Showing papers on "Homotopy analysis method published in 2009"
TL;DR: In this article, the basic ideas and current developments of the homotopy analysis method, an analytic approach to get convergent series solutions of strongly nonlinear problems, which recently attracts interests of more and more researchers, are described.
835 citations
TL;DR: In this paper, the homotopy analysis method is applied to solve linear and nonlinear fractional initial-value problems (fIVPs), and the fractional derivatives are described by Caputo's sense.
414 citations
TL;DR: A new analytic approximate technique for addressing nonlinear problems, namely the Optimal Homotopy Asymptotic Method (OHAM), is proposed and used in an application to the steady flow of a fourth-grade fluid.
273 citations
TL;DR: In this article, the authors used the homotopy analysis method to detect the fin efficiency of convective straight fins with temperature-dependent thermal conductivity, compared with those of the exact solution and Adomian's decomposition method.
188 citations
TL;DR: This paper is an elementary introduction to the concepts of the homotopy perturbation method and gives an intuitive grasp for the solution procedure throughout the paper.
Abstract: This paper is an elementary introduction to the concepts of the homotopy perturbation method. Particular attention is paid to giving an intuitive grasp for the solution procedure throughout the paper.
183 citations
TL;DR: In this article, an analysis has been performed to study magneto-hydrodynamic (MHD) squeeze flow between two parallel infinite disks where one disk is impermeable and the other is porous with either suction or injection of the fluid.
Abstract: An analysis has been performed to study magneto-hydrodynamic (MHD) squeeze flow between two parallel infinite disks where one disk is impermeable and the other is porous with either suction or injection of the fluid. We investigate the combined effect of inertia, electromagnetic forces, and suction or injection. With the introduction of a similarity transformation, the continuity and momentum equations governing the squeeze flow are reduced to a single, nonlinear, ordinary differential equation. An approximate solution of the equation subject to the appropriate boundary conditions is derived using the homotopy perturbation method (HPM) and compared with the direct numerical solution (NS). Results showing the effect of squeeze Reynolds number, Hartmann number and the suction/injection parameter on the axial and radial velocity distributions are presented and discussed. The approximate solution is found to be highly accurate for the ranges of parameters investigated. Because of its simplicity, versatility and high accuracy, the method can be applied to study linear and nonlinear boundary value problems arising in other engineering applications.
179 citations
TL;DR: In this paper, the homotopy analysis method (HAM) is implemented for finding approximate solutions of the Burger and regularized long wave (RLW) equations, and the results of the proposed method are compared with the results obtained by HPM.
170 citations
TL;DR: In this paper, the authors adopt the homotopy analysis method (HAM) to obtain solutions of linear and nonlinear fractional diffusion and wave equation, where the fractional derivative is described in the Caputo sense.
166 citations
TL;DR: In this article, the stagnation point flow of an incompressible micropolar fluid over a non-linear stretching surface is studied, and the resulting nonlinear system of equations is solved analytically using homotopy analysis method (HAM).
Abstract: The two-dimensional magnetohydrodynamic (MHD) stagnation-point flow of an incompressible micropolar fluid over a non-linear stretching surface is studied. The resulting non-linear system of equations is solved analytically using homotopy analysis method (HAM). The convergence of the obtained series solutions is explicitly discussed and given in the form of recurrence formulas. The influence of various pertinent parameters on the velocity, microrotation and skin-friction are shown in the tables and graphs. Comparison is also made with the corresponding numerical results of viscous ( K = 0 ) [R. Cortell, Viscous flow and heat transfer over a nonlinearly stretching sheet, Appl. Math. Comput. 184 (2007) 864–873] and hydrodynamic micropolar fluid ( M = 0 ) [R. Nazar, N. Amin, D. Filip, I. Pop, Stagnation point flow of a micropolar fluid towards a stretching sheet, Internat. J. Non-Linear Mech. 39 (2004) 1227–1235] for linear and non-linear stretching sheet. An excellent agreement is found.
160 citations
TL;DR: In this article, a new analytic technique is proposed to solve non-linear Riccati differential equation with fractional order, which provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter.
Abstract: In this paper, based on the homotopy analysis method (HAM), a new analytic technique is proposed to solve non-linear Riccati differential equation with fractional order. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter ℏ . Besides, it is proved that well-known Adomian’s decomposition method is a special case of the homotopy analysis method when ℏ = −1. This work illustrates the validity and great potential of the homotopy analysis method for the non-linear differential equations with fractional order. The basic ideas of this approach can be widely employed to solve other strongly non-linear problems in fractional calculus.
159 citations
TL;DR: In this paper, the homotopy perturbation method was applied to solve both linear and nonlinear boundary value problems for fourth-order parabolic partial differential equations with variable coefficients.
Abstract: In this work, the homotopy perturbation method proposed by Ji-Huan He [1] is applied to solve both linear and nonlinear boundary value problems for fourth-order partial differential equations The numerical results obtained with minimum amount of computation are compared with the exact solution to show the efficiency of the method The results show that the homotopy perturbation method is of high accuracy and efficient for solving the fourth-order parabolic partial differential equation with variable coefficients The results show also that the introduced method is a powerful tool for solving the fourth-order parabolic partial differential equations
TL;DR: In this article, an analysis is performed to find the series solution of the boundary layer Falkner-Skan equation for wedge, which takes into account a special form of the chosen magnetic field.
TL;DR: In this paper, the steady two-dimensional magnetohydrodynamic flow of an upper-convected Maxwell fluid near a stagnation point over a stretching surface is analyzed and the governing nonlinear partial differential equation for the flow are reduced to an ordinary differential equation by using similarity transformations.
Abstract: The present analysis comprises the steady two-dimensional magnetohydrodynamic flow of an upper-convected Maxwell fluid near a stagnation-point over a stretching surface. The governing non-linear partial differential equation for the flow are reduced to an ordinary differential equation by using similarity transformations. The analytic solution of nonlinear system is constructed in the series form using Homotopy analysis method. Convergence of the obtained series is discussed explicitly. The effects of the sundry parameters on the velocity profile is shown through graphs. The values of skin-friction coefficient for different parameters is tabulated.
TL;DR: In this paper, the authors discuss the selection of the initial approximation, auxiliary linear operator, auxiliary function, and convergence control parameter in the application of the homotopy analysis method, in a fairly general setting.
TL;DR: In this paper, an analytical technique, namely the homotopy analysis method (HAM), is applied to obtain an approximate analytical solution of the Burgers-Huxley equation.
Abstract: In this paper, an analytical technique, namely the homotopy analysis method (HAM) is applied to obtain an approximate analytical solution of the Burgers–Huxley equation. This paper introduces the two theorems which provide us with a simple and convenient way to apply the HAM to the nonlinear PDEs with the power-law nonlinearity. The homotopy analysis method contains the auxiliary parameter ħ , which provides us with a simple way to adjust and control the convergence region of solution series.
TL;DR: In this paper, a homotopy perturbation method is introduced to obtain exact solutions to some linear and nonlinear partial differential equations, which is a powerful device for solving a wide variety of problems.
Abstract: In this paper, we introduce a homotopy perturbation method to obtain exact solutions to some linear and nonlinear partial differential equations. This method is a powerful device for solving a wide variety of problems. Using the homotopy perturbation method, it is possible to find the exact solution or an approximate solution of the problem. Convergence of the method is proved. Some examples such as Burgers’, Schrodinger and fourth order parabolic partial differential equations are presented, to verify convergence hypothesis, and illustrating the efficiency and simplicity of the method.
TL;DR: In this paper, a detailed study of some relatively new techniques which are originated by He for solving diversified nonlinear problems of physical nature is presented, focusing on the variational iteration method (VIM) and its modifications, the homotopy perturbation method (HPM), the parameter expansion method and exp-function method.
Abstract: This paper outlines a detailed study of some relatively new techniques which are originated by He for solving diversified nonlinear problems of physical nature. In particular, we will focus on the variational iteration method (VIM) and its modifications, the homotopy perturbation method (HPM), the parameter expansion method, and exp-function method. These relatively new but very reliable techniques proved useful for solving a wide class of nonlinear problems and are capable to cope with the versatility of the physical problems. Several examples are given to reconfirm the efficiency of these algorithms. Some open problems are also suggested for future research work.
TL;DR: In this paper, a homotopy perturbation method (HPM) is proposed to solve non-linear systems of second-order boundary value problems, which yields solutions in convergent series forms with easily computable terms.
Abstract: A homotopy perturbation method (HPM) is proposed to solve non-linear systems of second-order boundary value problems. HPM yields solutions in convergent series forms with easily computable terms, and in some cases, yields exact solutions in one iteration. Moreover, this technique does not require any discretization, linearization or small perturbations and therefore reduces the numerical computations a lot. Some numerical results are also given to demonstrate the validity and applicability of the presented technique. The results reveal that the method is very effective, straightforward and simple.
TL;DR: In this article, the homotopy analysis method has been employed to obtain solutions of a system of nonlinear fractional partial differential equations, which indicates the validity and great potential of the Homotopy Analysis method for solving system of fractional PDEs.
TL;DR: In this paper, the He's homotopy perturbation method is applied to solve systems of Volterra integral equations of the second kind, and the results reveal that the method is very effective and simple.
Abstract: In this paper, the He’s homotopy perturbation method is applied to solve systems of Volterra integral equations of the second kind. Some examples are presented to illustrate the ability of the method for linear and non-linear such systems. The results reveal that the method is very effective and simple.
TL;DR: In this article, an analytic method for strongly nonlinear problems, namely the homotopy analysis method (HAM), is applied to give convergent series solution of non-similarity boundary-layer flows.
TL;DR: In this paper, the homotopy analysis method (HAM) was applied to the case of two-dimensional and axisymmetric shrinking and the convergence of the obtained series solution was discussed explicitly.
Abstract: This work is concerned with the magnetohydrodynamic (MHD) viscous flow due to a shrinking sheet. The cases of two dimensional and axisymmetric shrinking have been discussed. Exact series solution is obtained using the homotopy analysis method (HAM). The convergence of the obtained series solution is discussed explicitly. The obtained HAM solution is valid for all values of the suction parameter and Hartman number.
TL;DR: In this article, the homotopy perturbation method (HPM) is employed to derive analytical solutions for nonlinear fractional Schrödinger equations, and the results reveal that the technique introduced here is very effective and convenient for solving nonlinear partial differential equations of fractional order.
Abstract: In this study, we present a framework to obtain analytical solutions to nonlinear fractional Schrödinger equations. The homotopy perturbation method (HPM) is employed to derive analytical solutions for these equations. Some examples are tested and the results reveal that the technique introduced here is very effective and convenient for solving nonlinear partial differential equations of fractional order.
TL;DR: In this article, the homotopy perturbation method (HPM) is applied for solving linear and nonlinear partial differential equations. But the proposed algorithm is quite efficient and is practically well suited for use in these problems.
Abstract: We apply a relatively new technique which is called the homotopy perturbation method (HPM) for solving linear and nonlinear partial differential equations. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the HPM solves nonlinear problems without using Adomian’s polynomials can be considered as a clear advantage of this technique over the decomposition method.
TL;DR: In this article, the authors used the homotopy analysis method (HAM) to evaluate the analytical approximate solutions and efficiency of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient.
TL;DR: Alizadeh-Pahlavan et al. as mentioned in this paper investigated the effects of parameters such as elasticity number, magnetic number, radiative heat transfer, Prandtl number, and Eckert number on the temperature field above the sheet.
TL;DR: In this article, the homotopy perturbation method is used to construct numerical solutions of the space-time fractional advection-dispersion equation in the form of a rapidly convergent series with easily computable components.
TL;DR: In this paper, the generalized Emden-Fowler type equations in the second-order ordinary differential equations (ODEs) are obtained by homotopy perturbation method (HPM).
Abstract: In this paper, approximate and/or exact analytical solutions of the generalized Emden–Fowler type equations in the second-order ordinary differential equations (ODEs) are obtained by homotopy-perturbation method (HPM). The homotopy-perturbation method (HPM) is a coupling of the perturbation method and the homotopy method. The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve. In this work, HPM yields solutions in convergent series forms with easily computable terms, and in some cases, only one iteration leads to the high accuracy of the solutions. Comparisons with the exact solutions and the solutions obtained by the Adomian decomposition method (ADM) show the efficiency of HPM in solving equations with singularity.
TL;DR: The purpose of this study is to introduce a modification of the homotopy perturbation method using Laplace transform and Pade approximation to obtain closed form solutions of nonlinear coupled systems of partial differential equations.
Abstract: The purpose of this study is to introduce a modification of the homotopy perturbation method using Laplace transform and Pade approximation to obtain closed form solutions of nonlinear coupled systems of partial differential equations. Two test examples are given; the coupled nonlinear system of Burger equations and the coupled nonlinear system in one dimensional thermoelasticity. The results obtained ensure that this modification is capable of solving a large number of nonlinear differential equations that have wide application in physics and engineering.
TL;DR: In this paper, the homotopy analysis method (HAM) is used to investigate non-linear vibration behavior of Euler-Bernoulli beams subjected to axial loads, and the effect of vibration amplitude on the nonlinear frequency and buckling load is discussed.