Abstract
In this paper, we suggest and analyze a technique by combining the Shehu transform method and the homotopy perturbation method. This method is called the Shehu transform homotopy method (STHM). This method is used to solve the time-fractional partial differential equations (TFPDEs) with proportional delay. The fractional derivative is described in Caputo's sense. The solutions proposed in the series converge rapidly to the exact solution. Some examples are solved to show the STHM is easy to apply.
Highlights
Due to its broad variety of applications in various practical fields such as fluid dynamics, signal processing, electrical grids, diffusion, reaction processes and others in science and engineering [5, 12, 18], fractional differential equation has become very important among researchers
Different types of vigorous techniques have been developed recent years to find an approximate solution to this type of fractional model differential equations, such as general differential transform method [13], Variational iteration method [21, 24], Adomian decomposition method [20], Homotopy perturbation method [19, 25], Homotopy perturbation Sumudu transform method [14, 28], Homotopy analysis method [23], Local fractional variational iteration method [37], Variaitional homotopy perturbation method [15] and Fractional reduced differential transform method [26, 31, 32, 33]
The main object of this paper is to suggest by employing Shehu transform homotopy method (STHM) anlternative approximate solution of the initial valued autonomous method of time-fractional partial differential equations (TFPDEs) with proportional delay [25, 29, 30]
Summary
Due to its broad variety of applications in various practical fields such as fluid dynamics, signal processing, electrical grids, diffusion, reaction processes and others in science and engineering [5, 12, 18], fractional differential equation has become very important among researchers. We get the numerical solution of the initial valued autonomous system of TFPDEs with proportional delay [25,29 ,30] defined by. A further well known model, Klein-Gordon time fractional nonlinear equation with proportional delay, describes aries in quantum field theory as nonlinear wave interaction. To the best of my knowledge, a little literature on numerical methods used to solve the TFPDE with proportional delay, among them, Chebyshev pseudo spectral method [40], spectral collocation & waveform relaxation methods [41], iterated pseudo spectral method [17], Differential transform method [1, 2], Variational iteration method [6] and Homotopy perturbation method [4, 25, 27, 29]. The main object of this paper is to suggest by employing STHM anlternative approximate solution of the initial valued autonomous method of TFPDE with proportional delay [25, 29, 30].
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