A Comparative Study of Two Legendre-Collocation Schemes Applied to Fractional Logistic Equation

Abstract

In this work, two families of shifted-Legendre polynomials consist of fractional and non-fractional basis functions are utilized to obtain approximate solutions of the fractional-order logistic differential equation. Indeed, two numerical algorithms based on Legendre-collocation spectral methods are developed for this model problem. Our main objective is to present a comparative study of these polynomials and to asses their performances as well as accuracies applied to the logistic equation. By means of the fractional derivative in the Caputo sense and the collocation points, the proposed methods transforms the model problem into a system of nonlinear algebraic equations. The efficiency and accuracy of the proposed schemes are examined on a range of problems with the aid of error functions. A comparison shows that the both schemes are simple and give satisfactory results in solving the logistic equation. However, the work affirms the superiority of the fractional-collocation scheme to the logistic equation on the point view of accuracy.