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Advances in Applied Mathematics and Mechanics 2017年第1期9卷 23-42页

作者： Mir Masoud Seyyed Fakhrabadi School ofMechanical Engineering College of EngineeringUniversity of TehranTehranIran

The paper presents the size-dependant behaviors of the carbon nanotubes under electrostatic actuation using the modified couple stress theory and Homotopy perturbation method.Due to the less accuracy of the classical elasticity theorems,the modified couple stress theory is applied in order to capture the size-dependant properties of the carbon nanotubes.Both of the static and dynamic behaviors under static DC and step DC voltages are discussed.The effects of various dimensions and boundary conditions on the deflection and pull-in voltages of the carbon nanotubes are to be investigated in detail via application of the Homotopy perturbation method to solve the nonlinear governing equations semi-analytically.

关键词： Carbon nanotubes modified couple stress theory Homotopy perturbation method electrostatic actuation

Nonlinear Vibration Analysis of Functionally GradedNanobeam Using Homotopy perturbation method

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Advances in Applied Mathematics and Mechanics 2017年第1期9卷 144-156页

作者： Majid Ghadiri Mohsen Safi Faculty of Engineering Department of MechanicsImam Khomeini International UniversityQazvinIran

In this paper, He’s Homotopy perturbation method is utilized to obtainthe analytical solution for the nonlinear natural frequency of functionally gradednanobeam. The functionally graded nanobeam is modeled using the Eringen’s nonlocalelasticity theory based on Euler-Bernoulli beam theory with von Karman nonlinearityrelation. The boundary conditions of problem are considered with both sidessimply supported and simply supported-clamped. The Galerkin’s method is utilizedto decrease the nonlinear partial differential equation to a nonlinear second-order ordinarydifferential equation. Based on numerical results, homotopy perturbationmethodconvergence is illustrated. According to obtained results, it is seen that the second termof the Homotopy perturbation method gives extremely precise solution.

关键词： Homotopy perturbation method Lindstedt-Poincare method analytical solution nonlocal nonlinear free vibration functionally graded nanobeam

On Exact Solutions to Partial Differential Equations by the Modified Homotopy perturbation method

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Acta Mathematicae Applicatae Sinica 2012年第1期28卷 91-98页

作者： Gang YANG Ru-yun CHEN Luo-gen YAO School of Information Hunan University of CommerceChangsha 410205China School of Science Guangdong Ocean UniversityZhanjiang 524088China

Based on the modified Homotopy perturbation method （MHPM）, exact solutions of certain partial differential equations are constructed by separation of variables and choosing the finite terms of a series in p as exact solutions. Under suitable initial conditions, the PDE is transformed into an ODE. Some illustrative examples reveal the efficiency of the proposed method.

关键词： Homotopy perturbation method modified Homotopy perturbation method analytic solution Fokker-Planck equation

Homotopy perturbation method for Time-Fractional Shock Wave Equation

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Advances in Applied Mathematics and Mechanics 2011年第6期3卷 774-783页

作者： Mithilesh Singh Praveen Kumar Gupta Department of Mathematics Dehradun Institute of TechnologyDehradunIndia Department of Applied Mathematics Institute of TechnologyBanaras Hindu UniversityVaranasiIndia

A scheme is developed to study numerical solution of the time-fractional shock wave equation and wave equation under initial conditions by the Homotopy perturbation method(HPM).The fractional derivatives are taken in the Caputo sense.The solutions are given in the form of series with easily computable terms.Numerical results are illustrated through the graph.

关键词： Partial differential equation fractional derivative shock wave equation Homotopy perturbation method

homotopy Analysis method for Solving (2+1)-dimensional Navier-Stokes Equations with perturbation Terms

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Communications in Mathematical Research 2018年第1期34卷 1-14页

作者： Ji Juan-juan Zhang Lan-fang School of Physics and Electrical Engineering Anqing Normal University

In this paper homotopy Analysis method(HAM) is implemented for obtaining approximate solutions of(2+1)-dimensional Navier-Stokes equations with perturbation terms. The initial approximations are obtained using linear systems of the Navier-Stokes equations; by the iterations formula of HAM, the first approximation solutions and the second approximation solutions are successively obtained and Homotopy perturbation method(HPM) is also used to solve these equations; finally,approximate solutions by HAM of(2+1)-dimensional Navier-Stokes equations without perturbation terms and with perturbation terms are compared. Because of the freedom of choice the auxiliary parameter of HAM, the results demonstrate that the rapid convergence and the high accuracy of the HAM in solving Navier-Stokes equations; due to the effects of perturbation terms, the 3 rd-order approximation solutions by HAM and HPM have great fluctuation.

关键词： Navier-Stokes equation homotopy analysis method Homotopy perturbation method perturbation term

A New Approximation to the Linear Matrix Equation AX = B by Modification of He’s Homotopy perturbation method

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Advances in Linear Algebra & Matrix Theory 2016年第2期6卷 23-30页

作者： Amir Sadeghi Young Researchers and Elite Club Robat Karim Branch Islamic Azad University Tehran Iran

It is well known that the matrix equations play a significant role in engineering and applicable sciences. In this research article, a new modification of the Homotopy perturbation method (HPM) will be proposed to obtain the approximated solution of the matrix equation in the form AX = B. Moreover, the conditions are deduced to check the convergence of the homotopy series. Numerical implementations are adapted to illustrate the properties of the modified method.

关键词： Matrix Equation Homotopy perturbation method Convergence Diagonally Dominant Matrix Regular Splitting

New generalized fuzzy transform computations for solving fractional partial differential equations arising in oceanography

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Journal of Ocean Engineering and Science 2023年第1期8卷 55-78页

作者： Saima Rashid Rehana Ashraf Zakia Hammouch Department of Mathematics Government College UniversityFaisalabadPakistan Department of Mathematis Lahore College for Women UniversityLahorePakistan Division of Applied Mathematics Thu Dau Mot University Binh Duong ProvinceVietnam Department of Medical Research China Medical University HospitalTaichungTaiwan Department of Sciences Ecole Normal SuperieureMoulay Ismail University of Meknes50000 Morocco

This paper presents a study of nonlinear waves in shallow water.The Korteweg-de Vries(KdV)equa-tion has a canonical version based on oceanography theory,the shallow water waves in the oceans,and the internal ion-acoustic waves in plasma.Indeed,the main goal of this investigation is to employ a semi-analytical method based on the homotopy perturbation transform method(HPTM)to obtain the numerical findings of nonlinear dispersive and fifth order KdV models for investigating the behaviour of magneto-acoustic waves in plasma via fuzziness.This approach is connected with the fuzzy generalized integral transform and HPTM.Besides that,two novel results for fuzzy generalized integral transforma-tion concerning fuzzy partial gH-derivatives are presented.Several illustrative examples are illustrated to show the effectiveness and supremacy of the proposed method.Furthermore,2D and 3D simulations de-pict the comparison analysis between two fractional derivative operators(Caputo and Atangana-Baleanu fractional derivative operators in the Caputo sense)under generalized gH-differentiability.The projected method(GHPTM)demonstrates a diverse spectrum of applications for dealing with nonlinear wave equa-tions in scientific domains.The current work,as a novel use of GHPTM,demonstrates some key differ-ences from existing similar methods.

关键词： Fuzzy set theory Hukuhara differentiability KdV Equation Generalized integral transform Caputo fractional derivative AB-Fractional operator Homotopy perturbation method

A novel fractional case study of nonlinear dynamics via analytical approach

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Applied Mathematics(A Journal of Chinese Universities) 2024年第2期39卷 276-290页

作者： Hassan Khan Adnan Khan Rasool Shah Dumitru Baleanu Department of Mathematics Abdul Wali khan UniversityMardan 23200Pakistan Department of Mathematics Faculty of Arts and SciencesCankaya UniversityAnkara 06530Turkey Institute of Space Sciences Magurele-BucharestRomania

The present work describes the fractional view analysis of Newell-Whitehead-Segal equations,using an innovative technique.The work is carried with the help of the Caputo operator of fractional derivative.The analytical solutions of some numerical examples are presented to confirm the reliability of the proposed method.The derived results are very consistent with the actual solutions to the problems.A graphical representation has been done for the solution of the problems at various fractional-order derivatives.Moreover,the solution in series form has the desired rate of convergence and provides the closed-form solutions.It is noted that the procedure can be modified in other directions for fractional order problems.

关键词： Homotopy perturbation method Shehu transform Newell-Whitehead-Segel Equation Caputo operator

Dynamical Behaviors of Nonlinear Coronavirus (COVID-19) Model with Numerical Studies

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Computers, Materials & Continua 2021年第4期67卷 675-686页

作者： Khaled A.Gepreel Mohamed S.Mohamed Hammad Alotaibi Amr M.S.Mahdy Department of Mathematics College of ScienceTaif UniversityTaif21944Saudi Arabia Department of Mathematics Zagazig UniversityZagazig44519Egypt Department of Mathematics Al-Azher UniversityNasr City11884Egypt

The development of mathematical modeling of infectious diseases is a key research area in various elds including ecology and epidemiology.One aim of these models is to understand the dynamics of behavior in infectious diseases.For the new strain of coronavirus(COVID-19),there is no vaccine to protect people and to prevent its spread so far.Instead,control strategies associated with health care,such as social distancing,quarantine,travel restrictions,can be adopted to control the pandemic of COVID-19.This article sheds light on the dynamical behaviors of nonlinear COVID-19 models based on two methods:the Homotopy perturbation method(HPM)and the modied reduced differential transform method(MRDTM).We invoke a novel signal ow graph that is used to describe the COVID-19 model.Through our mathematical studies,it is revealed that social distancing between potentially infected individuals who are carrying the virus and healthy individuals can decrease or interrupt the spread of the virus.The numerical simulation results are in reasonable agreement with the study predictions.The free equilibrium and stability point for the COVID-19 model are investigated.Also,the existence of a uniformly stable solution is proved.

关键词： Nonlinear COVID-19 model equilibrium point stability existence of uniformly stable signal ow graph Homotopy perturbation method reduced differential transform method

Fractional Analysis of Thin Film Flow of Non-Newtonian Fluid

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Computer Modeling in Engineering & Sciences 2020年第9期124卷 825-845页

作者： Farnaz Ismail Mubashir Qayyum Syed Inayat Ali Shah Department of Mathematics Islamia College UniversityPeshawarPakistan Department of Sciences and Humanities National University of Computer&Emerging Sciences FASTLahorePakistan

Modeling and analysis of thin film flow with respect to magneto hydro dynamical effect has been an important theme in the field of fluid dynamics,due to its vast industrial applications.The analysis involves studying the behavior and response of governing equations on the basis of various parameters such as thickness of the film,film surface profile,shear stress,liquid velocity,volumetric flux,vorticity,gravity,viscosity among others,along with different boundary conditions.In this article,we extend this analysis in fractional space using a homotopy based scheme,considering the case of a Non-Newtonian Pseudo-Plastic fluid for lifting and drainage on a vertical wall.After applying similarity transformations,the given problems are reduced to highly non-linear and inhomogeneous ordinary differential equations.Moreover,fractional differential equations are obtained using basic definitions of fractional calculus.The Homotopy perturbation method(HPM),along with fractional calculus is used for obtaining approximate solutions.Physical quantities such as the velocity profile,volume flux and average velocity respectively for lift and drainage cases have been calculated.To the best of our knowledge,the given problems have not been attempted before in fractional space.Validity and convergence of the obtained solutions are confirmed by finding residual errors.From a physical perspective,a comprehensive study of the effects of various parameters on the velocity profile is also performed.Study reveals that Stokes number St,non-Newtonian parameterand magnetic parameter M have inverse relationship with fluid velocity in lifting case.In the drainage case,Stokes number St and non-Newtonian parameterhave direct relationship with fluid velocity,but magnetic parameter M shows inverse relationship with velocity.The investigation also shows that the fractional parameterhas direct relationship with the fluid velocity in lifting case,while it has inverse relationship with velocity in the drainage case.

关键词： Pseudo-plastic fluid magneto hydro dynamic fractional differential equation Homotopy perturbation method