Application of the enhanced homotopy perturbation method to solve the fractional-order Bagley–Torvik differential equation

Zolfaghari, M; Ghaderi, Reza; Sheikholeslami, A.; Ranjbar, A.; HosseinNia, S. Hassan; Momani, Shaher; Sadati, Jalil

doi:10.1088/0031-8949/2009/t136/014032

Cited by 17 publications

(11 citation statements)

References 19 publications

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“…y(t) stands for the displacement of the plate and it should be solved. The Bagley-Torvik equation (1) has been generalized as α ∈ (0, 2) and many methods have been proposed for its general solution and the special solutions with initial value conditions [5][6][7][8][9][10][11][12][13][14][15].…”

Section: Ay (T) + B D α Y(t) + Cy(t) = F(t)mentioning

confidence: 99%

Approximate Solution of Bagley–Torvik Equations with Variable Coefficients and Three-point Boundary-value Conditions

Huang

Zhong

Guo

2015

Int. J. Appl. Comput. Math

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The fractional Bagley-Torvik equation with variable coefficients is investigated under three-point boundary-value conditions. By using the integration method, the considered problems are transformed into Fredholm integral equations of the second kind. It is found that when the fractional order is 1 < α < 2, the obtained Fredholm integral equation is with a weakly singular kernel. When the fractional order is 0 < α < 1, the given Fredholm integral equation is with a continuous kernel or a weakly singular kernel depending on the applied boundary-value conditions. The uniqueness of solution for the obtained Fredholm integral equation of the second kind with weakly singular kernel is addressed in continuous function spaces. A new numerical method is further proposed to solve Fredholm integral equations of the second kind with weakly singular kernels. The approximate solution is made and its convergence and error estimate are analyzed. Several numerical examples are computed to show the effectiveness of the solution procedures.

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Section: Ay (T) + B D α Y(t) + Cy(t) = F(t)mentioning

confidence: 99%

Approximate Solution of Bagley–Torvik Equations with Variable Coefficients and Three-point Boundary-value Conditions

Huang

Zhong

Guo

2015

Int. J. Appl. Comput. Math

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“…Ray and Bera [11] obtained an analytical solution by Adomian decomposition method. Zolfaghari et al [12] used the enhanced homotopy perturbation method to find more accurate solution. Çenesiz et al [13] employed the generalized Taylor collocation method to solving the Bagley-Torvik equation.…”

Section: Introductionmentioning

confidence: 99%

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

Liu

et al. 2019

Journal of Computational and Nonlinear Dynamics

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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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“…The control of FO systems in the presence of parameter perturbations and (or) external disturbances has been investigated, but it is restricted to identical‐order FO systems. Nevertheless, multi‐order FO differential equations have been used to model various types of systems, such as the Bagley‐Torvik and Basset equations. How to control such systems with unmodeled dynamics and external perturbations is essential, but to the best of the authors knowledge, limited attention has been paid to this topic .…”

Section: Introductionmentioning

confidence: 99%

Stabilization of Uncertain Multi‐Order Fractional Systems Based on the Extended State Observer

Chen

et al. 2017

Asian Journal of Control

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The extended state observer (ESO) based controller has been used successfully with integer-order systems involving large uncertainties. In this paper, the robust control of uncertain multi-order fractional-order (FO) systems based on ESO is investigated. First, we transform the multi-order FO system into an equivalent system in the form of a same-order state-space equation. Then, the ESO for the new system is established for estimating both the state and the total disturbance. Sufficient conditions for bounded-input and bounded-output stability are derived, and the asymptotic stability of the closed loop system is analyzed, based on whether the states are available or not. Finally, numerical simulations are presented to demonstrate the validity and feasibility of the proposed methodology.

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Application of the enhanced homotopy perturbation method to solve the fractional-order Bagley–Torvik differential equation

Cited by 17 publications

References 19 publications

Approximate Solution of Bagley–Torvik Equations with Variable Coefficients and Three-point Boundary-value Conditions

Approximate Solution of Bagley–Torvik Equations with Variable Coefficients and Three-point Boundary-value Conditions

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

Stabilization of Uncertain Multi‐Order Fractional Systems Based on the Extended State Observer

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