Exact finite-difference schemes for first order differential equations having three distinct fixed-points

Abstract

We construct nonstandard finite-difference (NSFD) schemes that provide exact numerical methods for a first-order differential equation having three distinct fixed-points. An explicit, but also nonexact, NSFD scheme is also constructed. It has the feature of preserving the critical properties of the original differential equation such as the positivity of the solutions and the stability behavior of the three fixedpoints.

General nonstandard finite-difference schemes for differential equations with three fixed-points

“…General NSFD method for (9) In this section, we construct general NSFD schemes for the differential equation (9). They are as follows:…”

“…In this manuscript, we extend the previous results and construct more general NSFD schemes for the two differential Eqs. (8) and (9) and the schemes are elementary stable and stable with respect to the monotonicity of solutions for y ≥ 0. The schemes are constructed so that the difference equations preserve the positivity of solutions (the set [0, ∞) for y is positively invariant).…”

“…In [9], an NSFD scheme for Eq. (9) that is elementary stable and preserves positivity of solutions was given.…”

“…In Section 3, the NSFD method in [9] for differential equation (9) is revisited and we construct more general schemes for Eq. (9).…”

“…In Section 3, the NSFD method in [9] for differential equation (9) is revisited and we construct more general schemes for Eq. (9). Finally, in Section 4, we discuss our results and provide several general comments on related issues for future study.…”