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Mathematical Biosciences and Engineering

2022, Volume 19, Issue 7: 6962-6984. doi: 10.3934/mbe.2022328
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On reducing and finding solutions of nonlinear evolutionary equations via generalized symmetry of ordinary differential equations

  • AGH University of Science and Technology, Faculty of Applied Mathematics, aleja Adama Mickiewicza 30, 30-059 Kraków, Poland

  • Academic Editor: Masood Khalique
  • Received: 30 December 2021 Revised: 26 April 2022 Accepted: 03 May 2022 Published: 10 May 2022

  • We study symmetry reductions of nonlinear partial differential equations that can be used for describing diffusion processes in heterogeneous medium. We find ansatzes reducing these equations to systems of ordinary differential equations. The ansatzes are constructed using generalized symmetries of second-order ordinary differential equations. The method applied gives the possibility to find exact solutions which cannot be obtained by virtue of the classical Lie method. Such solutions are constructed for nonlinear diffusion equations that are invariant with respect to one-parameter and two-parameter Lie groups of point transformations. We prove a theorem relating the property of invariance of a found solution to the dimension of the Lie algebra admitted by the corresponding equation. We also show that the method is applicable to non-evolutionary partial differential equations and ordinary differential equations.

    Citation: Ivan Tsyfra, Wojciech Rzeszut. On reducing and finding solutions of nonlinear evolutionary equations via generalized symmetry of ordinary differential equations[J]. Mathematical Biosciences and Engineering, 2022, 19(7): 6962-6984. doi: 10.3934/mbe.2022328

    Related Papers:

  • We study symmetry reductions of nonlinear partial differential equations that can be used for describing diffusion processes in heterogeneous medium. We find ansatzes reducing these equations to systems of ordinary differential equations. The ansatzes are constructed using generalized symmetries of second-order ordinary differential equations. The method applied gives the possibility to find exact solutions which cannot be obtained by virtue of the classical Lie method. Such solutions are constructed for nonlinear diffusion equations that are invariant with respect to one-parameter and two-parameter Lie groups of point transformations. We prove a theorem relating the property of invariance of a found solution to the dimension of the Lie algebra admitted by the corresponding equation. We also show that the method is applicable to non-evolutionary partial differential equations and ordinary differential equations.


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