A search for a spectral technique to solve nonlinear fractional differential equations

TL;DR

This paper explores a spectral decomposition method to solve nonlinear fractional differential equations, demonstrating its application to specific fractional equations and analyzing the accuracy of the solutions.

Contribution

It extends spectral techniques to fractional nonlinear differential equations and evaluates their effectiveness on several fractional models.

Findings

Solutions reduce to classical cases when fractional order is 1

Derived solutions exhibit small systematic errors

Method provides a new approach to fractional nonlinear equations

Abstract

A spectral decomposition method is used to obtain solutions to a class of nonlinear differential equations. We extend this approach to the analysis of the fractional form of these equations and demonstrate the method by applying it to the fractional Riccati equation, the fractional logistic equation and a fractional cubic equation. The solutions reduce to those of the ordinary nonlinear differential equations, when the order of the fractional derivative is $\alpha =1$. The exact analytic solutions to the fractional nonlinear differential equations are not known, so we evaluate how well the derived solutions satisfy the corresponding fractional dynamic equations. In the three cases we find a small, apparently generic, systematic error that we are not able to fully interpret.