The presented dataset is a result of numerical analysis of a generalized Burgers–Huxley partial differential equation. An analyzed diffusive partial differential equation consist with nonlinear advection and reaction. The reaction term is a generalized form of the reaction law of the Hodgkin–Huxley model, while the advection is a generalized form of the corresponding term in the classical Burgers equation. This generalized Burgers–Huxley equation possesses travelling-wave solutions that are positive and bounded. Moreover, such solutions are spatially monotone at each instant of time, and temporally monotone at each spatial point. Unfortunately, only a few travelling-wave solutions of such model are known in exact form, therefore, the construction of a suitable numerical method is highly desirable.
We provide a Mickens-type, nonlinear, finite-difference discretization of this model. The conditionally monotone scheme approximates solutions of the generalized Burgers–Huxley model, and preserves the positive and the bounded characters of initial approximations. Such mathematical features of the constructed finite-difference scheme are important characteristics of the travelling-wave solutions of interest.
The construction of numerical method, conditions that guarantee the existence and the uniqueness of monotone and bounded solutions of the scheme, are available in the paper: Existence and uniqueness of monotone and bounded solutions for a finite-difference discretization à la Mickens of the generalized Burgers–Huxley equation, Journal of Difference Equations and Applications (2014) , Vol. 20, No. 7, 989–1004,http://dx.doi.org/10.1080/10236198.2013.877457.
The dataset consists of illustrative simulations (480 .tif files) which demonstrate the capability of the method to preserve the positivity, the boundedness and the monotonicity of solutions, and to provide good approximations to the known exact solutions bounded within [0,1] or within [0,γ^(1/p)]. The graphs show the results along time T=100 (for boundedness [0,1]) and T=60 (for boundedness [0,γ^(1/p)]) with different values of steps satisfying the convergence conditions. We perform calculations with model parameters α = 1, γ = 0.8, p = 2
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- Fields of science:
- Mathematics (Natural sciences)
- 10.34808/xdp5-xm94 open in new tab
- Verified by:
- Gdańsk University of Technology
- numerical analysis
- generalized Burgers–Huxley equation
- Mickens-type numerical method
- finite-difference discretization
- travelling-wave solutions
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