Showing papers on "Homotopy analysis method published in 2008"
TL;DR: In this paper, a modification of He's homotopy perturbation method is presented, which extends the application of the method to solve nonlinear differential equations of fractional order, which does not require a small parameter in an equation.
Abstract: In this paper, a modification of He’s homotopy perturbation method is presented. The new modification extends the application of the method to solve nonlinear differential equations of fractional order. In this method, which does not require a small parameter in an equation, a homotopy with an imbedding parameter p ∈ [0, 1] is constructed. The proposed algorithm is applied to the quadratic Riccati differential equation of fractional order. The results reveal that the method is very effective and convenient for solving nonlinear differential equations of fractional order.
397 citations
TL;DR: In this paper, the Optimal Homotopy Asymptotic Method (OHAM) is used to solve nonlinear equations arising in heat transfer, which provides a convenient way to control the convergence of approximation series and adjust convergence regions when necessary.
375 citations
TL;DR: In this paper, the effect of radiation on the boundary layer flow and heat transfer of a viscous fluid over an exponentially stretching sheet is studied and the homotopy analysis method (HAM) is employed to determine the convergent series expressions of velocity and temperature.
357 citations
Journal Article•
TL;DR: The homotopy perturbation method as discussed by the authors decomposes a complex problem under study into a series of simple problems that are easy to be solved, and thus is accessible to non-mathematicians and engineers.
Abstract: The homotopy perturbation method is extremely accessible to non-mathematicians and engineers. The method decomposes a complex problem under study into a series of simple problems that are easy to be solved. This note gives an elementary introduction to the basic solution procedure of the homotopy perturbation method. Particular attention is paid to constructing a suitable homotopy equation.
285 citations
TL;DR: The solution of a delay differential equation is presented by means of a homotopy perturbation method and then some numerical illustrations are given to reveal that the proposed method is very effective and simple to perform.
281 citations
TL;DR: In this paper, the homotopy analysis method (HAM) was used to solve the quadratic Riccati differential equation (QRDE) by means of an analytic technique.
277 citations
TL;DR: In this paper, the Optimal Homotopy Asymptotic Method (OHAM) has been applied to thin film flow of a fourth grade fluid down a vertical cylinder and the results reveal that the proposed method is very accurate, effective and easy to use.
Abstract: A new approximate analytical technique to address for non-linear problems, namely Optimal Homotopy Asymptotic Method (OHAM) is proposed and has been applied to thin film flow of a fourth grade fluid down a vertical cylinder. This approach however, does not depend upon any small/large parameters in comparison to other perturbation method. This method provides a convenient way to control the convergence of approximation series and allows adjustment of convergence regions where necessary. The series solution has been developed and the recurrence relations are given explicitly. The results reveal that the proposed method is very accurate, effective and easy to use.
231 citations
TL;DR: In this article, the homotopy perturbation method (HPM) is used for solving the integro-differential equation with time-periodic coefficients, and the results of applying this procedure to the integral differential equation have shown the high accuracy, simplicity and efficiency of this method.
Abstract: In this research, an integro-differential equation which describes the charged particle motion for certain configurations of oscillating magnetic fields is considered.The homotopy perturbation method (HPM) is used for solving this equation.HPM is an analytical procedure for finding the solutions of problems which is based on the constructing a homotopy with an imbedding parameter p that is considered as a small parameter.The results of applying this procedure to the integro-differential equation with time-periodic coefficients show the high accuracy, simplicity and efficiency of this method.
220 citations
TL;DR: In this paper, the homotopy analysis method (HAM) is applied to obtain the approximate solution of the nonlinear model of diffusion and reaction in catalyst pellets for the case of nth-order reactions.
190 citations
TL;DR: In this article, series solutions of the Lane-Emden equation based on either a Volterra integral equation formulation or the expansion of the dependent variable in the original ordinary differential equation are presented and compared with series solutions obtained by means of integral or differential equations based on a transformation of dependent variables.
Abstract: Series solutions of the Lane–Emden equation based on either a Volterra integral equation formulation or the expansion of the dependent variable in the original ordinary differential equation are presented and compared with series solutions obtained by means of integral or differential equations based on a transformation of the dependent variables. It is shown that these four series solutions are the same as those obtained by a direct application of Adomian’s decomposition method to the original differential equation, He’s homotopy perturbation technique, and Wazwaz’s two implementations of the Adomian method based on either the introduction of a new differential operator that overcomes the singularity of the Lane–Emden equation at the origin or the elimination of the first-order derivative term of the original equation. It is also shown that Adomian’s decomposition technique can be interpreted as a perturbative approach which coincides with He’s homotopy perturbation method. An iterative technique based on Picard’s fixed-point theory is also presented and its convergence is analyzed. The convergence of this iterative approach depends on the independent variable and, therefore, this technique is not as convenient as the series solutions derived by the four methods presented in this paper, He’s homotopy perturbation technique, and Adomian’s decomposition method.
153 citations
TL;DR: In this article, the authors analyzed the thin film flow problem with a third grade fluid on an inclined plane and solved the governing non-linear equation for the velocity field using the traditional perturbation technique as well as the recently introduced homotopy perturbing method and compared the results.
Abstract: The present paper analyses the thin film flow problem with a third grade fluid on an inclined plane. The governing non-linear equation is solved for the velocity field using the traditional perturbation technique as well as the recently introduced homotopy perturbation method and the results are compared. The expressions for volume flux and average film velocity are also given.
TL;DR: In this article, the homotopy perturbation method was extended to solve nonlinear fractional partial differential equations with initial conditions, and the results reveal that the proposed method is very effective and simple for obtaining approximate solutions.
Abstract: In the paper, we extend the homotopy perturbation method to solve nonlinear fractional partial differential equations. The time- and space-fractional KdV-Burgers equations with initial conditions are chosen to illustrate our method. As a result, we successfully obtain some available approximate solutions of them. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of nonlinear fractional partial differential equations.
TL;DR: The homotopy perturbation method is applied to solve both linear and nonlinear boundary value problems for fourth-order integro-differential equations and shows that it is of high accuracy, more convenient and efficient for solving Integro- differential equations.
Abstract: In this study, the homotopy perturbation method proposed by Ji-Huan He is applied to solve both linear and nonlinear boundary value problems for fourth-order integro-differential equations. The analysis is accompanied by numerical examples. The results show that the homotopy perturbation method is of high accuracy, more convenient and efficient for solving integro-differential equations.
TL;DR: In this paper, the steady two-dimensional mixed convection flow of a micropolar fluid over a non-linear stretching sheet is investigated and the governing nonlinear equations and their associated boundary conditions are transformed into coupled nonlinear ordinary differential equations.
TL;DR: In this article, the flow of a viscous incompressible fluid between two parallel plates due to the normal motion of the plates is investigated, and the unsteady Navier-Stokes equations are reduced to a nonlinear fourth-order differential equation by using similarity solutions.
Abstract: The flow of a viscous incompressible fluid between two parallel plates due to the normal motion of the plates is investigated. The unsteady Navier-Stokes equations are reduced to a nonlinear fourth-order differential equation by using similarity solutions. Homotopy analysis method (HAM) is used to solve this nonlinear equation analytically. The convergence of the obtained series solution is carefully analyzed. The validity of our solutions is verified by the numerical results obtained by fourth-order Runge-Kutta.
TL;DR: In this paper, a similarity transformation is used to reduce the partial differential equations modeling the flow, to a single fourth-order nonlinear differential equation containing the Reynolds number and the magnetic field strength as parameters.
Abstract: The present paper analyses the unsteady 2‐dimensional flow of a viscous MHD fluid between two parallel infinite plates. The two infinite plates are considered to be approaching each other symmetrically, causing the squeezing flow. A similarity transformation is used to reduce the partial differential equations modeling the flow, to a single fourth‐order non‐linear differential equation containing the Reynolds number and the magnetic field strength as parameters. The velocity functions are obtained for a range of values of both parameters by using the homotopy perturbation method. The total resistance to the upper plate is presented.
TL;DR: In this article, an analytic technique, the homotopy analysis method (HAM), is applied to obtain the soliton solution of the Fitzhugh-Nagumo equation.
TL;DR: In this article, the mass transfer of the steady two-dimensional magnetohydrodynamic boundary layer flow of an upper-convected Maxwell (UCM) fluid past a porous shrinking sheet in the presence of chemical reaction is investigated.
TL;DR: In this article, a similar solution for the nano boundary layer with nonlinear Navier boundary condition is presented, where three types of flows are considered: the flow past a wedge, the flow in a convergent channel and the flow driven by an exponentially-varying outer flows.
TL;DR: This paper applies the homotopy perturbation method for solving the sixth-order boundary value problems by reformulating them as an equivalent system of integral equations by using a suitable transformation.
Abstract: In this paper, we apply the homotopy perturbation method for solving the sixth-order boundary value problems by reformulating them as an equivalent system of integral equations. This equivalent formulation is obtained by using a suitable transformation. The analytical results of the integral equations have been obtained in terms of convergent series with easily computable components. Several examples are given to illustrate the efficiency and implementation of the homotopy perturbation method. Comparisons are made to confirm the reliability of the homotopy perturbation method. We have also considered an example where the homotopy perturbation method is not reliable.
TL;DR: This article implements the new iterative method (NIM) proposed by Daftardar-Gejji and Jafari to solve linear/nonlinear partial differential equations of integer and fractional order and the results obtained are compared with other iterative methods such as adomian decomposition, homotopy perturbation and variational iteration methods.
TL;DR: In this article, the equations of motion and energy of a second grade fluid for the developed flow over a stretching sheet with slip condition are presented, where the electrically conducting fluid occupies the semi-infinite porous space.
TL;DR: In this article, the results of the problem considered in example 2 of (Rajabi, Ganji, Therian, Application of homotopy perturbation method in nonlinear heat conduction and convection equations, Phys. Lett. A 360 (2007) 570−573).
Abstract: Recently, Rajabi et al. (Application of homotopy perturbation method in nonlinear heat conduction and convection equations, Phys. Lett. A 360 (2007) 570–573.) discussed the solutions of temperature distribution in lumped system of combined convection–radiation. They solved a nonlinear equation of the steady conduction in a slab with variable thermal conductivity using both perturbation and homotopy perturbation methods. They claim that homotopy perturbation method (HPM) does not require any small parameter. However, this statement is not true always. Moreover, HPM have no criteria for establishing the convergence of the series solution. In this letter we have explicitly shown that the results of the problem considered in example 2 of (Rajabi, Ganji, Therian, Application of homotopy perturbation method in nonlinear heat conduction and convection equations, Phys. Lett. A 360 (2007) 570–573.) are valid only for 0 ⩽ e ⩽ 1 . We have used the homotopy analysis method for finding the more meaningful solution.
TL;DR: In this article, the authors presented analytical solutions to the Lane-Emden equation y = 0 describing the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules.
TL;DR: In this paper, the rotating flow of a second grade fluid past a porous shrinking surface was studied and the governing partial differential equations were first reduced into ordinary differential equations and then solved by homotopy analysis method (HAM).
TL;DR: In this article, the hydrodynamic elastico-viscous fluid over a stretching surface is reduced to ordinary differential equations, which are analytically solved by applying an efficient technique namely the homotopy analysis method (HAM).
Abstract: This article looks at the hydrodynamic elastico-viscous fluid over a stretching surface. The equations governing the flow are reduced to ordinary differential equations, which are analytically solved by applying an efficient technique namely the homotopy analysis method (HAM). The solutions for the velocity components are computed. The numerical values of wall skin friction coefficients are also tabulated. The present HAM solution is compared with the known exact solution for the two-dimensional flow and an excellent agreement is found.
TL;DR: In this article, a modification to homotopy perturbation method for solving linear Fredholm integral equations is presented and compared with the standard HPM and the modified one, which reveals that the proposed method is very effective and simple and gives the exact solution.
Abstract: In this paper we present a modification to homotopy perturbation method for solving linear Fredholm integral equations. Comparisons are made between the standard HPM and the modified one. The results reveal that the proposed method is very effective and simple and gives the exact solution.
TL;DR: In this article, an analysis is carried out to study the unsteady magnetohydrodynamic (MHD) two-dimensional boundary layer flow of a second grade viscoelastic fluid over an oscillatory stretching surface, which is induced due to an infinite elastic sheet which is stretched back and forth in its own plane.
Abstract: An analysis is carried out to study the unsteady magnetohydrodynamic (MHD) two-dimensional boundary layer flow of a second grade viscoelastic fluid over an oscillatory stretching surface. The flow is induced due to an infinite elastic sheet which is stretched back and forth in its own plane. For the investigated problem, the governing equations are reduced to a non-linear partial differential equation by means of similarity transformations. This equation is solved both by a newly developed analytic technique, namely homotopy analysis method (HAM) and by a numerical method employing the finite difference scheme, in which a coordinate transformation is employed to transform the semi-infinite physical space to a bounded computational domain. The results obtained by means of both methods are then compared and show an excellent agreement. The effects of various parameters like visco-elastic parameter, the Hartman number and the relative frequency amplitude of the oscillatory sheet to the stretching rate on the velocity field are graphically illustrated and analysed. The values of wall shear stress for these parameters are also tabulated and discussed.
TL;DR: In this paper, the deformation of a cantilever beam under point load at the free tip is investigated by an analytic method, namely the homotopy analysis method (HAM).
TL;DR: The governing nonlinear partial differential equation has been reduced to the nonlinear ordinary differential equation by means of suitable transformations and the developed nonlinear equation is solved analytically by using the homotopy analysis method (HAM).