An asymptotic analysis of a self-similar solution for the double nonlinear reaction-diffusion system
https://doi.org/10.17586/2220-8054-2015-6-6-793-802
Abstract
We study the solution for a system of reaction-diffusion equations with double nonlinearity in the presence of a source. A self-similar approach is used for the treatment of qualitative properties of a nonlinear reactiondiffusion system. It is shown that there exist some parameter values for which the effect of finite velocity of perturbation of distribution (FSPD), localization of solution, onside localization can occur. The problem for choosing the appropriate initial approximation for the iteration process used in numerical analysis is solved.
About the Authors
M. Aripov
National University of Uzbekistan named after M. Ulugbek
Uzbekistan
Uzbekistan
Tashkent
Sh. Sadullaeva
Tashkent University of information technology
Uzbekistan
Uzbekistan
Tashkent
References
1. Samarskii A.A., Galaktionov V.A., Kurduomov S.P., Mikhajlov A.P. Blowe-up in quasilinear parabolic equations. Berlin, 4, Walter de Grueter, 1995, p.535.
2. Fujita H. On the blowing up of solutions to the Cauchy problem for ut = ∆u + u1+α, J. Fac. Sci. Univ. Tokyo Sect. I 13, 1966, P. 109{124.
3. Hayakowa K. On nonexistence of global sol bal solutions of some semi linear parabolic differential equations, Proc. Japan Acad. Ser. A Math. Sci., 1973, 49, P. 503{505.
4. Granik I.C. To localization of a temperature perturbation in a nonlinear medium with absorption. Journal of Comp. Math. Math. Phys. 1978, 18(3), P. 770{774.
5. Deng K. and Levine H.A. The role of critical exponents in blow-up theorems: The sequel, J.Math. Anal. Appl. 2000, 243, P. 85{126.
6. Levine H.A. The role of critical exponents in blow-up theorems, SIAM Rer., 1990, 32, P. 262{288.
7. Aripov M. Standard Equation’s Methods for Solutions to Nonlinear problems (Monograph), Tashkent, FAN, 1988, 137 p.
8. Victor A. Galaktionov and J. L. Vazquez. The problem of blow-up in nonlinear parabolic equations. Discrete and continuous dynamical systems, April 2002, 8(2), P. 399{433.
9. Tedeyev A.F. Conditions for the existence and nonexistence of a compact support in time of solutions of the Cauchy problem for quasilinear degenerate parabolic equations. Siberian Math. Jour., 2004, 45(1), P. 189{200.
10. V´azquez J.L. The porous medium equation. Mathematical theory. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007, 430 p.
11. Chien-Hong Cho. On the computation of the numerical blow-up time. Japan Journal of Industrial and Applied Mathematics, 2013, 30(2), P. 331{349.
12. Pan Zheng, Chunlai Mu. A complete upper estimate on the localization for the degenerate parabolic equation with nonlinear source. Mathematical methods in the Applied Sciences. Online publication date: 1-Jan-2014.
13. M. Aripov, S. A. Sadullaeva. To properties of solutions to reaction-diffusion equation with double nonlinearity with distributed parameters. J. Siberian Federal University. Math. Phys., 2013, 6(2), P. 157{167.
14. Aripov M. Asymptotes of the Solutions of the Non-Newton Polytrophic Filtration equation. ZAMM, 2000, 80(3), P. 767{768.
15. Martynenko A.V. and A.F. Tedeev. The Cauchy problem for a quasilinear parabolic equation with a source and inhomogeneous density, Comput. Math. Math. Phys., 2007, 47(2), P. 238{248.
16. Martynenko A.V. and A. F. Tedeev, On the behavior of solutions to the Cauchy problem for a degenerate parabolic equation with inhomogeneous density and a source, Comput. Math. Math. Phys. 2008, 48(7), P. 1145{1160.
17. Afanas’eva N.V. and Tedeev A.F. Fujita type theorems for quasilinear parabolic equations with initial data slowly decaying to zero. Sbornik: Math., 2004, 195(1), P. 3{22.
18. Escobedo M., Levine H. A. Explosion et existence global pour un system habiliment couple deration direction diffusion. C. R. Acad. Sci. Ser.1, 1992, 314(10), P. 735{739.
19. Chien-Hong Cho. On the computation of the numerical blow-up time. Japan Journal of Industrial and Applied Mathematics, 2013, 30(2), P. 331{349.
20. Pan Zheng, Chunlai Mu, Iftikhar Ahmed. Cauchy problem for the non-Newtonian polytrophic filtration equation with a localized reaction. Applicable Analysis 1-16. Online publication date: 20-Feb-2014.
21. Pan Zheng, Chunlai Mu. A complete upper estimate on the localization for the degenerate parabolic equation with nonlinear source. Mathematical methods in the Applied Sciences. Online publication date: 1-Jan-2014.
22. Aripov M. Approximate Self-similar Approach for Solving of the Quasilinear Parabolic Equation. Experimentation, Modeling and Computation in Flow Turbulence and Combustion. Willey&Sons, 1997, 2, P.9{26.
23. Aripov M., Sadullaeva Sh. A. Properties of the solutions of one parabolic equation of non-divergent type. Proceedings of CAIM, 2003, 2, P. 130{136.
24. Gilding B. H., Pelletier L. A. On a class of similarity of the porous media equation 2. J. Math. Anal. And Appl., 1977, 57, P. 522{538.
25. Knerr B. F. The behavior of the support of solutions of the equation of nonlinear heat conduction with absorption in one dimension. Trans. of Amer. Math. Soc., 1979, 249, P. 409{424.
26. Kombe I. Double nonlinear parabolic equations with singular lower order term, Nonlinear Analysis, 2004, 56, P. 185{199.
27. Kolmogorov A.N., Petrovskii I.G., Piskunov N..S. Investigation of the equation of the diffusion connected to increase of quantity of substance and its application to one biological problem. Acta the Moscow State University, ser. Mathematics and mechanics, 1937, 1, P. 1{25.
28. Kurduomov S.P., Kurkina E.S., Telkovskii. Blow up in two componential media. Mathematical Modeling, 1989, 5.
29. Aripov M. Self-Similar and Approximately Self-Similar Method to Solving of Problems of Nonlinear Unsteady a Filtration. Proceedings of the 1-st Turkish world mathematics symposium. Elazig/Turkey, 1999, P. 239{248.
30. Marri Dj. Nonlinear diffusion equations in biology. M., Mir, 1983, 397 p.
31. Dimova S.N., Kastchiev M.S., Koleva M.G., Vasileva D.P., Numerical analysis of the blow-up regimes of combustion of two-component nonlinear heat-conducting medium. JVM and MF, 1995, 35(3), P. 303{ 319.
32. Aripov M., Muhammadiev J. Asymptotic behavior of automodel solutions for one system of quasilinear equations of parabolic type. Buletin Stiintific-Universitatea din Pitesti, Seria Matematica si Informatica, 1999, 3.
33. Holodnyok M., Klich A., Kubichek M., Marec M. Methods of analysis of dynamical models. Moscow, Mir, 1991, 365 p.
Review
For citations:
Aripov M., Sadullaeva Sh. An asymptotic analysis of a self-similar solution for the double nonlinear reaction-diffusion system. Nanosystems: Physics, Chemistry, Mathematics. 2015;6(6):793-802. https://doi.org/10.17586/2220-8054-2015-6-6-793-802
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