A new numerical algorithm based on the first kind of modified Bessel function to solve population growth in a closed system

• Published in: International journal of computer mathematics 2014-06, Vol.91 (6), p.1239-1254
• Main Authors: Parand, K., Rad, J.A., Nikarya, M.
• Format: Article
• Language: English
• Subjects: Algebra; Algorithms; Bessel functions; collocation method; Computation; Differential equations; Dynamical systems; Integrals; integro-differential equation; Mathematical analysis; mathematical ecology; Mathematical models; nonlinear ODE; Numerical analysis; Population growth; semi-infinite interval; the first kind of modified Bessel functions; Toxicity; Voltera's Population Model
• ISSN: 0020-7160; 1029-0265
• DOI: 10.1080/00207160.2013.829917

Volterra's model for population growth in a closed system includes an integral term to indicate accumulated toxicity in addition to the usual terms of the logistic equation. In this research, a new numerical algorithm is introduced for solving this model. The proposed numerical approach is based on the modified Bessel function of the first kind and the collocation method. In this method, we aim to solve the problems on the semi-infinite domain without any domain truncation, variable transformation in basis functions and shifting the problem to a finite domain. Accordingly, we employ two different collocation approaches, one by computing through Volterra's population model in the integro-differential form and the other by computing by converting this model to an ordinary differential form. These methods reduce the solution of a problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of these methods, we compare the numerical results of the present methods with some well-known results in other to show that the new methods are efficient and applicable.