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Generalized Symmetries and Recursive Operators of Some Diffusive Equations

  • Sameerah JamalEmail author
  • A. Mathebula
Article

Abstract

This paper considers different routes to generalized symmetries for some ecological equations that arise in spatial theory. Two primary methods for the derivation of generalized symmetries are the standard Lie invariance condition with vector fields dependent on derivatives and, secondly, a recursive operator. The former is less efficient especially if it includes derivatives that become increasingly higher in order, and this necessarily complicates the nature of the computations. The latter involves a nontrivial analysis to define a recursion operator, if one exists, but is successful in providing higher-order analogs of the equation or equivalently, higher-order symmetries. A linear Kierstead–Slobodkin and Skellam model is shown to possess a recursion operator that renders the equation completely integrable, by verifying the presence of infinitely many higher-order symmetries. Moreover, we apply the scheme of the characteristic approach to establish nontrivial conserved vectors from multipliers \({\varLambda }(t,x,u,u_x,u_t),\) that are analogous to integrating factors.

Keywords

Diffusion equations Lie symmetries Higher-order symmetries Recursion operators 

Mathematics Subject Classification

37L20 35K57 70G65 58J72 

References

  1. 1.
    Olver, P.J.: Evolution equations possessing infinitely many symmetries. J. Math. Phys. 18(6), 1212–1215 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Lax, P.D.: Periodic solutions of the KdV equation. Commun. Pure Appl. Math. 28, 141–188 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Paliathanasis, A., Krishnakumar, K., Tamizhmani, K.M., Leach, P.G.L.: Lie symmetry analysis of the Black–Scholes–Merton model for European options with stochastic volatility. Mathematics 4(2), 1–14 (2016)zbMATHGoogle Scholar
  4. 4.
    Nucci, M.C., Sanchini, G.: Noether symmetries quantization and superintegrability of biological models. Symmetry 8, 1–9 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Paliathanasis, A., Tsamparlis, M.: The reduction of Laplace equation in certain Riemannian spaces and the resulting Type II hidden symmetries. J. Geom. Phys. 76, 107–123 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Belmonte-Beitia, J., Pérez-García, V.M., Vekslerchik, V., Torres, P.J.: Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities. Phys. Rev. Lett. 98, 064102 (2007)CrossRefGoogle Scholar
  7. 7.
    Champagne, B., Hereman, W., Winternitz, P.: The computer calculation of Lie point symmetries of large systems of differential equations. Comput. Phys. Commun. 66, 319–340 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Baumann, G.: Symmetry Analysis of Differential Equations with Mathematica. Springer, New York (2000)CrossRefzbMATHGoogle Scholar
  9. 9.
    Dimas, S., Tsoubelis, D.: SYM: A New Symmetry-Finding Package for Mathematica in Group Analysis of Differential Equations. University of Cyprus, Nicosia (2005)Google Scholar
  10. 10.
    Steudel, H.: Über die Zuordnung zwischen Invarianzeigenschaften und Erhaltungssätzen. Zeitschrift für Naturforschung 17, 129–132 (1962)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Jamal, S., Kara, A.H.: New higher-order conservation laws of some classes of wave and Gordon-type equations. Nonlinear Dyn. 67, 97–102 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Morris, R., Kara, A.H., Biswas, A.: Soliton solution and conservation laws of the Zakharov equation in plasmas with power law nonlinearity. Nonlinear Anal.: Model. Control 18(2), 153–159 (2013)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Jamal, S., Kara, A.H., Bokhari, A.H., Zaman, F.D.: The symmetries and conservation laws of some Gordon-type equations in Milne space-time. Pramana J. Phys. 80(5), 739–755 (2013)CrossRefGoogle Scholar
  14. 14.
    Naz, R.: Conservation laws for some systems of nonlinear partial differential equations via multiplier approach. J. Appl. Math. 871253, 1–13 (2012)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Jamal, S., Kara, A.H.: Higher-order symmetries and conservation laws of multi-dimensional Gordon-type equations. Pramana J. Phys. 77(3), 1–14 (2011)CrossRefGoogle Scholar
  16. 16.
    Skellam, J.G.: Random dispersal in theoretical populations. Biometrika 38, 196–218 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Holmes, E.E., Lewis, M.A., Banks, J.E., Veit, R.R.: Partial differential equations in ecology: spatial interactions and population dynamics. Ecology 75(1), 17–29 (1994)CrossRefGoogle Scholar
  18. 18.
    Goldstein, S.: On diffusion by discontinuous movements, and on the telegraph equation. Q. J. Mech. Appl. Mech. 6, 129–156 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugen. 7, 355–369 (1937)CrossRefzbMATHGoogle Scholar
  20. 20.
    Dobzhansky, T., Wright, S.: Genetics of natural populations. X. Dispersion rates in Drosophila pseudoobscura. Genetics 28, 304–340 (1943)Google Scholar
  21. 21.
    Okubo, A.: Diffusion and Ecological Problems: Mathematical Models. Springer, Berlin (1980)zbMATHGoogle Scholar
  22. 22.
    Helland, I.S., Hoff, J.M., Anderbrant, G.: Attraction of bark beetles (Coleoptera: Scolytidae) to a pheromone trap: experiment and mathematical models. J. Chem. Ecol. 10, 723–752 (1984)CrossRefGoogle Scholar
  23. 23.
    Bluman, G.W.: Simplifying the form of Lie groups admitted by a given differential equation. J. Math. Anal. Appl. 145, 52–62 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gurney, W.S.C., Nisbet, R.M.: The regulation of inhomogeneous populations. J. Theor. Biol. 52, 441–457 (1975)CrossRefGoogle Scholar
  25. 25.
    LeVeque, R.J.: Numerical Methods for Conservation Laws. Birkhauser-Verlag, Basel (1992)CrossRefzbMATHGoogle Scholar
  26. 26.
    Hereman, W.: Symbolic computation of conservation laws of nonlinear partial differential equations in multidimensions. Int. J. Quantum Chem. 106, 278–299 (2006)CrossRefzbMATHGoogle Scholar
  27. 27.
    Kara, A.H.: An analysis of the symmetries and conservation laws of the class of Zakharov–Kuznetsov equations. Math. Comput. Appl. 15(4), 658–664 (2010)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Anco, S., Bluman, G.: Direct construction method for conservation laws of partial differential equations Part I: examples of conservation law classifications. Eur. J. Appl. Math. 13, 545–566 (2002)zbMATHGoogle Scholar
  29. 29.
    Kierstead, H., Slobodkin, L.B.: The size of water masses containing plankton blooms. J. Mar. Res. 12, 141–147 (1953)Google Scholar
  30. 30.
    Fokas, A.S.: Symmetries and integrability. Stud. Appl. Math. 77, 253–299 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.School of Mathematics and Centre for Differential Equations, Continuum Mechanics and ApplicationsUniversity of the WitwatersrandJohannesburgSouth Africa

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