In this paper, we construct and analyze a Legendre spectral-collocation method for the numerical solution of distributed-order fractional initial value problems. We first introduce three-term recurrence relations for the fractional integrals of the Legendre polynomial. We then use the properties of the Caputo fractional derivative to reduce the problem into a distributed-order fractional integral equation. We apply the Legendre–Gauss quadrature formula to compute the distributed-order fractional integral and construct the collocation scheme. The convergence of the proposed method is discussed. Numerical results are provided to give insights into the convergence behavior of our method.
A Spectral Numerical Method for Solving Distributed-Order Fractional Initial Value Problems
Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 6, 2018; final manuscript received July 21, 2018; published online August 22, 2018. Assoc. Editor: Brian Feeny.
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Zaky, M. A., Doha, E. H., and Tenreiro Machado, J. A. (August 22, 2018). "A Spectral Numerical Method for Solving Distributed-Order Fractional Initial Value Problems." ASME. J. Comput. Nonlinear Dynam. October 2018; 13(10): 101007. https://doi.org/10.1115/1.4041030
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