Advertisement

Differential Equations

, Volume 54, Issue 1, pp 106–120 | Cite as

On Exact Multidimensional Solutions of a Nonlinear System of Reaction–Diffusion Equations

  • A. A. Kosov
  • E. I. Semenov
Partial Differential Equations

Abstract

We study a nonlinear reaction–diffusion system modeled by a system of two parabolic-type equations with power-law nonlinearities. Such systems describe the processes of nonlinear diffusion in reacting two-component media. We construct multiparameter families of exact solutions and distinguish the cases of blow-up solutions and exact solutions periodic in time and anisotropic in spatial variables that can be represented in elementary functions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Polyanin, A.D., Kutepov, A.M., Vyazmin, A.V., and Kazenin, D.A., Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, London–New York: CRC, 2002.Google Scholar
  2. 2.
    Kaptsov, O.V., Metody integrirovaniya uravnenii s chastnymi proizvodnymi (Methods for Integrating Partial Differential Equations), Moscow: Fizmatlit, 2009.Google Scholar
  3. 3.
    Zhuravlev, V.M., On a class of exactly solvable models of autowaves in active media with diffusion admitting exact solutions, JETP Lett., 1997, vol. 65, no. 3, pp. 300–370.CrossRefGoogle Scholar
  4. 4.
    Shmidt, A.V., Exact solutions of systems of reaction–diffusion-type equations, Vychisl. Tekhn., 1998, vol. 3, no. 4, pp. 87–94.Google Scholar
  5. 5.
    Polyanin, A.D., Nonlinear Systems of Two Parabolic-Type Equations (http://eqworld.ipmnet.ru/en/ solutions/syspde/spde2114.pdf) (electronic resource).Google Scholar
  6. 6.
    Cherniha, R. and King, J.R., Non-linear reaction-diffusion systems with variable diffusivities; Lie symmetries, ansatzs and exact solutions, J. Math. Anal. Appl., 2005, vol. 308, pp. 11–35.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Samarskii, A.A., Elenin, G.G., Zmitrenko, N.V., et al., Burning of a nonlinear medium in the form of complex structures, Dokl. Akad. Nauk SSSR, 1977, vol. 237, no. 6, pp. 1330–1333.Google Scholar
  8. 8.
    Elenin, G.G., Kurdyumov, S.P., and Samarskii, A.A., Non-stationary dissipative structures in a nonlinear heat-conducting medium, USSR. Comput. Math. Math. Phys., 1983, vol. 23, no. 2, pp. 80–86.CrossRefzbMATHGoogle Scholar
  9. 9.
    Titov, S.S. and Ustinov, V.A., Study of polynomial solutions of gas filtration equations with integer adiabatic index, in Priblizhennye metody resheniya kraevykh zadach mekhaniki sploshnoi sredy. Sb. nauchn. tr. (Approximate Methods for Solving Boundary Value Problems of Continuum Mechanics: Collection of Scientific Papers), Ural Branch, USSR Acad. Sci, Inst. Math. Mech., 1985, pp. 64–70.Google Scholar
  10. 10.
    Titov, S.S., Method of finite-dimensional rings for solving nonlinear equations of mathematical physics, in Aerodinamika (Aerodynamics), Saratov, 1988, pp. 104–110.Google Scholar
  11. 11.
    Galaktionov, V.A., Dorodnitsyn, V.A., Elenin, G.G., et al., Quasilinear heat conduction equation: blowup, symmetry, localization, asymptotics, exact solutions, in Itogi Nauki Tekh. Sovrem. Probl. Mat. Novye Dostizheniya, Moscow: VINITI, 1986, vol. 28, pp. 95–205.zbMATHGoogle Scholar
  12. 12.
    Galaktionov, V.A. and Posashkov, S.S., New exact solutions of parabolic equations with quadratic nonlinearities, USSR Comput. Math. Math. Phys., 1989, vol. 29, no. 4, pp. 112–119.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dorodnitsyn, V.A., Knyazeva, I.V., and Svirshchevskii, S.R., Group properties of heat equation with a source in the two-dimensional and three-dimensional cases, Differ. Uravn., 1983, vol. 19, no. 7, pp. 1215–1223.MathSciNetGoogle Scholar
  14. 14.
    Rudykh, G.A. and Semenov, G.A, Construction of exact solutions of multidimensional quasilinear heatconduction equations, Comput. Math. Math. Phys., 1993, vol. 33, no. 8, pp. 1087–1097.MathSciNetGoogle Scholar
  15. 15.
    King, J.R., Exact multidimensional solutions to some nonlinear diffusion equations, Quart. J. Mech. Appl. Math., 1993, pp. 46, no. 3, pp. 419–436.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Andreev, V.K., Kaptsov, O.V., Pukhnachev, V.V., and Rodionov, A.A., Primenenie teoretiko-gruppovykh metodov v gidrodinamike (Application of Theoretical Group Methods in Hydrodynamics), Novosibirsk: Nauka, 1994.zbMATHGoogle Scholar
  17. 17.
    Pukhnachev, V.V., Multidimensional exact solutions of equations of nonlinear diffusion, Prikl. Mekh. Tekh. Fiz., 1995, vol. 36, no. 2, pp. 23–31.MathSciNetzbMATHGoogle Scholar
  18. 18.
    Polyanin, A.D. and Zaitsev, A.D., Spravochnik po nelineinym uravneniyam matematicheskoi fiziki: tochnye resheniya (Handbook on Nonlinear Equations of Mathematical Physics: Exact Solutions), Moscow: Fizmatlit, 2002.Google Scholar
  19. 19.
    Polyanin, A.D., Zaitsev, A.D., and Zhurov, A.I., Metody resheniya nelineinykh uravnenii matematicheskoi fiziki i mekhaniki (Methods for Solving Nonlinear Equations of Mathematical Physics and Mechanics) Moscow: Fizmatlit, 2005.Google Scholar
  20. 20.
    Galactionov, V.A. and Svirshchevskii, S.R., Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Boca Raton: Chapman & Hall, 2007.Google Scholar
  21. 21.
    Rudykh, G.A. and Semenov, E.I., Non-self-similar solutions of multidimensional nonlinear diffusion equations, Math. Notes, 2000, vol. 67, no. 2, pp. 200–206.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rudykh, G.A. and Semenov, E.I., Sushchestvovanie i kachestvennyi analiz tochnykh neavtomodel’nykh reshenii mnogomernogo uravneniya nelineinoi diffuzii, nelineinyi analiz i nelineinye differentsial’nye uravneniya (Existence and Qualitative Analysis of Exact Nonself-Similar Solutions of Multidimensional Nonlinear Diffusion Equation, Nonlinear Analysis and Nonlinear Differential Equations), Trenogin, V.A. and Filippov, A.F., Eds., Moscow: Fizmatlit, 2003.Google Scholar
  23. 23.
    Rudykh, G.A. and Semenov, E.I., Exact nonnegative solutions of multidimensional nonlinear diffusion equation, Sib. Math. J., 1998, vol. 39, no. 5, pp. 977–985.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rudykh, G.A. and Semenov, E.I., Existence and construction of anisotropic solutions to the multidimensional equation of nonlinear diffusion. I, Sib. Math. J., 2000, vol. 41, no. 5, pp. 940–959.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Rudykh, G.A. and Semenov, E.I., Existence and construction of anisotropic solutions to the multidimensional equation of nonlinear diffusion. II, Sib. Math. J., 2001, vol. 42, no. 1, pp. 157–175.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Samarskii, A.A., Galaktionov, V.A., Kurdyumov, S.P., and Mikhailov, A.P., Rezhimy s obostreniem v zadachakh dlya kvazilineinykh parabolicheskikh uravnenii (Regimes with Blow-Up in Problems for Quasilinear Parabolic Equations), Moscow: Nauka, 1987.Google Scholar
  27. 27.
    Gantmakher, F.R., Teoriya matrits (Theory of Matrices), Moscow: Nauka, 1967; 1988.zbMATHGoogle Scholar
  28. 28.
    Kamke, E., Differentialgleichungen: Losungsmethoden und Losungen, Leipzig: Teubner, 1977.CrossRefzbMATHGoogle Scholar
  29. 29.
    Prudnikov, A.P., Brychkov, Yu.A., and Marichev, O.I., Integraly i ryady. Elementarnye funktsii (Integrals and Series: Elementary Functions), Moscow: Nauka, 1981.zbMATHGoogle Scholar
  30. 30.
    Polyanin, A.D. and Zaitsev, V.F., Spravochnik po obyknovennym differentsial’nym uravneniyam (Handbook on Ordinary Differential Equations), Moscow: Gos. Izd. Fiz. Mat. Lit., 2001.zbMATHGoogle Scholar
  31. 31.
    Belousov, B.P., Periodicheski deistvuyushchaya reaktsiya i ee mekhanism, Avtovolnovye protsessy v sistemakh s diffusiei (Periodically Acting Reaction and Its Mechanism. Autowave Processes in Systems with Diffusion), Gorkii, 1951.Google Scholar
  32. 32.
    Zhabotinskii, A.M., Kontsentratsionnye avtokolebaniya (Concentration Self-Excited Oscillations), Moscow: Nauka, 1974.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Matrosov Institute for System Dynamics and Control Theory, Siberian BranchRussian Academy of SciencesIrkutskRussia

Personalised recommendations