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

Authors

  • Soheyla Ansari Yazd University
  • Mohammad Hossein Akrami Yazd University

DOI:

https://doi.org/10.30495/jme.v18i0.2941

Keywords:

Fractional initial value problems‎, ‎Picard iteration method‎, ‎Shifted Legendre polynomials‎, ‎Fractional Bratu's problem‎, ‎Fractional Riccati equation‎.

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.

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Published

2024-08-04

Issue

Vol. 18

Section

Vol. 18, No. 3, (2024)

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