We introduce an efficient algorithm for computing fractional integrals andderivatives and apply it for solving problems of the calculus of variations offractional order. The proposed approximations are particularly useful forsolving fractional boundary value problems. As an application, we solve aspecial class of fractional Euler-Lagrange equations. The method is based onHale and Townsend algorithm for finding the roots and weights of the fractionalGauss-Jacobi quadrature rule and the predictor-corrector method introduced byDiethelm for solving fractional differential equations. Illustrative examplesshow 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 algorithmfor evaluating fractional directional integrals.
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