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Research Papers

A Novel Method for Solving the Bagley-Torvik Equation as Ordinary Differential Equation

[+] Author and Article Information
Yong Xu, Jike Liu, Yanmao Chen

Department of Mechanics,
Sun Yat-sen University,
No. 135 Xingang Road,
Guangzhou 510275, China

Qixian Liu

Department of Mechanics,
Sun Yat-sen University,
No. 135 Xingang Road,
Guangzhou 510275, China
e-mail: liu_qixian@qq.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 6, 2019; final manuscript received April 9, 2019; published online May 13, 2019. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 14(8), 081005 (May 13, 2019) (5 pages) Paper No: CND-19-1008; doi: 10.1115/1.4043525 History: Received January 06, 2019; Revised April 09, 2019

We present a novel method to solve the Bagley-Torvik equation by transforming it into ordinary differential equations (ODEs). This method is based on the equivalence between the Caputo-type fractional derivative (FD) of order 3/2 and the solution of a diffusion equation subjected to certain initial and boundary conditions. The key procedure is to approximate the infinite boundary condition by a finite one, so that the diffusion equation can be solved by separation of variables. By this procedure, the Bagley-Torvik and the diffusion equations together are transformed to be a set of ODEs, which can be integrated numerically by the Runge-Kutta scheme. The presented method is tested by various numerical cases including linear, nonlinear, nonsmooth, or multidimensional equations, respectively. Importantly, high computational efficiency is achieved as this method is at the expense of linearly increasing computational cost with the solution domain being enlarged.

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Figures

Grahic Jump Location
Fig. 1

Comparison of the results obtained by the PC algorithm and the presented method for system (1), respectively

Grahic Jump Location
Fig. 2

Comparison of the respective CPU running time for the PC algorithm and the presented method in obtaining the results in Fig. 1

Grahic Jump Location
Fig. 3

Absolute errors of x at t=0.95 for Eq. (1) with m=1, c1=0.2, c2=0.1, k=1, and f(t)=sin t.

Grahic Jump Location
Fig. 4

Comparison of numerical results of system (16) with g(x)=2x3 obtained by the PC algorithm and the presented method, respectively

Grahic Jump Location
Fig. 5

Comparison of numerical results of system (16) with a nonsmooth nonlinearity obtained by the PC algorithm and the presented method, respectively

Grahic Jump Location
Fig. 6

Comparison of numerical results of Eq. (17) obtained by the PC algorithm and the presented method, respectively.

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