‎An effective approach for solving nonlinear fractional initial value problems‎: ‎The fractional Legendre-Picard iteration method‎

Abstract

The purpose of this research is to present an effective numerical technique for solving nonlinear fractional differential equations‎. ‎The proposed approach‎, ‎known as the fractional Legendre-Picard iteration method‎, ‎utilizes the shifted Legendre polynomial and the Picard iteration method‎. ‎The Picard method is a recursive algorithm commonly used to solve initial value problems‎. ‎However‎, ‎the main challenge of this method is computing the integral of the complex and nonlinear function‎. ‎In this study‎, ‎we aim to approximate the function within the integral using Legendre polynomials‎, ‎thereby resolving this issue‎. ‎Furthermore‎, ‎the fractional integrals of the shifted Legendre polynomials are easily calculated at each step‎. ‎Additionally‎, ‎we provide a detailed explanation of the proposed method in the form of a vector matrix‎, ‎which reduces CPU time‎. ‎The convergence analysis of the method is conducted‎, ‎and numerical simulations are performed to demonstrate the effectiveness and accuracy of the proposed approach‎.