A fractional Gauss-Jacobi quadrature rule for approximating fractional integrals and derivatives. (English) Zbl 1422.65057
Summary: 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 [G. Pang et al., Comput. Math. Appl. 66, No. 5, 597–607 (2013; Zbl 1350.65019)], which uses the Golub-Welsch algorithm for evaluating fractional directional integrals.
MSC:
| 65D32 | Numerical quadrature and cubature formulas |
| 26A33 | Fractional derivatives and integrals |
| 49K05 | Optimality conditions for free problems in one independent variable |
Keywords:
fractional integrals; fractional derivatives; Gauss-Jacobi quadrature rule; fractional differential equations; fractional variational calculusCitations:
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