The Nonstandard Finite-Difference Algorithm

Nonstandard finite-difference schemes are, primarily, a generalization of the customary models of differential equations [ 1-4]. While the latter cannot satisfactorily model cumbersome physical phenomena, the particular forms maintain the underlying properties of their continuous analogs and evade elementary instabilities. In this manner, they have gained a noteworthy popularity in several scientific fields [5], like fluid mechanics, electromagnetics, photoconductivity, nonlinear optics, and mathematical biology to name a few. 

Based on elaborate analytical and numerical investigations, a set of rules that accumulate the focal characteristics of a prospective framework has been introduced in [1]. These maybe summarized as 

Rule 1 The order of the discrete derivatives must be equal to the order of the corresponding 

Rule 2 Denominator functions for the discrete derivatives must be generally defined by means of more sophisticated expressions of spatial and temporal increments than those of the conventional ones. 

Rule 3 Nonlinear terms must be represented in a nonlocal discrete sense. 

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