Implication of locally supported radial basis functions for solving a general form of fractional Volterra's population model

Abstract

A model of the accumulated effect of toxins on a population living in a closed system as a widely used model in ecology was proposed for the first time by Vito Volterra in the early 1900. The current paper investigates a numerical method for solving a type of nonlinear fractional integro-differential equations as an extension of the Volterra population's model with non-integer order derivatives. The method is based on the discrete collocation method with locally supported radial basis functions as a basis by combining the Gauss-Legendre quadrature rule over the influence domains to estimate the integrals. Since the proposed scheme is based on a small set of data instead of all points in the solution domain, it uses much less computer memory and volume computing in comparison with global cases. Moreover, it seems that the algorithm of the presented approach is attractive and easy to be implement on computers because it requires no cell structures. The error analysis of the method is provided. Numerical results are included to show the validity and efficiency of the new technique with the high convergence rates.