Jacobi-Predictor-Corrector Approach for the Fractional Ordinary Differential Equations

TL;DR

This paper introduces a new Jacobi-predictor-corrector numerical method for solving fractional ordinary differential equations, achieving high convergence order with efficient computation and validated by thorough error analysis and numerical experiments.

Contribution

The paper proposes a novel Jacobi-predictor-corrector approach utilizing polynomial interpolation and Gauss-Lobatto quadrature for fractional ODEs, with proven efficiency and accuracy.

Findings

Computational cost is O(N) for the method.

Convergent order is proportional to the number of interpolating points.

Numerical experiments confirm theoretical accuracy and robustness.

Abstract

We present a novel numerical method, called {\tt Jacobi-predictor-corrector approach}, for the numerical solution of fractional ordinary differential equations based on the polynomial interpolation and the Gauss-Lobatto quadrature w.r.t. the Jacobi-weight function $\omega(s)=(1-s)^{\alpha-1}(1+s)^0$. This method has the computational cost O(N) and the convergent order $IN$, where $N$ and $IN$ are, respectively, the total computational steps and the number of used interpolating points. The detailed error analysis is performed, and the extensive numerical experiments confirm the theoretical results and show the robustness of this method.