Research Papers

A Spectral Numerical Method for Solving Distributed-Order Fractional Initial Value Problems

[+] Author and Article Information
M. A. Zaky

Department of Applied Mathematics,
National Research Centre Dokki,
Giza 12622, Egypt
e-mail: ma.zaky@yahoo.com

E. H. Doha

Department of Mathematics,
Faculty of Science,
Cairo University,
Giza 12613, Egypt
e-mail: eiddoha@sci.cu.edu.eg

J. A. Tenreiro Machado

Department of Electrical Engineering,
Institute of Engineering,
Polytechnic of Porto,
Rua Dr. António Bernardino de Almeida,
Porto 431 4249-015, Portugal
e-mail: jtm@isep.ipp.pt

1Corresponding author.

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.

J. Comput. Nonlinear Dynam 13(10), 101007 (Aug 22, 2018) (6 pages) Paper No: CND-18-1007; doi: 10.1115/1.4041030 History: Received January 06, 2018; Revised July 21, 2018

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.

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Grahic Jump Location
Fig. 1

Convergence of problem (33)

Grahic Jump Location
Fig. 2

Convergence of problem (34)

Grahic Jump Location
Fig. 3

Convergence of problem (35) for varying α



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