A Fractional Gauss-Jacobi quadrature rule for approximating fractional integrals and derivatives
For researchers working on fractional calculus and boundary value problems, this provides a more accurate numerical method, though it is an incremental improvement over existing techniques.
The paper introduces a fractional Gauss-Jacobi quadrature rule for computing fractional integrals and derivatives, and applies it to solve fractional Euler-Lagrange equations. The method is shown to be more accurate than a previous approach from 2013.
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.