An Investigation of Fractional Bagley-Torvik Equation.

Azhar Ali Zafar,Jan Awrejcewicz,Grzegorz Kudra

doi:10.3390/e22010028

Azhar Ali Zafar, Jan Awrejcewicz + Show 1 more

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https://doi.org/10.3390/e22010028

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Abstract

In this article, we will solve the Bagley–Torvik equation by employing integral transform method. Caputo fractional derivative operator is used in the modeling of the equation. The obtained solution is expressed in terms of generalized G function. Further, we will compare the obtained results with other available results in the literature to validate their usefulness. Furthermore, examples are included to highlight the control of the fractional parameters on he dynamics of the model. Moreover, we use this equation in modelling of real free oscillations of a one-degree-of-freedom mechanical system composed of a cart connected with the springs to the support and moving via linear rolling bearing block along a rail.

Highlights

The concept of fractional calculus (FC) and entropy are very important for the investigation of complex dynamical systems and got the attention of researchers, physicists and mathematicians
By graphical illustrations, we will testify the agreement of our results with the exiting solutions of Bagley–Torvik equation (BTE) obtained by different methods in the literature, and the control of the non-integer order parameter on the model equation
Regarding the control of the fractional parameters, it is reported that the motion of the plate is an increasing function of the fractional parameters and their influence is sensitive to the applied force to the plate

Summary

Introduction

The concept of fractional calculus (FC) and entropy are very important for the investigation of complex dynamical systems and got the attention of researchers, physicists and mathematicians.Machado [1] investigated the importance of entropy for the analysis of complex dynamical systems.Lopes and Machado [2] used the FC tools for the study of complex systems. The concept of fractional calculus (FC) and entropy are very important for the investigation of complex dynamical systems and got the attention of researchers, physicists and mathematicians. Machado [1] investigated the importance of entropy for the analysis of complex dynamical systems. Lopes and Machado [2] used the FC tools for the study of complex systems. Proposed the entropy functions based on FC for the analysis of dynamical systems. For more about FC and entropy we refer [6,7,8]. The non-local nature of fractional derivatives allows to describe changes in an interval. This important property makes these derivatives suitable to simulate more physical and complex phenomena. The non-integer order derivative parameter used in the modelling of dynamical systems behaves as the rheological parameter and influence the properties of the dynamical systems, it is seen that different interdisciplinary problems can likewise be solved with good accuracy by the aid of non-integer order derivatives [12]

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