An analytical approach for systems of fractional differential equations by means of the innovative homotopy perturbation method

Abstract

Abstract. We have applied the new approach of homotopy perturbation method (NAHPM) for partial differential system equations featuring time-fractional derivative. The Caputo-type of fractional derivative is considered in this paper. A combination of NAHPM and multiple fractional power series form has been used the first time to present analytical solution. In order to illustrate the simplicity and ability of the suggested approach, some specific and clear examples have been given. All numerical calculations in this manuscript have been carried out with M athematica.

Homotopy Perturbation ρ-Laplace Transform Method and Its Application to the Fractional Diffusion Equation and the Fractional Diffusion-Reaction Equation

“…In this section, we analyze the stability and the convergence of the homotopy perturbation ρ-Laplace transform method. The solution of the fractional diffusion equation (Equation (13)) and the fractional diffusion-reaction equation (Equation 29) converge when the series defined in Equation (24) and (40) converge. To study the convergence of the homotopy perturbation ρ-Laplace transform method, we defined the error function expressed as:…”

“…See also Yavuz et al [19]. Darzi et al [40] proposed a new version of the homotopy perturbation method. The authors in Reference [40] introduced the fractional power series in the homotopy perturbation method.…”

“…Darzi et al [40] proposed a new version of the homotopy perturbation method. The authors in Reference [40] introduced the fractional power series in the homotopy perturbation method. In our method, we proposed the ρ-Laplace transform for solving the fractional differential equations.…”

On coupled systems of fractional impulsive differential equations by using a new Caputo-Fabrizio fractional derivative

“…A fractional order derivative is important for developing mathematical models in many scientific and engineering disciplines (see [7]). Several qualitative results for different classes of differential equations with different types of fractional integral and derivatives were obtained in [1,3,17,23,25,27,28].…”

On systems of fractional differential equations with the ψ‐Caputo derivative and their applications

“…We refer the reader to monographs and papers. [1][2][3][4][5][6] It is brought about by its applications in the modelling of many phenomena in various fields of science and engineering, such as control theory, electric circuits, viscoelasticity, mechanics, fluid mechanics, electrochemistry, epidemiological models, neural networks, and multi-agent systems [7][8][9][10][11][12][13][14][15][16][17][18][19][20] ; see also a review paper. 21 Fractional derivatives are excellent mathematical tools for the description of memory and hereditary properties of various processes and materials.…”