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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

Analytical Approach to Fractional Zakharov-Kuznetsov Equations by He's homotopy perturbation method

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Communications in Theoretical Physics 2010年第6期53卷 1005-1010页

作者： Ahmet Yildirim Yagmur Gülkanat Ege University Department of Mathematics

The aim of this paper is to obtain the approximate analytical solution of a fractional Zakharov-Kuznetsov equation by using homotopy perturbation method (HPM). The fractional derivatives are described in the Caputo sense. Several examples are given and the results are compared to exact solutions. The results reveal that the method is very effective and simple.

关键词： homotopy perturbation method fractional Zakharov-Kuznetsov equations fractional calculus Caputo sense

Solution to the MHD flow over a non-linear stretching sheet by homotopy perturbation method

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Science China(Physics,Mechanics & Astronomy) 2011年第2期54卷 342-345页

作者： RAFTARI Behrouz MOHYUD-DIN Syed Tauseef YILDIRIM Ahmet Department of Mathematics Islamic Azad UniversityKermanshah branch Taxila Cantt HITEC University Department of Mathematics Ege University

In this study,by means of homotopy perturbation method(HPM) an approximate solution of the magnetohydrodynamic(MHD) boundary layer flow is *** main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to *** produces analytical expressions for the solution to nonlinear differential *** obtained analytic solution is in the form of an infinite power *** this work,the analytical solution obtained by using only two terms from HPM *** with the exact solution and the solution obtained by the Pade approximants and shooting method show the high accuracy,simplicity and efficiency of this method.

关键词： boundary layer flow magnetohydrodynamics flow(MHD) nonlinear differential equations homotopy perturbation method

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

Application of Modified Couple Stress Theory and homotopy perturbation method in Investigation of Electromechanical Behaviors of Carbon Nanotubes

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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 *** 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 *** of the static and dynamic behaviors under static DC and step DC voltages are *** 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

Application of homotopy perturbation method to solve two models of delay differential equation systems

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International Journal of Biomathematics 2017年第6期10卷 99-113页

作者： Suayip Yfizbasi Murat Karacayir Department of Mathematics Akdeniz University Antalya 07058 Turkey

In this paper, two delay differential systems are considered, namely, a famous model from mathematical biology about the spread of HIV viruses in blood and the advanced Lorenz system from mathematical physics. We then apply the homotopy perturbation method （HPM） to find their approximate solutions. It turns out that the method gives rise to easily obtainable solutions. In addition, residual error functions of the solutions are graphed and it is shown that increasing the parameter n in the method improves the results in both cases.

关键词： Systems of delay differential equations homotopy perturbation method HIVinfection of CD4+ T-cells advanced Lorenz system.

The homotopy perturbation method and the Adomian Decomposition method for the Nonlinear Coupled Equations

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Journal of Partial Differential Equations 2009年第4期22卷 334-351页

作者： ZAYED E. M. E. ABDEL RAHMAN H. M. Department of Mathematics Faculty of Science Zagazig University Zagazig Egypt. Department of Basic Sciences Higher Technological Institute Tenth of Ramadan City Egypt.

In this paper, we have used the homotopy perturbation and the Adomian decomposition methods to study the nonlinear coupled Kortewge-de Vries and shallow water equations. The main objective of this paper is to propose alternative methods of solutions, which do not require small parameters and avoid linearization and physical unrealistic assumptions. The proposed methods give more general exact solutions without much extra effort and the results reveal that the homotopy perturbation and the Adomian decomposition methods are very effective, convenient and quite accurate to the systems of coupled nonlinear equations.

关键词： homotopy perturbation method Adomian decomposition method coupledKorteweg-de Vries equations shallow water equations.

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

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