Derivation of New Non-Standard Finite Difference Schemes for Non-autonomous Ordinary Differential equation

Abstract

In this paper we derive Non-standard finite difference schemes for the solution of some IVPs emanating from non-autonomous Ordinary Differential Equation. A new technique based on the method of Non-local approximation and renormalization of the denominator function was employed. Numerical experiments were used to verify the reliability of the new Finite Difference schemes proposed for the IVPs and the results obtained shows that the schemes are computationally reliable.

Non-standard Numerical Approaches for Solutions of Some Second Order Ordinary Differential Equations

“…It has been observed in the tested cases that the use of ߰ = ൫ ഊ ିଵ൯ ఒ , λ ϵ R is better than ߰ =sinሺℎሻ (see Table 1 for Schemes A2 and A3, Table 2 for Schemes A2 and A3, Table 3 for Schemes B2 and B3).This is not unconnected with the opportunity to choose λ appropriately to satisfy the condition ߰ሺℎሻ → ℎ + 0ሺℎ ଶ ሻ ‫ݏܽ‬ ℎ → 0 . This is also because the step size is dynamically varied during the iterations .This confirms some earlier results (for example see [7] and [8]). It can be that using a fixed h during iterations makes each of the schemes perform poorly.…”

“…It has been shown by [3,4] that the central difference scheme allows for the existence of chaotic orbits for all positive time-steps for the Logistic differential equation. Notable work on this problem has been done by other researchers including [5,6,7,8]. The major conclusion is that the use of the central difference scheme forces all the fixed-points to become unstable.…”

Nonlinear Elimination of Drugs in One-Compartment Pharmacokinetic Models: Nonstandard Finite Difference Approach for Various Routes of Administration

“…The nonstandard finite difference (NSFD) scheme was developed by Mickens as an alternative method providing an approximate solution to a wide range of differential equations and catering for the numerical instabilities that occur when using standard methods. NSFD methods have been well reported in recent years, mainly because they are efficient and and preserve qualitative properties, see for example Villatoro [7], Roeger [8], Ibijola & Obayomi [9], Manning & Margrave [10], Mickens [11][12][13][14][15][16] and Sunday [17] which give the relevant background materials on this topic. Some of these authors deal with the exact finite difference scheme which is a special NSFD method.…”

Initial-Value Problems for Ordinary Differential Equations

“…Following [15] (see also [11] and [12] for related work), we assume that the theoretical solution () yx to the initial value problem (1.1) can be locally…”