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

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Muhammad Nadeem ◽  
Ji-Huan He ◽  
Asad Islam
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Content Type ◽  
Homotopy Perturbation ◽  
Fractional Pdes ◽  
Specific Objective ◽  
Multiple Behaviors ◽  
Fractional Differential ◽  
Positive Integers

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

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Muhammad Nadeem ◽  
Ji-Huan He
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Perturbation Method ◽  
Fractional Differential Equations ◽  
Homotopy Perturbation Method ◽  
Solution Process ◽  
Solid State Physics ◽  
Content Type ◽  
Homotopy Perturbation ◽  
Fractional Differential

Purpose The purpose of this paper is to find an approximate solution of a fractional differential equation. The fractional Newell–Whitehead–Segel equation (FNWSE) is used to elucidate the solution process, which is one of the nonlinear amplitude equation, and it enhances a significant role in the modeling of various physical phenomena arising in fluid mechanics, solid-state physics, optics, plasma physics, dispersion and convection systems. Design/methodology/approach In Part 1, the authors adopted Mohand transform to find the analytical solution of FNWSE. In this part, the authors apply the fractional complex transform (the two-scale transform) to convert the problem into its differential partner, and then they introduce the homotopy perturbation method (HPM) to bring down the nonlinear terms for the approximate solution. Findings The HPM makes numerical simulation for the fractional differential equations easy, and the two-scale transform is a strong tool for fractal models. Originality/value The HPM with the two-scale transform sheds a bright light on numerical approach to fractional calculus.

scholarly journals Modified Homotopy Perturbation Method for Solving Fractional Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
A. A. Hemeda
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Numerical Comparison ◽  
Rapid Convergence ◽  
Homotopy Perturbation ◽  
Fractional Differential ◽  
Applied Fields

The modified homotopy perturbation method is extended to derive the exact solutions for linear (nonlinear) ordinary (partial) differential equations of fractional order in fluid mechanics. The fractional derivatives are taken in the Caputo sense. This work will present a numerical comparison between the considered method and some other methods through solving various fractional differential equations in applied fields. The obtained results reveal that this method is very effective and simple, accelerates the rapid convergence of the series solution, and reduces the size of work to only one iteration.

scholarly journals Application of the Generalized Laplace Homotopy Perturbation Method to the Time-Fractional Black–Scholes Equations Based on the Katugampola Fractional Derivative in Caputo Type

Computation ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 33
Author(s):  
Sirunya Thanompolkrang ◽  
Wannika Sawangtong ◽  
Panumart Sawangtong
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Perturbation Method ◽  
Fractional Differential Equations ◽  
Homotopy Perturbation Method ◽  
Call Option ◽  
Homotopy Perturbation ◽  
European Call Option ◽  
Black Scholes ◽  
Fractional Differential

In the finance market, the Black–Scholes equation is used to model the price change of the underlying fractal transmission system. Moreover, the fractional differential equations recently are accepted by researchers that fractional differential equations are a powerful tool in studying fractal geometry and fractal dynamics. Fractional differential equations are used in modeling the various important situations or phenomena in the real world such as fluid flow, acoustics, electromagnetic, electrochemistry and material science. There is an important question in finance: “Can the fractional differential equation be applied in the financial market?”. The answer is “Yes”. Due to the self-similar property of the fractional derivative, it can reply to the long-range dependence better than the integer-order derivative. Thus, these advantages are beneficial to manage the fractal structure in the financial market. In this article, the classical Black–Scholes equation with two assets for the European call option is modified by replacing the order of ordinary derivative with the fractional derivative order in the Caputo type Katugampola fractional derivative sense. The analytic solution of time-fractional Black–Scholes European call option pricing equation with two assets is derived by using the generalized Laplace homotopy perturbation method. The used method is the combination of the homotopy perturbation method and generalized Laplace transform. The analytic solution of the time-fractional Black–Scholes equation is carried out in the form of a Mittag–Leffler function. Finally, the effects of the fractional-order in the Caputo type Katugampola fractional derivative to change of a European call option price are shown.

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