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A Predictor-Corrector Algorithm in the Framework of Conformable Fractional Differential Equations

Mohamed Echchehira, Youness Assebbane, Mustapha Atraoui, Mohamed Bouaouid

TL;DR

The paper addresses numerical solution of conformable fractional differential equations using a predictor-corrector approach. It derives a conformable Adams-Bashforth–Adams-Moulton scheme by applying the conformable integral $I_alpha$ to $T_alpha y=f(t,y)$ and discretizing $ abla_0^{t} x^{alpha-1} f(x,y(x)) dx$, yielding an explicit predictor and a semi-implicit corrector with coefficients $a_j$. Key contributions include the explicit predictor $y^p_{n+1}= y_0 + rac{h^{ ext{alpha}}}{ ext{alpha}} sum_{j=0}^{n} ig((j+1)^{ ext{alpha}}-j^{ ext{alpha}}ig) f(t_j,y_j)$ and a conformable-Moulton type corrector, plus numerical validation on linear and nonlinear problems with known exact solutions. The results indicate good accuracy and practical viability, supported by example problems and a Python implementation in Appendix A.

Abstract

This work proposes a conformable fractional predictor-corrector algorithm for solving conformable fractional differential equations. Fractional calculus is finding applications in various scientific fields, but existing numerical methods might have limitations. This work addresses that gap by introducing a new algorithm specifically designed for the conformable fractional derivative using Adams-Bashforth and Adams-Moulton methods.

A Predictor-Corrector Algorithm in the Framework of Conformable Fractional Differential Equations

TL;DR

The paper addresses numerical solution of conformable fractional differential equations using a predictor-corrector approach. It derives a conformable Adams-Bashforth–Adams-Moulton scheme by applying the conformable integral to and discretizing , yielding an explicit predictor and a semi-implicit corrector with coefficients . Key contributions include the explicit predictor and a conformable-Moulton type corrector, plus numerical validation on linear and nonlinear problems with known exact solutions. The results indicate good accuracy and practical viability, supported by example problems and a Python implementation in Appendix A.

Abstract

This work proposes a conformable fractional predictor-corrector algorithm for solving conformable fractional differential equations. Fractional calculus is finding applications in various scientific fields, but existing numerical methods might have limitations. This work addresses that gap by introducing a new algorithm specifically designed for the conformable fractional derivative using Adams-Bashforth and Adams-Moulton methods.

Paper Structure

This paper contains 7 sections, 5 theorems, 36 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Description of the Adams-Bashforth and Adams-Moulton method
  4. The calssical Adams-Bashforth and Adams-Moulton method
  5. A predictor-corrector algorithm in the framework of Caputo type fractional derivative of order $\alpha$
  6. A conformable fractional Predictor-Corrector Algorithm
  7. Numerical application

Key Result

Theorem 2.1

khalil2014new If a function $f:[0, \infty) \longrightarrow \mathbb{R}$ is $\alpha$-differentiable at $t_0>0, \alpha \in(0,1]$, then $f$ is continuous at $t_0$.

Figures (3)

  • Figure 1: The plot of $y(t)$, the solution of the conformable fractional initial value problem (\ref{['FIVPexample1']}), for $\alpha=0.5$.
  • Figure 2: The plot of $y(t)$, the solution of the conformable fractional initial value problem (\ref{['FIVPexample2']}), for $\alpha=0.5$.
  • Figure 3: The plot of $y(t)$, the solution of the conformable fractional initial value problem (\ref{['FIVPexample3']}), for $\alpha=0.7$.

Theorems & Definitions (10)

  • Definition 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.1
  • Theorem 2.3
  • Theorem 2.4
  • Example 5.1
  • Lemma 5.1
  • Example 5.2
  • Example 5.3