A new reliable method and its convergence for nonlinear second-order fractional differential equations

Abstract

The main goal of this article is to propose a new reliable method to solve nonlinear second-order fractional differential equations in particular, nonlinear fractional Bratu-type equation. This method called the modified Taylor fractional series method (MTFSM). The fractional derivative is defined in the Liouville-Caputo sense. Simplicity, rapid convergence, and high accuracy are the advantages of this method. In addition, the MTFSM reduces the size of calculations by not requiring linearization, discretization, perturbation or any other restriction. Three numerical examples are exhibited to demonstrate the reliability and efficiency of the proposed method, and the solutions are considered as an infinite series that converge rapidly to the exact solutions. The results display that the MTFSM is very effective and accurate to solve this type of nonlinear fractional problems.