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Research Papers

An Efficient Method for Numerical Solutions of Distributed-Order Fractional Differential Equations

[+] Author and Article Information
N. Jibenja

Department of Mathematics and Statistics,
Faculty of Science,
Prince of Songkla University,
Songkhla 90112, Thailand

B. Yuttanan

Algebra and Applications Research Unit,
Department of Mathematics and Statistics,
Faculty of Science,
Prince of Songkla University,
Songkhla 90112, Thailand

M. Razzaghi

Department of Mathematics and Statistics,
Mississippi State University,
Mississippi, MS 39762
e-mail: razzaghi@math.msstate.edu

1Present address: This research was supported by the Faculty of Science Research Fund, Prince of Songkla University, Hat Yai, Songkhla 90110, Thailand.

2Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 10, 2017; final manuscript received July 11, 2018; published online August 27, 2018. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 13(11), 111003 (Aug 27, 2018) (10 pages) Paper No: CND-17-1454; doi: 10.1115/1.4040951 History: Received October 10, 2017; Revised July 11, 2018

This paper presents an efficient numerical method for solving the distributed fractional differential equations (FDEs). The suggested framework is based on a hybrid of block-pulse functions and Taylor polynomials. For the first time, the Riemann–Liouville fractional integral operator for the hybrid of block-pulse functions and Taylor polynomials has been derived directly and without any approximations. By taking into account the property of this operator, the problem under consideration is converted into a system of algebraic equations. The present method can be applied to both linear and nonlinear distributed FDEs. Easy implementation, simple operations, and accurate solutions are the essential features of the proposed hybrid functions. Illustrative examples are examined to demonstrate the performance and effectiveness of the developed approximation technique, and a comparison is made with the existing results.

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Figures

Grahic Jump Location
Fig. 1

The graph of the function f(t) for t ∈ [0,10]

Grahic Jump Location
Fig. 2

Exact and approximate solution for s ∈ [0,10]

Grahic Jump Location
Fig. 3

The graph of the function f(t) for t ∈ [0,30]

Grahic Jump Location
Fig. 4

N = 12 and M = 5

Grahic Jump Location
Fig. 5

N = 15 and M = 5

Grahic Jump Location
Fig. 6

Absolute error for N = 15 and M = 5

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