On the numerical solution of a population growth model of a species living in a closed system based on the moving least squares scheme

Abstract

In this research paper, we introduce a numerical approach to solve a particular type of nonlinear integro-differential equations derived from Volterra's population model. This model characterizes the growth of a biological species in a closed system and includes an integral term to consider the influence of toxin accumulation on the species, along with the conventional terms found in the logistic equation. The proposed technique estimates the solution of integro-differential equations utilizing the discrete Galerkin scheme using the moving least squares (MLS) algorithm. The locally weighted least squares polynomial fitting, known as the MLS method, is a valuable approach for approximating unknown functions. Since the offered scheme does not require any cell structures, it can be known as a meshless local discrete Galerkin method. Moreover, we obtain the error estimate of the proposed approach. The validity and efficiency of the newly developed technique are assessed over several nonlinear integro-differential equations.

Keywords:

• Volterra's population model
• integro-differential equation
• moving least squares
• discrete Galerkin method
• error analysis

MSC (2020) CLASSIFICATIONS::

• 45J05
• 45L05
• 92-08
• 92D25

Acknowledgments

The authors express their sincere gratitude to the reviewers for their valuable comments and suggestions, which have significantly enhanced the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).