An Effective Approach for Solving Nonlinear Fractional Initial Value Problems: The Fractional Legendre-Picard Iteration Method

The purpose of this manuscript 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.