolution equations with convection term," Euro. J. Appl. Math., 9, 527-542 (1998).
20. G. W. Bluman and I. D. Cole, "The general similarity solution of the heat equation," J. Math. Mech., 18, 1025-1042 (1969).
21. P. J. Olver and P. Rosenau, "Group-invariant solutions of differential equations," SIAM J.Appl.Math., 47,263-278(1987).
22. R. M. Cherniha, "New symmetries and exact solutions of nonlinear reaction-diffusion-convection equations," in: Proceedings of the International Workshop "Similarity Methods, " Univ. Stuttgart, Stuttgart (1998), pp. 323-336.
23. R. M. Cherniha and J. Fehribach, "New exact solutions for a free boundary system," J.Phys.A: Math.Gen., 31, 3815-3829 (1998).
24. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin (1981).
25. M. Ablowitz and A. Zeppetella, "Explicit solutions of Fisher's equation for a special wave speed," Bull. Math. Biol., 41, 835-840(1979).