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Research Papers

Computing Numerical Solutions of Delayed Fractional Differential Equations With Time Varying Coefficients

[+] Author and Article Information
Venkatesh Suresh Deshmukh

Department of Mechanical Engineering,
Villanova University,
800 Lancaster Avenue,
Villanova, PA 19085
e-mail: venkatesh.deshmukh@villanova.edu

Manuscript received March 5, 2013; final manuscript received April 21, 2014; published online September 12, 2014. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 10(1), 011004 (Sep 12, 2014) (5 pages) Paper No: CND-13-1056; doi: 10.1115/1.4027489 History: Received March 05, 2013; Revised April 21, 2014

Fractional differential equations with time varying coefficients and delay are encountered in the analysis of models of metal cutting processes such as milling and drilling with viscoelastic damping elements. Viscoelastic damping is modeled as a fractional derivative. In the present paper, delayed fractional differential equations with bounded time varying coefficients in four different forms are analyzed using series solution and Chebyshev spectral collocation. A fractional differential equation with a known exact solution is then solved by the methodology presented in the paper. The agreement between the two is found to be excellent in terms of point-wise error in the trajectories. Solutions to the described fractional differential equations are computed next in state space and second order forms.

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Figures

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Fig. 1

Comparison of exact and approximate solutions of Eq. (22)

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Fig. 2

The trajectories of underlying ordinary differential equation (ODE), delay differential equation (DDE), fractional ordinary differential equation(FODE) of Eq. (23), and fractional delay differential equation (FDDE (23))

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Fig. 3

Displacements of underlying ODE, DDE, FODE of Eq. (24), and the FDDE (24) Eq. (1b)

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Fig. 4

Displacements of underlying ODE, DDE, FODE of Eq. (24), and the FDDE (24)

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Fig. 5

Displacements of underlying ODE, DDE, FODE of Eq. (24), and the FDDE (24) Eq. (1d)

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