Skip to main content
SpringerLink
Log in

Q-Conditional Symmetries of the First Type and Exact Solutions of Nonlinear Reaction-Diffusion Systems

  • Chapter
  • First Online:
Nonlinear Reaction-Diffusion Systems

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2196))

  • 1404 Accesses

Abstract

Two classes of two-component nonlinear reaction-diffusion systems are studied in order to find Q-conditional symmetries of the first type (a special subset of nonclassical symmetries), to construct exact solutions, and to show their applicability. The first class involves systems with constant coefficient of diffusivity, while the second contains systems with variable diffusivities only. The main theoretical results are given in the form of two theorems presenting exhaustive lists (up to the given sets of point transformations) of the reaction-diffusion systems belonging to the above classes and admitting Q-conditional symmetries of the first type. The reaction-diffusion systems obtained allow one to extract specific systems occurring in real-world models. A few examples are presented, including a modification of the classical prey–predator system with diffusivity and a system modelling the gravity-driven flow of thin films of viscous fluid. Exact solutions with attractive properties are found for these nonlinear systems and their possible biological and physical interpretations are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Archilla, J., Romero, J., Romero, F., Palmero, F.: Lie symmetries and multiple solutions in α −ω reaction-diffusion systems. J. Phys. A Math. Gen. 30, 185–194 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Archilla, J., Romero, J., Romero, F., Palmero, F.: Spiral wave solutions ineaction-diffusion equations with symmetries. Analysis through specific models. J. Phys. A Math. Gen. 30, 4259–4271 (1997)

    Article  MATH  Google Scholar 

  3. Arrigo, D.J., Broadbridge, P., Hill, J.M.: Nonclassical symmetry solutions and the methods of Bluman–Cole and Clarkson–Kruskal. J. Math. Phys. 34, 4692–4703 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barannyk, T.: Symmetry and exact solutions for systems of nonlinear reaction-diffusion equations. Proc. Inst. Math. Nat. Acad. Sci. Ukraine 43, 80–85 (2002)

    MathSciNet  MATH  Google Scholar 

  5. Barannyk, T.A., Nikitin, A.G.: Solitary wave solutions for heat equations. Proc. Inst. Math. Nat. Acad. Sci. Ukraine 50, 34–39 (2004)

    MathSciNet  MATH  Google Scholar 

  6. Bertsch, M., Kersner, R., Peletier, L.A.: Positivity versus localization in degenerate diffusion equations. Nonliniar Anal. TMA 9, 987–1008 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bluman, G.W., Cole, J.D.: The general similarity solution of the heat equation. J. Math. Mech. 18, 1025–1042 (1969)

    MathSciNet  MATH  Google Scholar 

  8. Boussinesq, J.: Recherches théoriques sur l’écoulement des nappes d’eau infiltrées dans le sol et sur débit de sources (in French). J. Math. Pures Appl. 10, 5–78 (1904)

    MATH  Google Scholar 

  9. Carini, M., Fusco, D., Manganaro, N.: Wave-like solutions for a class of parabolic models. Nonlinear Dyn. 32, 211–222 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cherniha, R.M.: On exact solutions of a nonlinear diffusion-type system. In: Symmetry Analysis and Exact Solutions of Equations of Mathematical Physics. Inst. Math., Acad. Sci. of USSR, Kyiv, pp. 49–53 (1988)

    Google Scholar 

  11. Cherniha, R.: Symmetry and exact solutions of heat-and-mass transfer equations in Tokamak plasma (in Ukrainian). Proc. Acad. Sci. Ukraine 4, 17–21 (1995)

    Google Scholar 

  12. Cherniha, R.: A constructive method for construction of new exact solutions of nonlinear evolution equations. Rep. Math. Phys. 38, 301–312 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cherniha, R.: New non-Lie ansätze and exact solutions of nonlinear reaction-diffusion-convection equations. J. Phys. A Math. Gen. 31, 8179–8198 (1998)

    Article  MATH  Google Scholar 

  14. Cherniha, R.: Lie symmetries of nonlinear two-dimensional reaction-diffusion systems. Rept. Math. Phys. 46, 63–76 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cherniha, R.: Nonlinear Galilei-invariant PDEs with infinite-dimensional Lie symmetry. J. Math. Anal. Appl. 253, 126–141 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cherniha, R., Davydovych, V.: Conditional symmetries and exact solutions of nonlinear reaction-diffusion systems with non-constant diffusivities. Commun. Nonlinear Sci. Numer. Simul. 17, 3177–3188 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cherniha, R., Davydovych, V.: Reaction-diffusion systems with constant diffusivities: conditional symmetries and form-preserving transformations. In: Algebra, Geometry and Mathematical Physics, vol. 85, pp. 533–553. Springer, Berlin (2014)

    Google Scholar 

  18. Cherniha, R., Davydovych, V.: Nonlinear reaction-diffusion systems with a non-constant diffusivity: conditional symmetries in no-go case. Appl. Math. Comput. 268, 23–34 (2015)

    MathSciNet  Google Scholar 

  19. Cherniha, R., Dutka, V.: A diffusive Lotka–Volterra system: Lie symmetries, exact and numerical solutions. Ukr. Math. J. 56, 1665–1675 (2004)

    Article  MathSciNet  Google Scholar 

  20. Cherniha, R., King, J.R.: Lie symmetries of nonlinear multidimensional reaction-diffusion systems: I. J. Phys. A Math. Gen. 33, 267–282, 7839–7841 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cherniha, R., King, J.R.: Lie symmetries of nonlinear multidimensional reaction-diffusion systems: II. J. Phys. A Math. Gen. 36, 405–425 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Cherniha, R., King, J.R.: Nonlinear reaction-diffusion systems with variable diffusivities: Lie symmetries, ansätze and exact solutions. J. Math. Anal. Appl. 308, 11–35 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cherniha, R.M., Serov, M.I.: Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection term II. Euro. J. Appl. Math. 17, 597–605 (2006)

    Article  MATH  Google Scholar 

  24. Cherniha, R., Waniewski, J.: Exact solutions of a mathematical model for fluid transport in peritoneal dialysis. Ukr. Math. J. 57, 1316–1324 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Cherniha, R., Serov, M., Rassokha, I.: Lie symmetries and form-preserving transformations of reaction-diffusion-convection equations. J. Math. Anal. Appl. 342, 1363–1379 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Clarkson, P.A.: New similarity solutions for the modified Boussinesq equation. J. Phys. A Math. Gen. 22, 2355 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Clarkson, P.A., Kruskal, M.D.: New similarity reductions of the Boussinesq equation. J. Math. Phys. 30, 2201–2213 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  28. Fushchych, W.I., Zhdanov, R.Z.: Antireduction and exact solutions of nonlinear heat equations. J. Nonlinear Math. Phys. 1 60–64 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  29. Galaktionov, V.A.: Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities. Proc. Roy. Soc. Edinb. Sect. A 125 225–246 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. Galaktionov, V.A., Svirshchevskii, S.R.: Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics. Chapman & Hall/CRC, Boca Raton (2007)

    MATH  Google Scholar 

  31. Gazeau, J.P., Winternitz, P.: Symmetries of variable coefficient Korteweg-de Vries equations. J. Math. Phys. 33, 4087–4102 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. Gudkov, V.V.: Exact solutions of the type of propagating waves for certain evolution equations. Dokl. Ros. Akad. Nauk 353, 439–441 (1997)

    MathSciNet  MATH  Google Scholar 

  33. Hashemi, M.S., Nucci, M.C.: Nonclassical symmetries for a class of reaction-diffusion equations: the method of heir-equations. J. Nonlinear Math. Phys. 20, 44–60 (2013)

    Article  MathSciNet  Google Scholar 

  34. Hung, L.-C.: Exact traveling wave solutions for diffusive Lotka–Volterra systems of two competing species. Jpn. J. Indust. Appl. Math. 29, 237–251 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ji, L.: The method of linear determining equations to evolution system and application for reaction-diffusion system with power diffusivities. Symmetry 8, 157 (2016)

    Article  MathSciNet  Google Scholar 

  36. Kamke, E.: Differentialgleichungen. Lösungsmethoden und Lösungen. I: Gewöhnliche (in German). Stuttgart (1977)

    Google Scholar 

  37. Kaptsov, O.V.: Construction of exact solutions of systems of diffusion equations (in Russian). Math. Model. 7, 107–115 (1995)

    MathSciNet  MATH  Google Scholar 

  38. Kaptsov, O.V.: Determining equations and differential constraints. J. Nonlinear Math. Phys. 2, 283–291 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kaptsov, O.V., Verevkin. V.: Differential constraints and exact solutions of nonlinear diffusion equations. J. Phys. A Math. Gen. 36, 1401 (2003)

    Google Scholar 

  40. King, J.R.: Exact results for the nonlinear diffusion equations ∂u∕∂t = (∂∕∂x)(u −4∕3 ∂u∕∂x) and ∂u∕∂t = (∂∕∂x)(u −2∕3 ∂u∕∂x). J. Phys. A Math. Gen. 24, 5721–5745 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  41. King, J.R.: Some non-self-similar solutions to a nonlinear diffusion equation. J. Phys. A Math. Gen. 25, 4861–4868 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  42. King, J.R.: Exact multidimensional solutions to some nonlinear diffusion equations. Quart. J. Mech. Appl. Math. 46, 419–436 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  43. King, J.R.: Exact polynomial solutions to some nonlinear diffusion equations. Physica D 64, 35–65 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  44. Kingston, J.G., Sophocleous, C.: On form-preserving point transformations of partial differential equations. J. Phys. A 31, 1597–1619 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  45. Kudryashov, N.A.: Comment on: “A novel approach for solving the Fisher equation using Exp-function method” [Phys. Lett. A 372 3836 (2008)] [MR2418599]. Phys. Lett. A 373, 1196–1197 (2009)

    MATH  Google Scholar 

  46. Ma, W.X., Huang, T., Zhang, Y.: A multiple exp-function method for nonlinear differential equations and its applications. Phys. Scr. 82, 065003 (2010)

    Article  MATH  Google Scholar 

  47. Malfliet, W.: The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations. J. Comp. Appl. Math. 164, 529–541 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  48. Murray, J.D.: Mathematical Biology. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  49. Myroniuk, L.P., Cherniha, R.M.: Reduction and solutions of a class of nonlinear reaction-diffusion systems with the power nonliniarities (in Ukrainian). Proc. Inst. Math. NAS Ukraine 3, 217–224 (2006)

    Google Scholar 

  50. Nucci, M.C.: Iterations of the non-classical symmetries method and conditional Lie-Bäcklund symmetries. J. Phys. A Math. Gen. 29, 8117–8122 (1996)

    Article  MATH  Google Scholar 

  51. Oron, A. Rosenau, P.: Some symmetries of the nonlinear heat and wave equations. Phys. Lett. A 118, 172–176 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  52. Ovsiannikov, L.V.: The Group Analysis of Differential Equations. Academic, New York (1980)

    MATH  Google Scholar 

  53. Polyanin, A.D.: Exact solutions of nonlinear sets of equations of the theory of heat and mass trannsfer in reactive media and mathematical bilogy. Theor. Found. Chem. Eng. 38, 622–635 (2004)

    Article  Google Scholar 

  54. Polyanin, A.D., Zaitsev, V.F.: Handbook of Exact Solutions for Ordinary Differential Equations. CRC Press, London (2003)

    MATH  Google Scholar 

  55. Polyanin, A.D., Zaitsev, V.F.: Handbook of Nonlinear Partial Differential Equations, 2nd edn. CRC Press, Boca Raton (2012)

    MATH  Google Scholar 

  56. Popovych, R.O.: On a class of Q-conditional symmetries and solutions of evolution equations (in Ukrainian). In: Symmetry and Analytic Methods in Mathematical Physics. Proc. Inst. Math.of Ukraine, Kyiv, vol. 19, pp. 194–199 (1998)

    Google Scholar 

  57. Rodrigo, M., Mimura, M.: Exact solutions of a competition-diffusion system. Hiroshima Math. J. 30, 257–270 (2000)

    MathSciNet  MATH  Google Scholar 

  58. Sidorov, A.F., Shapeev, V.P., Yanenko, N.N.: Method of Differential Relations and its Application to Gas Dynamics (in Russian). Nauka, Novosibirsk (1984)

    MATH  Google Scholar 

  59. Svirshchevskii, S.: Invariant linear spaces and exact solutions of nonlinear evolution equations. J. Nonlinear Math. Phys. 3, 164–79 (1996)

    Article  MathSciNet  Google Scholar 

  60. Titov, S.S.: Solutions of non-linear partial differential equations in the form of polynomials with respect to one of variables (in Russian). Chislennye Metody Mekhaniki Sploshnoy Sredy 8, 144–149 (1977)

    Google Scholar 

  61. Vyaz’mina, E.A., Polyanin, A.D.: New classes of exact solutions to general nonlinear diffusion-kinetic equations. Theor. Found. Chem. Eng. 40, 595–603 (2006)

    MATH  Google Scholar 

  62. Wang, J., Ji, L.: Conditional Lie–Bäcklund symmetry, second-order differential constraint and direct reduction of diffusion systems. J. Math. Anal. Appl. 427, 1101–1118 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  63. Wazwaz, A.M.: The extended tanh method for the Zakharo–Kuznetsov (ZK) equation, the modified ZK equation, and its generalized forms. Commun. Nonlinear Sci. Numer. Simulat. 13, 1039–1047 (2008)

    Article  MATH  Google Scholar 

  64. Yanenko, N.N.: Compatibility theory and methods of integration of systems of nonlinear partial differential equations (in Russian). In: Proceedings of Fourth All-Union Mathematical Conference, pp. 613–621. Nauka, Leningrad (1964)

    Google Scholar 

  65. Zhdanov, R.Z., Lahno, V.I.: Conditional symmetry of a porous medium equation. Phys. D 122, 178–186 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, National Academy of Science, Kyiv, Ukraine

    Roman Cherniha & Vasyl’ Davydovych

Authors
  1. Roman Cherniha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vasyl’ Davydovych
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Cherniha, R., Davydovych, V. (2017). Q-Conditional Symmetries of the First Type and Exact Solutions of Nonlinear Reaction-Diffusion Systems. In: Nonlinear Reaction-Diffusion Systems. Lecture Notes in Mathematics, vol 2196. Springer, Cham. https://doi.org/10.1007/978-3-319-65467-6_4

Download citation

Publish with us

Policies and ethics

Search

Navigation