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Abstract
In this work, we present a finite-difference scheme that preserves the non-negativity and the boundedness of some solutions of a FitzHugh–Nagumo equation. The method is explicit, and it approximates the solutions of the nonlinear, parabolic partial differential equation under study with a consistency of order 𝒪 (Δ t+(Δ x)2) in the Dirichlet regime investigated. We give sufficient conditions in terms of the computational and the model parameters, in order to guarantee the non-negativity and the boundedness of the approximations. We also provide analyses of consistency, linear stability and convergence of the method. Our simulations establish that the properties of non-negativity and boundedness are actually preserved by the scheme when the proposed constraints are satisfied. Finally, a comparison against some second-order accurate methods reveals that our technique is easier to implement computationally, and it is better at preserving the properties of non-negativity and boundedness of the solutions of the FitzHugh–Nagumo equation under study.
Acknowledgements
The authors want to thank the editor and the anonymous reviewers for their invaluable comments, which led to improve the overall quality of this manuscript. One of us (JEMD) wishes to express his deepest gratitude to Dr. F. J. Álvarez-Rodríguez, dean of the Faculty of Sciences of the Universidad Autónoma de Aguascalientes (UAA), and to Dr. F. J. Avelar-González, director of the Office for Research and Graduate Studies of the same university, for support in the form of computational resources. This article summarizes the results of project PIM09-1 at UAA.