Uniformly convergent non-standard finite difference methods for singularly perturbed differential-difference equations with delay and advance

Abstract

SUMMARYA new class of fitted operator finite difference methods are constructed via non-standard finite difference methods ((NSFDM)s) for the numerical solution of singularly perturbed differential difference equations having both delay and advance arguments. The main idea behind the construction of our method(s) is to replace the denominator function of the classical second-order derivative with a positive function derived systematically in such a way that it captures significant properties of the governing differential equation and thus provides the reliable numerical results. Unlike other FOFDMs constructed in standard ways, the methods that we present in this paper are fairly simple to construct (and thus enrich the class of fitted operator methods by adding these new methods). These methods are shown to be -uniformly convergent with order two which is the highest possible order of convergence obtained via any fitted operator method for the problems under consideration. This paper further clarifies several doubts, e.g. why a particular scheme is not suitable for the whole range of values of the associated parameters and what could be the possible remedies. Finally, we provide some numerical examples which illustrate the theoretical findings.

A fitted numerical method for a system of partial delay differential equations

“…We have noticed that when κ/λ < 1, the solutions do always go to the unique stable trivial attractor (0, 0) (see Figs. 5,6,13,14,21,22,29 and 30). When the ratio κ/λ passes the unity, a stable or a stable periodic solution bifurcates from κ/λ = 1 (see Figs.…”

“…These FOFDMs, which are based on the philosophy of non-standard finite difference methods of Mickens [12,13] are widely used for singularly perturbed twopoint boundary value problems. See for instance [14][15][16]. However, their extensions for the coupled PDEs whose solutions possess oscillatory dynamics, have not been seen in the literature.…”

An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection

“…Applications of these NSFDMs for singularly perturbed problems can be seen in the work of Kadalbajoo et al (2006), Lubuma and Patidar (2006;2007c;2007a;2007b), Munyakazi and Patidar (2010), Patidar and Sharma (2006a;2006b), or Patidar (2008. However, an exhaustive account of work that use such methods is provided in the survey article by Patidar (2005).…”

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

“…The first scheme is based on the nonstandard finite difference (NSFD) method which originated from the work of Mickens, [13,14]. 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.…”