An Efficient Analytical Approach for the Solution of Certain Fractional-Order Dynamical Systems

Ya Qin,Maysaa Al Qurashi,Dumitru Baleanu,Adnan Khan,Hassan Khan,Izaz Ali,Rasool Shah

doi:10.3390/en13112725

Abstract

Mostly, it is very difficult to obtained the exact solution of fractional-order partial differential equations. However, semi-analytical or numerical methods are considered to be an alternative to handle the solutions of such complicated problems. To extend this idea, we used semi-analytical procedures which are mixtures of Laplace transform, Shehu transform and Homotopy perturbation techniques to solve certain systems with Caputo derivative differential equations. The effectiveness of the present technique is justified by taking some examples. The graphical representation of the obtained results have confirmed the significant association between the actual and derived solutions. It is also shown that the suggested method provides a higher rate of convergence with a very small number of calculations. The problems with derivatives of fractional-order are also solved by using the present method. The convergence behavior of the fractional-order solutions to an integer-order solution is observed. The convergence phenomena described a very broad concept of the physical problems. Due to simple and useful implementation, the current methods can be used to solve problems containing the derivative of a fractional-order.

Highlights

Coupled schemes of fractional-order partial differential equations (PDEs) are commonly applied in phenomena that occur in biomechanics and engineering
Some systems of FPDEs are solved by the homotopy perturbation method along with Laplace and Shehu transformations
The derivatives with fractional-order are expressed in term of the Caputo operator

Summary

Introduction

Coupled schemes of fractional-order partial differential equations (PDEs) are commonly applied in phenomena that occur in biomechanics and engineering. Researchers have shown that several engineering and practical phenomena can be described well by FDEs systems as compared to classical differential equation systems and that equivalent FDEs and fractional integral equations give better precise and practical insights into the systems under discussion [34,35,36,37,38] Many of these engineering challenging problems are addressed by using deterministic mathematical models that are represented by either partial differential equations of integer order or fractional-order. The HPTM results were compared with the actual solution to the problems and confirmed a higher degree of accuracy This technique has been used to solve address non-linear wave equations [44], bifurcation of nonlinear problems [45], and boundary value problems [46]. The present technique has shown a sufficient degree of accuracy

Preliminaries

Homotopy Perturbation Shehu Transform Method

Applications

Conclusions

Methods