Non-standard finite-differences schemes for generalized reaction–diffusion equations

Abstract

a b s t r a c tReaction-diffusion equations are commonly used in different science and engineering fields to describe spatial patterns arising from the interaction of chemical or biochemical reactions and diffusive transport mechanisms. In this work we design, in a systematic way, non-standard finite-differences (FD) schemes for a class of reaction-diffusion equations of the form 1, where σ is the shape power that accounts for the complexity of the domain geometry. The proposed FD scheme, that is derived from a Green's function formulation, replicates the underlying geometry and reduces to traditional FD schemes for sufficiently small values of the grid spacing. Numerical results show that the non-standard FD scheme offers smaller approximation errors with respect to traditional schemes, specially for coarse grids.

Comparison between standard and non-standard finite difference methods for solving first and second order ordinary differential equations

“…Standard method: (2). In this example, we choose the numerator and denominator functions as follows: (12) In figure 1 the results of equations (9) and (12)…”

“…Where () h  and () h  are known, respectively, as the numerator and denominator functions, having the properties The full details about these procedures are given in [11][12][13][14][15][16]. The nonstandard finite difference scheme has developed as an alternative method for solving a wide range of problems whose mathematical models involve algebraic, differential, biological models and chaotic systems [1], [2] and [4][5][6][7][8][9][10]. In this work, we compare non-standard finite difference (NSFD) and standard finite difference (FD) schemes for solving ordinary differential equations.…”

Coupling finite volume and nonstandard finite difference schemes for a singularly perturbed Schrödinger equation

“…We highlight that NSFD schemes have been efficient in tackling the deficiency of classical finite difference schemes for the approximation of solutions of several differential equation models, see for example [16,17,18,19,20] and the literature therein. Here we also highlight the work [1,6], which differs from the approach adopted here in the sense that their NSFD schemes are derived by adopting a Green's function approach which offers a good platform to develop nontrivial schemes for a range of reaction diffusion equations. For more on the application and earlier developments on the NSFD the reader can consult [14,15].…”

Solving Dynamic Equation by the Non-Standard Finite Difference Method

“…The full details of these procedures are given by some studies [1][2][3][4][5][6]. The non-standard finite difference schemes have been developed as an alternative method for solving a wide range of problems such as the mathematical models of biology and chaotic systems [7][8][9][10][11][12][13][14].…”

“…8 we compared the non-standard finite difference scheme (19) and standard finite difference scheme (20) for various values of h.…”