Existence and uniqueness of monotone and bounded solutions for a finite-difference discretization à la Mickens of the generalized Burgers–Huxley equation |
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Authors: | JE Macías-Díaz Anna Szafrańska |
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Institution: | 1. Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Aguascalientes 20131, Mexicojemacias@correo.uaa.mx;3. Department of Applied Physics and Mathematics, Gdańsk University of Technology, 80-233 Gdańsk, Poland |
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Abstract: | Departing from a generalized Burgers–Huxley partial differential equation, we provide a Mickens-type, nonlinear, finite-difference discretization of this model. The continuous system is a nonlinear regime for which the existence of travelling-wave solutions has been established previously in the literature. We prove that the method proposed also preserves many of the relevant characteristics of these solutions, such as the positivity, the boundedness and the spatial and the temporal monotonicity. The main results provide conditions that guarantee the existence and the uniqueness of monotone and bounded solutions of our scheme. The technique was implemented and tested computationally, and the results confirm both a good agreement with respect to the travelling-wave solutions reported in the literature and the preservation of the mathematical features of interest. |
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Keywords: | existence and uniqueness of solutions Mickens-type finite-difference scheme generalized Burgers–Huxley model preservation of positivity and boundedness monotone method preservation of skew-symmetry |
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