A Spectral Parametric Iteration Method for Solving Volterra Population Model

Integral equations are widely used in various branches of mathematics and mathematical physics, and many problems of initial value and boundary value which are related to ordinary and partial differential equations can be converted to integral equations and then be solved. The explicit methods generally provide a good approximation of the answer to a stiff problem if there are too many node points. However, from the computational point of view, this is not acceptable nor cost-effective. Because it requires high computational costs and more time for evaluations, implicit methods are proposed, in which to obtain an approximate solution we must solve a nonlinear system of equations using the Jacobin method. In addition, by increasing the number of nodes and increasing the matrix dimension, examining convergence and stability is a serious problem. In this paper, a hybrid explicit method based on the parametric iteration method and the spectral collocation method was developed for simulating the solution of the nonlinear stiff Volterra’s model for population growth of a species within a closed system. The method derived here has the advantage that it does not require the solution of nonlinear systems of equations encountered in the Jacobian evaluation. The results obtained in the present work demonstrate excellent performance of the developed method.