The homotopy perturbation method for fractional differential equations: part 1 Mohand transform

Abstract

Purpose This study aims that very lately, Mohand transform is introduced to solve the ordinary and partial differential equations (PDEs). In this paper, the authors modify this transformation and associate it with a further analytical method called homotopy perturbation method (HPM) for the fractional view of Newell–Whitehead–Segel equation (NWSE). As Mohand transform is restricted to linear obstacles only, as a consequence, HPM is used to crack the nonlinear terms arising in the illustrated problems. The fractional derivatives are taken into the Caputo sense. Design/methodology/approach The specific objective of this study is to examine the problem which performs an efficient role in the form of stripe orders of two dimensional systems. The authors achieve the multiple behaviors and properties of fractional NWSE with different positive integers. Findings The main finding of this paper is to analyze the fractional view of NWSE. The obtain results perform very good in agreement with exact solution. The authors show that this strategy is absolutely very easy and smooth and have no assumption for the constriction of this approach. Research limitations/implications This paper invokes these two main inspirations: first, Mohand transform is associated with HPM, secondly, fractional view of NWSE with different positive integers. Practical implications In this paper, the graph of approximate solution has the excellent promise with the graphs of exact solutions. Social implications This paper presents valuable technique for handling the fractional PDEs without involving any restrictions or hypothesis. Originality/value The authors discuss the fractional view of NWSE by a Mohand transform. The work of the present paper is original and advanced. Significantly, to the best of the authors’ knowledge, no such work has yet been published in the literature.

The homotopy perturbation method for fractional differential equations: part 2, two-scale transform

“…Fractional calculus is a branch of applied mathematics that deals with derivatives and integrals of arbitrary orders. Many physical discontinuous phenomena are modeled by nonlinear fractional differential equations arising in porous media, unsmooth boundaries and lattice mechanics (Nadeem et al, 2021;Zuo and Liu, 2021;He et al, 2020dHe et al, , 2021Habib et al, 2020;Wang, 2020;He, 2014He, , 2019cHe, , 2020bHe, , 2021Wang et al, 2019;He and Latifizadeh, 2020;Nadeem et al, 2019). However, such problems are very difficult to solve either analytically or numerically, among all analytical methods [e.g.…”

“…However, such problems are very difficult to solve either analytically or numerically, among all analytical methods [e.g. the variational iteration method (Naveed and He, 2019;He, 2020d), the Taylor series method He, 2019aHe, , 2020c, the exp-function method (He, 2013)], the homotopy perturbation method is the most effective tool for fractional differential equations (Nadeem et al, 2021;He and El-Dib, 2020a, 2020b, 2020cAnjum and He, 2020a;He, 1999;Anjum and He, 2020b;He and Jin, 2020;Yu et al, 2019;He et al, 2019).…”

“…The two-scale approach observes the phenomenon using two different scales, one scale is generally used in the conventional continuum mechanics, and the other scale can reveal the hidden truth beyond the continuum assumption. In the first part of our series publications (Nadeem et al, 2021), we have showed the homotopy perturbation method is the best candidate to solve such problems, and Mohand transform makes the solution process easy. However, the Mohand transform is not accessible to all audience, a much simpler transform is therefore much needed to achieve the goal of a highly accurate solution with simple calculation.…”

Gamma function method for the nonlinear cubic-quintic Duffing oscillators

“…However, the explicit analytical solutions of nonlinear oscillator equations are few, and either numerical solutions or approximate analytical techniques are frequently used. Many scholars have made outstanding contributions and many different methods are obtained such as homotopy perturbation method, [1][2][3] variational approach, [4][5][6][7][8][9] variational iteration method, [10][11][12][13][14][15][16] He's frequency formulation, [17][18][19] Hamiltonian approach, 20 Taylor series method, 21 and so on. [22][23][24] The well-known Duffing oscillator equation was named after a German electrical engineer Georg Duffing who first proposed the equation in 1918, 25 and then, it is developed into different forms to describe many physical, mechanical engineering, circuits and biological processes in various areas of science.…”

A simplified He’s frequency–amplitude formulation for nonlinear oscillators

“…Therefore, researchers from different fields have explored the nature of nonlinear oscillators for decades. 1 Currently, more and more analytical and approximate methods have been developed to find approximate solutions of nonlinear oscillators, for instance, He's variational iteration method [2][3][4][5][6][7][8] ; the homotopy perturbation method [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] ; the variational method 25 ; the parameter-expanding method 26,27 and other methods. 28 Researchers worldwide have sought a concise and practical approach to solve various nonlinear oscillators for a long time.…”