Solving Volterra's population growth model of arbitrary order using the generalized fractional order of the Chebyshev functions (Q310520)

This paper formally introduces a numerical method to solve the Volterra-type integro-differential initial value problem \[ \kappa\frac{du}{dt}=u-u^2-u\int_0^tu(x)\,dx,\;u(0)=u_0. \] This initial value problem represents a population model in closed systems that involves a logistic term in addition to the accumulated toxicity. Such a problem has a closed form analytic solution that is not representing the population model accurately. Therefore, there are many approaches to solve this problem numerically, formally, or rigorously. The authors propose a numerical technique to solve the problem numerically, when the derivative is in the classical or fractional sense. For this reason, a family of Chebyshev-type functions is presented and the authors use their spectral expansion to approximate the solution. An extensive amount of numerical experiments is carried out, illustrating the results.