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Research Papers

An Extended Predictor–Corrector Algorithm for Variable-Order Fractional Delay Differential Equations

[+] Author and Article Information
B. Parsa Moghaddam

Department of Mathematics,
Lahijan Branch,
Islamic Azad University,
Lahijan 1616, Iran
e-mail: parsa.math@gmail.com

Sh. Yaghoobi

Department of Mathematics,
Lahijan Branch,
Islamic Azad University,
Lahijan 1616, Iran
e-mail: sholeyaghoobi1352@yahoo.com

J. A. Tenreiro Machado

Department of Electrical Engineering,
Institute of Engineering,
Rua Dr. Antonio Bernardino de Almeida, 431,
Porto 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 September 3, 2015; final manuscript received January 20, 2016; published online June 20, 2016. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 11(6), 061001 (Jun 20, 2016) (7 pages) Paper No: CND-15-1270; doi: 10.1115/1.4032574 History: Received September 03, 2015; Revised January 20, 2016

This article presents a numerical method based on the Adams–Bashforth–Moulton scheme to solve variable-order fractional delay differential equations (VFDDEs). In these equations, the variable-order (VO) fractional derivatives are described in the Caputo sense. The existence and uniqueness of the solutions are proved under Lipschitz condition. Numerical examples are presented showing the applicability and efficiency of the novel method.

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Figures

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Fig. 1

The numerical solution of model 1 for various values of α(t) and δ=0.5, with step size h=1/64

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Fig. 2

The numerical solution of model 1 for α(t)=| sin(t)| and δ=0.5, with step size h=1/64

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Fig. 3

The numerical solution of model 1 for α(t)=| sin(t)|,δ=0.5 and T = 30, with step size h=1/64, in the phase plane

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Fig. 4

The numerical solution of model 1 for various values of α(t) and δ=0.5, with step size h=1/64, in logarithmic scale

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Fig. 5

The numerical solution of model 2 for α(t)=1+cos(2t)/3 and δ = 1, with step size h=1/64

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Fig. 6

The numerical solution of model 2 for α(t)=1+cos(2t)/3, δ = 1 and T = 30, with step size h=1/64, in the phase plane

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Fig. 7

The numerical solution of model 2 for various values of α(t), δ = 1 and T = 10, with step size h=1/64, in logarithmic scale

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Fig. 8

The numerical solution of model 3 for value of α(t)=tan h(t+1), δ = 1 with step size h=1/64

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Fig. 9

The numerical solution of model 3 for value of α(t)=tan h(t+1), δ = 1 and T = 30 with step size h=1/64 in the phase plane

Grahic Jump Location
Fig. 10

The numerical solution of model 3 for various values of α(t) and δ=0.5 with step size h=1/64 in logarithmic scale

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