Generalized Symmetries and Recursive Operators of Some Diffusive Equations

  • Sameerah JamalEmail author
  • A. Mathebula


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.


Diffusion equations Lie symmetries Higher-order symmetries Recursion operators 

Mathematics Subject Classification

37L20 35K57 70G65 58J72 


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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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