Numerical approximations for Volterra's population growth model with fractional order via a multi-domain pseudospectral method

This paper presents a multi-domain Legendre-Gauss pseudospectral method for approximate solutions of the fractional Volterra’s model for population growth of a species in a closed system. Fractional Volterra’s model is a fractional Volterra integro-differential equation. The fractional derivative is considered in the Caputo sense. In this method, the fractional integro-differential equation is first replaced with a singular Volterra integro-differential equation (SVIDE). Then, by choosing a step-size, the replaced problem is converted into a sequence of SVIDEs in subintervals. The difficulty in SVIDEs due to singularity is overcome here by utilizing integration by parts. The obtained problems in subintervals are then step by step reduced to systems of algebraic equations using collocation. We give some numerical applications to show validity and high accuracy of the proposed technique.

Fractional Volterra's model, Fractional integro-differential equation, Caputo derivative, pseudospectral method, Shifted Legendre-Gauss points