Positivity and dynamics preserving discretization schemes for nonlinear evolution equations

Priyanka Saha; Nandadulal Bairagi; Gaston N'Guerekata

doi:10.26637/mjm1201/001

Abstract

Discretization of a continuous-time system of dierential equations becomes inevitable due to the lack of analytical solutions. Standard discretization techniques, however, have many things that could be improved, e.g., the positivity of the solution and dynamic consistency may be lost, and stability and convergence may depend on the step length. A nonstandard nite dierence (NSFD) scheme is sometimes used to avoid inconsistencies. There are two fundamental issues regarding the construction of NSFD models. First, how to construct the denominator function of the discrete first-order derivative? Second, how to discretize the nonlinear terms of a given dierential equation with nonlocal terms? We dene here a uniform technique for nonlocal discretization and construction of denominator function for NSFD models. We have discretized a couple of highly nonlinear continuous-time population models using these consistent rules. We give analytical proof in each case to show that the proposed NSFD model has identical dynamic properties to the continuous-time model. It is also shown that each NSFD system is positively invariant, and its dynamics do not depend on the step size. Numerical experiments have also been performed in favour of such claims.

Keywords:

Nonlocal discretization, denominator function, dynamic consistency, step-size independency, population models

Mathematics Subject Classification:

37N25, 39A30, 92B05, 92D25, 92D40

Authors Imprint References Funding Related Articles Downloads

Priyanka Saha Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata-700032, India. https://orcid.org/0000-0001-9695-7138
Nandadulal Bairagi Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata-700032, India. https://orcid.org/0000-0001-8867-7996
Gaston N'Guerekata NEERLab, Department of Mathematics, Morgan State University, Baltimore, MD 21251 USA. https://orcid.org/0000-0001-5765-7175

Pages: 1-20
Date Published: 01-01-2024
Vol. 12 No. 01 (2024): Malaya Journal of Matematik (MJM)

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