Abstract
This research paper represents a numerical approximation to three interesting equations of Fisher, which are linear, non-linear and coupled linear one dimensional reaction diffusion equations from population genetics. We studied accuracy in term of L∞ error norm by random selected grids along time levels for comparison with exact results. The test example demonstrates the accuracy, efficiency and versatility of the proposed schemes. It is shown that the numerical schemes give better solutions. Moreover, the schemes can be easily applied to a wide class of higher dimension non-linear reaction diffusion equations.
Highlights
Reaction diffusion equations arise as models for the densities of substances or organisms that disperse through space by Brownian motion, random walks, hydrodynamic turbulence, or similar mechanisms, and that react to each other and their surroundings in ways that affect their local densities [1]
The researchers studied some meaningful generalization of this equation, here we considered one generalization of this equation which is called as one component reaction diffusion equation
First we look at the linear Fisher’s equation by finite difference schemes as in Table 1, we used Forward in Time and Centre in Space (FTCS) explicit scheme with some variations in grid size and h is changed according to the grid sizes. this table explains the second order accuracy in term of L∞ norm, of the explicit numerical scheme Table 2 explains results for FTCS with different time steps (k)
Summary
Reaction diffusion equations arise as models for the densities of substances or organisms that disperse through space by Brownian motion, random walks, hydrodynamic turbulence, or similar mechanisms, and that react to each other and their surroundings in ways that affect their local densities [1]. Reaction diffusion models are in themselves deterministic, but they can be derived as limits of stochastic processes under suitable scaling. They provide a modelling approach that allows us to translate assumptions about stochastic local movement into deterministic descriptions of global densities [1] [2]. Reaction diffusion models are spatially explicit, describe population densities, and treat space and time as continuous [1] [2] [3]. Mashat of travelling wave fronts corresponding to biological invasions, and the formation of spatial patterns [1] [2] [3]
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