scholarly journals A Novel Homotopy Perturbation Method with Applications to Nonlinear Fractional Order KdV and Burger Equation with Exponential-Decay Kernel

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Shabir Ahmad ◽  
Aman Ullah ◽  
Ali Akgül ◽  
Manuel De la Sen
Keyword(s):  
Perturbation Method ◽  
Fractional Order ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Order Partial Differential Equation ◽  
The Novel ◽  
Homotopy Perturbation ◽  
Novel Approach ◽  
In Series ◽  
Novel Method

In this paper, we introduce the Yang transform homotopy perturbation method (YTHPM), which is a novel method. We provide formulae for the Yang transform of Caputo-Fabrizio fractional order derivatives. We derive an algorithm for the solution of Caputo-Fabrizio (CF) fractional order partial differential equation in series form and show its convergence to the exact solution. To demonstrate the novel approach, we include some examples with detailed solutions. We use tables and graphs to compare the exact and approximate solutions.

scholarly journals Application of Asymptotic Homotopy Perturbation Method to Fractional Order Partial Differential Equation

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2215
Author(s):  
Haji Gul ◽  
Sajjad Ali ◽  
Kamal Shah ◽  
Shakoor Muhammad ◽  
Thanin Sitthiwirattham ◽  
...  
Keyword(s):  
Perturbation Method ◽  
Fractional Order ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Order Partial Differential Equation ◽  
Two Dimensional ◽  
Numerical Treatment ◽  
Helmholtz Equations ◽  
Homotopy Perturbation ◽  
Linear Differential

In this article, we introduce a new algorithm-based scheme titled asymptotic homotopy perturbation method (AHPM) for simulation purposes of non-linear and linear differential equations of non-integer and integer orders. AHPM is extended for numerical treatment to the approximate solution of one of the important fractional-order two-dimensional Helmholtz equations and some of its cases . For probation and illustrative purposes, we have compared the AHPM solutions to the solutions from another existing method as well as the exact solutions of the considered problems. Moreover, it is observed that the symmetry or asymmetry of the solution of considered problems is invariant under the homotopy definition. Error estimates for solutions are also provided. The approximate solutions of AHPM are tabulated and plotted, which indicates that AHPM is effective and explicit.

scholarly journals Hypersingular integral equations of the first kind: A modified homotopy perturbation method and its application to vibration and active control

2019 ◽  
Vol 38 (2) ◽  
pp. 706-727 ◽  
Author(s):  
Reza Novin ◽  
Mohammad Ali Fariborzi Araghi
Keyword(s):  
Perturbation Method ◽  
Integral Equations ◽  
Homotopy Perturbation Method ◽  
The Novel ◽  
Validity And Reliability ◽  
Hypersingular Integral ◽  
Homotopy Perturbation ◽  
Hypersingular Integral Equations ◽  
Ideal Tool ◽  
Novel Method

This paper attempts to propose and investigate a modification of the homotopy perturbation method to study hypersingular integral equations of the first kind. Along with considering this matter, of course, the novel method has been compared with the standard homotopy perturbation method. This method can be conveniently fast to get the exact solutions. The validity and reliability of the proposed scheme are discussed. Different examples are included to prove so. According to the results, we further state that new simple homotopy perturbation method is so efficient and promises the exact solution. The modification of the homotopy perturbation method has been discovered to be the significant ideal tool in dealing with the complicated function-theoretic analytical structures within an analytical method.

scholarly journals On Semianalytical Study of Fractional-Order Kawahara Partial Differential Equation with the Homotopy Perturbation Method

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Muhammad Sinan ◽  
Kamal Shah ◽  
Zareen A. Khan ◽  
Qasem Al-Mdallal ◽  
Fathalla Rihan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Solitary Wave ◽  
Perturbation Method ◽  
Fractional Order ◽  
Homotopy Perturbation Method ◽  
Nonlinear Partial Differential Equation ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Homotopy Perturbation

In this study, we investigate the semianalytic solution of the fifth-order Kawahara partial differential equation (KPDE) with the approach of fractional-order derivative. We use Caputo-type derivative to investigate the said problem by using the homotopy perturbation method (HPM) for the required solution. We obtain the solution in the form of infinite series. We next triggered different parametric effects (such as x, t, and so on) on the structure of the solitary wave propagation, demonstrating that the breadth and amplitude of the solitary wave potential may alter when these parameters are changed. We have demonstrated that He’s approach is highly effective and powerful for the solution of such a higher-order nonlinear partial differential equation through our calculations and simulations. We may apply our method to an additional complicated problem, particularly on the applied side, such as astrophysics, plasma physics, and quantum mechanics, to perform complex theoretical computation. Graphical presentation of few terms approximate solutions are given at different fractional orders.

scholarly journals Convergence of the Generalized Homotopy Perturbation Method for Solving Fractional Order Integro-Differential Equations

2014 ◽  
Vol 11 (4) ◽  
pp. 1637-1648
Author(s):  
Baghdad Science Journal
Keyword(s):  
Theoretical Analysis ◽  
Differential Equations ◽  
Perturbation Method ◽  
Fractional Order ◽  
Infinite Series ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Powerful Method ◽  
Homotopy Perturbation ◽  
Fractional Order Integral

In this paper,the homtopy perturbation method (HPM) was applied to obtain the approximate solutions of the fractional order integro-differential equations . The fractional order derivatives and fractional order integral are described in the Caputo and Riemann-Liouville sense respectively. We can easily obtain the solution from convergent the infinite series of HPM . A theorem for convergence and error estimates of the HPM for solving fractional order integro-differential equations was given. Moreover, numerical results show that our theoretical analysis are accurate and the HPM can be considered as a powerful method for solving fractional order integro-diffrential equations.

ANALYTICAL COMPARISON BETWEEN THE HOMOTOPY PERTURBATION METHOD AND VARIATIONAL ITERATION METHOD FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

2008 ◽  
Vol 22 (23) ◽  
pp. 4041-4058 ◽  
Author(s):  
ZAID ODIBAT ◽  
SHAHER MOMANI
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Fractional Order ◽  
Iteration Method ◽  
Homotopy Perturbation Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Variational Iteration ◽  
Homotopy Perturbation ◽  
Different Types

Comparison of homotopy perturbation method (HPM) and variational iteration method (VIM) is made, revealing that the two methods can be used as alternative and equivalent methods for obtaining analytic and approximate solutions for different types of differential equations of fractional order. Furthermore, the former is more general and powerful than the latter. Numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order.

scholarly journals An efficient approach for solution of fractional-order Helmholtz equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nehad Ali Shah ◽  
Essam R. El-Zahar ◽  
Mona D. Aljoufi ◽  
Jae Dong Chung
Keyword(s):  
Perturbation Method ◽  
Fractional Order ◽  
Homotopy Perturbation Method ◽  
Hybrid Technique ◽  
Science And Engineering ◽  
Helmholtz Equations ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Suggested Technique ◽  
Present Technique

AbstractIn this article, a hybrid technique called the homotopy perturbation Elzaki transform method has been implemented to solve fractional-order Helmholtz equations. In the hybrid technique, the Elzaki transform method and the homotopy perturbation method are amalgamated. Three problems are solved to validate and demonstrate the efficacy of the present technique. It is also demonstrated that the results obtained from the suggested technique are in excellent agreement with the results by other techniques. It is shown that the proposed method is efficient, reliable and easy to implement for various related problems of science and engineering.

scholarly journals Analysis of Homotopy Perturbation Method for Solving Fractional Order Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 40 ◽  
Author(s):  
Shumaila Javeed ◽  
Dumitru Baleanu ◽  
Asif Waheed ◽  
Mansoor Shaukat Khan ◽  
Hira Affan
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Fractional Order ◽  
Homotopy Perturbation Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Fractional Order Differential Equations ◽  
Homotopy Perturbation ◽  
The Given ◽  
Ordinary Derivative

The analysis of Homotopy Perturbation Method (HPM) for the solution of fractional partial differential equations (FPDEs) is presented. A unified convergence theorem is given. In order to validate the theory, the solution of fractional-order Burger-Poisson (FBP) equation is obtained. Furthermore, this work presents the method to find the solution of FPDEs, while the same partial differential equation (PDE) with ordinary derivative i.e., for α = 1 , is not defined in the given domain. Moreover, HPM is applied to a complicated obstacle boundary value problem (BVP) of fractional order.

scholarly journals Second Approximate Solution of Duffing Equation with Strong Nonlinearity by Homotopy Perturbation Method

1970 ◽  
Vol 30 ◽  
pp. 59-75
Author(s):  
M Alhaz Uddin ◽  
M Abdus Sattar
Keyword(s):  
Approximate Solution ◽  
Nonlinear Systems ◽  
Perturbation Method ◽  
Differential System ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Second Order ◽  
Strong Nonlinearity ◽  
Analytical Technique ◽  
Homotopy Perturbation

 In this paper, the second order approximate solution of a general second order nonlinear ordinary differential system, modeling damped oscillatory process is considered. The new analytical technique based on the work of He’s homotopy perturbation method is developed to find the periodic solution of a second order ordinary nonlinear differential system with damping effects. Usually the second or higher order approximate solutions are able to give better results than the first order approximate solutions. The results show that the analytical approximate solutions obtained by homotopy perturbation method are uniformly valid on the whole solutions domain and they are suitable not only for strongly nonlinear systems, but also for weakly nonlinear systems. Another advantage of this new analytical technique is that it also works for strongly damped, weakly damped and undamped systems. Figures are provided to show the comparison between the analytical and the numerical solutions. Keywords: Homotopy perturbation method; damped oscillation; nonlinear equation; strong nonlinearity. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 59-75  DOI: http://dx.doi.org/10.3329/ganit.v30i0.8504

scholarly journals A Modified Techniques of Fractional-Order Cauchy-Reaction Diffusion Equation via Shehu Transform

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Mounirah Areshi ◽  
A. M. Zidan ◽  
Rasool Shah ◽  
Kamsing Nonlaopon
Keyword(s):  
Fractional Order ◽  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Transformation Method ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Closed Form Solutions ◽  
Homotopy Perturbation ◽  
In Series ◽  
The Given

In this article, the iterative transformation method and homotopy perturbation transformation method are applied to calculate the solution of time-fractional Cauchy-reaction diffusion equations. In this technique, Shehu transformation is combined of the iteration and the homotopy perturbation techniques. Four examples are examined to show validation and the efficacy of the present methods. The approximate solutions achieved by the suggested methods indicate that the approach is easy to apply to the given problems. Moreover, the solution in series form has the desire rate of convergence and provides closed-form solutions. It is noted that the procedure can be modified in other directions of fractional order problems. These solutions show that the current technique is very straightforward and helpful to perform in applied sciences.

Application of He’s Homotopy Perturbation Method for Solving Fractional Fokker-Planck Equations

2009 ◽  
Vol 64 (12) ◽  
pp. 788-794 ◽  
Author(s):  
Mohamed M. Mousa ◽  
Aidarkhan Kaltayev
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Planck Equation ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Homotopy Perturbation ◽  
External Force Field ◽  
Promising Tool ◽  
Transport Problems ◽  
Fokker Planck

Abstract The fractional Fokker-Planck equation (FFPE) has been used in many physical transport problems which take place under the influence of an external force field and other important applications in various areas of engineering and physics. In this paper, by means of the homotopy perturbation method (HPM), exact and approximate solutions are obtained for two classes of the FFPE initial value problems. The method gives an analytic solution in the form of a convergent series with easily computed components. The obtained results show that the HPM is easy to implement, accurate and reliable for solving FFPEs. The method introduces a promising tool for solving other types of differential equation with fractional order derivatives

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