A Fast Method for Solving the Bagley-Torvik Equation with Time Delay as Delay Differential Equations of Integer Order

Abstract

A method of solving the Bagley-Torvik equation with time delay has been presented in this article, which is based on the physical meaning of that equation and thus avoid the history dependence of it. The most important thing is that the fractional term of the Bagley-Torvik equation is transformed into a solution of a partial differential equation, which is then converted into a set of ordinary differential equations afterwards. An approximation of a boundary condition of the partial differential equations is used as a crucial point. Numerical results have indicated that the computational efficiency has improved significantly. We consider a numerical example with Hopf bifurcation caused by time delay of the Bagley-Torvik equation, which shows that the presented method is computationally more efficient than the predictor-corrector (PC) algorithm with the same time step length. Introduction Time delay is an unavoidable in the realistic mechanical systems. It often leads to stability switch, changing of the control performance and other complex dynamic behaviors [1-5]. On the other hand, the Bagley-Torvik equation is a mathematical model of viscoelastically damped structures with fractional derivative. Therefore, it is meaningful to study the Bagley-Torvik equation with time delay. Lots of analytical and numerical methods have been proposed since the model was proposed. Podlubny [6] studied the Bagley-Torvik equation both analytically and numerically in his book. Diethelm et al. [7-8] presented the predictor-corrector method (PC algorithm) to solve the equation and extend it to more general cases. Ray and Bera [9] used Adomian decomposition method to solve the equation. Çenesiz [10] et al. applied the generalized Taylor collocation method. Zolfaghari et al. [11] presented the homotopy perturbation method to solve the equation. Wang and Wang [12] used α-exponential functions. Haar wavelet method [13, 14] has been applied to solving the equation. Al-Mdallal [15] used a collocation-shooting method. Raja et al. [16] used a stochastic method. Yüzbaşı [17] used the Bessel collocation method. Atanackovic and Zorica [18] used an expansion formula for fractional derivative to solve the equation. Krishnasamy and Razzaghi [19] used Taylor vector approximation, while Gülsu et al. [20] used Taylor matrix method. And Arqub et al. [21] presented a kernel algorithm to solve the equation. Although some of the methods mentioned above can be extended to solving the Bagley-Torvik equation with time delay, there are still lots of problems in these methods, such as low computational efficiency. However, some studies on the physical meaning of the equation have provided us with a good deal of enlightenment. At first, Torvik and Bagley [5] have introduced the physical background of this equation. Fitt et al. [22] have also used it to deal with an engineering problem. Moreover, Xu et al. [23] have used it reversely to transform the Bagley-Torvik equation into ordinary differential equations. We extend their method to solve the equation with time delay by transform it into delay defferential equations (DDEs). International Conference on Modeling, Analysis, Simulation Technologies and Applications (MASTA 2019) Copyright © 2019, the Authors. Published by Atlantis Press. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/). Advances in Intelligent Systems Research, volume 168