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On Symmetries and Conservation Laws for a Generalized Fisher–Kolmogorov–Petrovsky–Piskunov Equation

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A Mathematical Modeling Approach from Nonlinear Dynamics to Complex Systems

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 22))

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Abstract

This chapter presents a generalized Fisher equation (GFE) from the point of view of the theory of symmetry reductions in partial differential equations. The GFE can be used to describe an ideal growth and spatial-diffusion phenomena. The reductions to ordinary differential equations are derived from the optimal system of subalgebras and new exact solutions are obtained. Conservation laws for this equation are constructed. The potential system has been achieved from the complete list of the conservation laws. Potential symmetries, which are not local symmetries, are carried out for the generalized Fisher equation, these symmetries lead to the linearization of the equation by non-invertible mappings.

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Acknowledgements

The authors warmly thank the anonymous referees for careful reading of the manuscript, and for providing excellent suggestions to improve the quality of the paper. They also acknowledge the financial support from Junta de Andalucía FQM-201 group, and from University of Cádiz Plan Propio de Investigación and project PR2016-097. M.S. Bruzón and M.L. Gandarias warmly thank the Organizing Committee of NSC-2016.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Cádiz, Cádiz, Spain

    María Luz Gandarias, María de los Santos Bruzón & María Rosa

Authors
  1. María Luz Gandarias
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  2. María de los Santos Bruzón
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  3. María Rosa
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Correspondence to María Luz Gandarias .

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  1. Instituto Nacional de Pesquisas Espaciais – INPE, São José dos Campos, Brazil

    Elbert E. N. Macau

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Gandarias, M.L., de los Santos Bruzón, M., Rosa, M. (2019). On Symmetries and Conservation Laws for a Generalized Fisher–Kolmogorov–Petrovsky–Piskunov Equation. In: Macau, E. (eds) A Mathematical Modeling Approach from Nonlinear Dynamics to Complex Systems . Nonlinear Systems and Complexity, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-78512-7_3

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  • DOI: https://doi.org/10.1007/978-3-319-78512-7_3

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  • Print ISBN: 978-3-319-78511-0

  • Online ISBN: 978-3-319-78512-7

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