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Research Papers

Numerical Analysis of Fractional Neutral Functional Differential Equations Based on Generalized Volterra-Integral Operators

[+] Author and Article Information
Xiao-Li Ding

Department of Mathematics,
Xi'an Polytechnic University,
Xi'an, Shaanxi 710048, China
e-mail: dingding0605@126.com

Juan J. Nieto

Departamento de Análisis Matemático,
Facultad de Matemáticas,
Universidad de Santiago de Compostela,
Santiago de Compostela 15782, Spain
e-mail: juanjose.nieto.roig@usc.es

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 2, 2016; final manuscript received November 10, 2016; published online January 11, 2017. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 12(3), 031018 (Jan 11, 2017) (7 pages) Paper No: CND-16-1171; doi: 10.1115/1.4035267 History: Received April 02, 2016; Revised November 10, 2016

We use waveform relaxation (WR) method to solve numerically fractional neutral functional differential equations and mainly consider the convergence of the numerical method with the help of a generalized Volterra-integral operator associated with the Mittag–Leffler function. We first give some properties of the integral operator. Using the proposed properties, we establish the convergence condition of the numerical method. Finally, we provide a new way to prove the convergence of waveform relaxation method for integer-order neutral functional differential equation, which is a special case of fractional neutral functional differential equation. Compared to the existing proof in the literature, our proof is concise and original.

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Figures

Grahic Jump Location
Fig. 1

The numerical solutions obtained by WR method with time step is 0.01 for Example 5.1

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