# A Fractional Gauss-Jacobi quadrature rule for approximating fractional integrals and derivatives

## TL;DR

This paper presents a new fractional Gauss-Jacobi quadrature rule that improves the accuracy of approximating fractional integrals and derivatives, facilitating solutions to fractional boundary value problems and Euler-Lagrange equations.

## Contribution

It introduces an efficient algorithm based on Hale and Townsend's method for roots and weights, enhancing accuracy over previous Golub-Welsch based approaches.

## Key findings

- The proposed method outperforms previous algorithms in accuracy.
- It effectively solves fractional boundary value problems.
- The approach is validated through illustrative examples.

## Abstract

We introduce an efficient algorithm for computing fractional integrals and derivatives and apply it for solving problems of the calculus of variations of fractional order. The proposed approximations are particularly useful for solving fractional boundary value problems. As an application, we solve a special class of fractional Euler-Lagrange equations. The method is based on Hale and Townsend algorithm for finding the roots and weights of the fractional Gauss-Jacobi quadrature rule and the predictor-corrector method introduced by Diethelm for solving fractional differential equations. Illustrative examples show that the given method is more accurate than the one introduced in [Comput. Math. Appl. 66 (2013), no. 5, 597--607], which uses the Golub-Welsch algorithm for evaluating fractional directional integrals.