Showing papers on "Homotopy analysis method published in 2007"
TL;DR: Based on homotopy, which is a basic concept in topology, a general analytic method was proposed to obtain series solutions of nonlinear differential equations in this article, where the authors showed that the second-order nonlinear oscillation equation can be replaced by an infinite number of (2κ)thorder linear differential equations, where κ≥ 1 can be any a positive integer.
Abstract: Based on homotopy, which is a basic concept in topology, a general analytic method (namely the homotopy analysis method) is proposed to obtain series solutions of nonlinear differential equations. Different from perturbation techniques, this approach is independent of small/large physical parameters. Besides, different from all previous analytic methods, it provides us with a simple way to adjust and control the convergence of solution series. Especially, it provides us with great freedom to replace a nonlinear differential equation of order n into an infinite number of linear differential equations of order k, where the order k is even unnecessary to be equal to the order n. In this paper, a nonlinear oscillation problem is used as example to describe the basic ideas of the homotopy analysis method. We illustrate that the second-order nonlinear oscillation equation can be replaced by an infinite number of (2κ)th-order linear differential equations, where κ≥ 1 can be any a positive integer. Then, the homotopy analysis method is further applied to solve a high-dimensional nonlinear differential equation with strong nonlinearity, i.e., the Gelfand equation. We illustrate that the second-order two or three-dimensional nonlinear Gelfand equation can be replaced by an infinite number of the fourth or sixth-order linear differential equations, respectively. In this way, it might be greatly simplified to solve some nonlinear problems, as illustrated in this paper. All of our series solutions agree well with numerical results. This paper illustrates that we might have much larger freedom and flexibility to solve nonlinear problems than we thought traditionally. It may keep us an open mind when solving nonlinear problems, and might bring forward some new and interesting mathematical problems to study.
432 citations
TL;DR: In this paper, the authors presented an efficient and reliable treatment of the homotopy perturbation method (HPM) for nonlinear partial differential equations with fractional time derivative, described in the Caputo sense.
368 citations
TL;DR: In this article, an analytic technique, namely the homotopy analysis method (HAM), is applied to solve a generalized Hirota-Satsuma coupled KdV equation, which provides a simple way to adjust and control the convergence region of solution series.
334 citations
TL;DR: In this paper, the variational iteration method and the homotopy perturbation method are used for solving the system of fraction differential equations (FDE) generated by a multi-order fraction differential equation.
262 citations
TL;DR: In this paper, a totally analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder is obtained using homotopy analysis method (HAM), and the series solution is developed and the recurrence relations are given explicitly.
254 citations
TL;DR: In this article, the homotopy analysis method (HAM) was applied to solve heat transfer problems with high nonlinearity order, and the results were compared with the numerical Runge-Kutta methods and homotropic perturbation method (HPM).
252 citations
TL;DR: In this paper, the homotopy perturbation method was introduced for Adomian's polynomials, and the solution procedure is very effective and straightforward, but the main demerit of this method is that it is difficult and complex to calculate.
Abstract: The Adomian method is widely used in approximate calculation, its main demerit is that it is very difficult and complex to calculate Adomian's polynomials. This paper introduces the homotopy perturbation method for overcoming completely the disadvantage. The solution procedure is very effective and straightforward.
224 citations
TL;DR: In this paper, the homotopy analysis method (HAM) is applied to obtain the soliton solution of the fifth-order KdV equation, which contains the auxiliary parameter ℏ, which provides a simple way to adjust and control the convergence region of series solution.
Abstract: An analytic technique, the homotopy analysis method (HAM), is applied to obtain the soliton solution of the fifth-order KdV equation. The homotopy analysis method (HAM) provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter ℏ, which provides us with a simple way to adjust and control the convergence region of series solution.
218 citations
TL;DR: The steady laminar flow and heat transfer of a second grade fluid over a radially stretching sheet is considered in this article, where axisymmetric flow is induced due to linear stretching of a sheet.
214 citations
TL;DR: In this article, the homotopy perturbation method (HPM) proposed in 1998 is only a special case of the HAM method, and it is shown that the solutions given by HPM (Siddiqui, A.M., Ahmed, M., Ghori, Q.K.).
Abstract: In this paper, we prove in general that the homotopy perturbation method (HPM) proposed in 1998 is only a special case of the homotopy analysis method (HAM) profound in 1992 when ħ = −1. Besides, by using the thin film flows of Sisko and Oldroyd 6-constant fluids on a moving belt as examples, we show that the solutions given by HPM (Siddiqui, A.M., Ahmed, M., Ghori, Q.K.: Chaos Solitons and Fractals (2006) in press) are divergent, and thus useless. However, by choosing a proper value of the auxiliary parameter ħ, we give convergent series solution by means of the HAM. These two examples also show that, different from the HPM and other traditional analytic techniques, the HAM indeed provides us with a simple way to ensure the convergence of the solution.
209 citations
TL;DR: In this article, a new scheme, deduced from He's homotopy perturbation method, is presented for solving Lane-Emden type singular IVPs problem, and only a few terms are required to obtain accurate computable solutions.
TL;DR: In this paper, the authors considered the problem of two-dimensional projectile motion in which the resistance acting on an object moving in air is proportional to the square of the velocity of the object (quadratic resistance law).
Abstract: We consider the problem of two-dimensional projectile motion in which the resistance acting on an object moving in air is proportional to the square of the velocity of the object (quadratic resistance law). It is well known that the quadratic resistance law is valid in the range of the Reynolds number: 1 × 103 ~ 2 × 105 (for instance, a sphere) for practical situations, such as throwing a ball. It has been considered that the equations of motion of this case are unsolvable for a general projectile angle, although some solutions have been obtained for a small projectile angle using perturbation techniques. To obtain a general analytic solution, we apply Liao's homotopy analysis method to this problem. The homotopy analysis method, which is different from a perturbation technique, can be applied to a problem which does not include small parameters. We apply the homotopy analysis method for not only governing differential equations, but also an algebraic equation of a velocity vector to extend the radius of convergence. Ultimately, we obtain the analytic solution to this problem and investigate the validation of the solution.
TL;DR: Comparison of the results obtained by the homotopy perturbation method with those obtaining by the variational iteration method reveals that the present methods are very effective and convenient.
Abstract: In this article, the homotopy perturbation method proposed by J.- H. He is adopted for solving linear fractional partial differential equations. The fractional derivatives are described in the Caputo sense. Comparison of the results obtained by the homotopy perturbation method with those obtained by the variational iteration method reveals that the present methods are very effective and convenient.
TL;DR: In this paper, the authors compare two distinct adaptations of He's homotopy perturbation method (HPM) for determining frequency-amplitude relation of the nonlinear oscillator with discontinuities.
Abstract: In nonlinear analysis, perturbation methods are well established tools to study diverse aspects of nonlinear problems. Surveys of the early literature with numerous references, and useful bibliographies, have been given by Nayfeh [1], Mickens [2], Jordan and Smith [3] and Hagedorn [4], However, the use of perturbation theory in many important practical problems is invalid, or it simply breaks down for parameters beyond a certain specified range. Therefore, new analytical techniques should be developed to overcome these shortcomings. Such a new technique should work over a large range of parameters and yield accurate analytical approximate solutions beyond the coverage and ability of the classical perturbation methods. For example, homotopy perturbation method introduced by He [5-14] can readily eliminate the limitations of the traditional perturbation techniques. Moreover, this method was first applied to nonlinear oscillators with discontinuities[15]. For more detailed information please refer to the comprehensive book[16] or the review articles[ 17,18] by He. There also exists a wide range of literature dealing with the approximate determination of periodic solutions for nonlinear problems by using a mixture of methodologies [19-37], The purpose of this paper is to make the comparison of two distinct adaptations of He's homotopy perturbation method (HPM) for determining frequency-amplitude relation of the nonlinear oscillator with discontinuities.
TL;DR: In this article, the authors presented the first available numerical solutions of the fractional KdV-Burgers-Kuramoto equation for nonlinear fractional differential equations.
TL;DR: In this paper, an analytical approximation to the solution of Schrodinger equations has been studied and the results reveal that the method is very effective and simple, and some examples are provided.
TL;DR: An approximate analytical solution is obtained of the steady, laminar three-dimensional flow for an incompressible, viscous fluid past a stretching sheet using the homotopy perturbation method (HPM) proposed by He.
Abstract: An approximate analytical solution is obtained of the steady, laminar three-dimensional flow for an incompressible, viscous fluid past a stretching sheet using the homotopy perturbation method (HPM) proposed by He. The flow is governed by a boundary value problem (BVP) consisting of a pair of non-linear differential equations. The solution is simple yet highly accurate and compares favorably with the exact solutions obtained early in the literature. The methodology presented in the paper is useful for solving the BVPs consisting of more than one differential equation.
TL;DR: In this article, Adomian's decomposition method is proposed to solve the well-known Blasius equation, which is of high accuracy compared with homotopy perturbation method and Howarth's numerical solution.
Abstract: In this paper, Adomian’s decomposition method is proposed to solve the well-known Blasius equation. Comparison with homotopy perturbation method and Howarth’s numerical solution reveals that the Adomian’s decomposition method is of high accuracy.
TL;DR: In this article, an application of He's homotopy perturbation method is applied to solve functional integral equations, and the results reveal that the He's HOP method is very effective and simple and gives the exact solution.
Abstract: In this paper, an application of He’s homotopy perturbation method is applied to solve functional integral equations. Comparisons are made between expansion method based on Lagrange interpolation formulae and the homotopy perturbation method. The results reveal that the He’s homotopy perturbation method is very effective and simple and gives the exact solution.
TL;DR: In this article, the steady magnetohydrodynamic (MHD) flow of a second grade fluid in the presence of radiation is analyzed by means of similarity transformation, and the arising nonlinear partial differential equations are reduced to a system of four coupled ordinary differential equations.
TL;DR: In this paper, the modified He's homotopy perturbation method (HPM) was applied to obtain solutions of linear and nonlinear fractional diffusion and wave equations.
TL;DR: In this paper, the homotopy perturbation method was applied for solving the fourth-order boundary value problems and the analytical results were obtained in terms of convergent series with easily computable components.
Abstract: We apply the homotopy perturbation method for solving the
fourth-order boundary value problems. The analytical results of the boundary value
problems 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
method. Homotopy method can be considered an alternative method to Adomian
decomposition method and its variant forms.
TL;DR: In this article, an analytical solution describing the magnetohydrodynamic boundary layer flow of a second grade fluid over a shrinking sheet is derived, both exact and series solutions have been determined.
Abstract: In this study, we derive an analytical solution describing the magnetohydrodynamic boundary layer flow of a second grade fluid over a shrinking sheet. Both exact and series solutions have been determined. For the series solution, the governing nonlinear problem is solved using the homotopy analysis method. The convergence of the obtained solution is analyzed explicitly. Graphical results have been presented and discussed for the pertinent parameters.
TL;DR: He's homotopy perturbation method is employed to compute an approximation to the solution of the system of nonlinear ordinary differential equations governing on the problem to show the reliability and simplicity of the method.
TL;DR: In this paper, the homotopy perturbation method is used for solving an inverse parabolic equation and computing an unknown time-dependent parameter, which is an analytical procedure for finding the solutions of differential equations which is based on the constructing of a homhotopy with an imbedding parameter p[0,1] that is considered as an expanding parameter.
Abstract: Inverse problems of parabolic type arise from many fields of physics and play a very important role in various branches of science and engineering. In the last few years, considerable efforts have been expended in formulating accurate and efficient methods to solve these equations. In this research, the homotopy perturbation method is used for solving an inverse parabolic equation and computing an unknown time-dependent parameter. The homotopy perturbation technique is an analytical procedure for finding the solutions of differential equations which is based on the constructing of a homotopy with an imbedding parameter p[0,1] that is considered as an 'expanding parameter'. In this paper, a very brief introduction to the applications of the used technique and its new development is given. The results of applying this procedure to the studied parabolic inverse problem show the high accuracy, simplicity and efficiency of the approach.
TL;DR: In this article, homotopy perturbation method is used to solve for the temperature distribution in a lumped system of combined convection-radiation and also a nonlinear equation of the steady conduction in a slab with variable thermal conductivity.
TL;DR: It is shown that an analytical solution is possible by employing a homotopy analysis method (HAM) and the convergence of the obtained solution is also taken into account.
Abstract: The steady flow of a second grade fluid in a porous channel is considered. The constitutive equations are those used for a second grade fluid. The fluid is electrically conducting in the presence of a uniform magnetic field applied in the transverse direction to the flow. It is shown that an analytical solution is possible by employing a homotopy analysis method (HAM). The convergence of the obtained solution is also taken into account. Assessment for the influence of various parameters of interest on the velocity is undertaken.
TL;DR: In this paper, the homotopy perturbation method combined with modified Lindstedt-Poincare method is applied to the search for traveling wave solutions of Korteweg-de Vries (KdV) equation.
Abstract: He's Homotopy Perturbation Method combined with modified Lindstedt-Poincare method is applied to the search for traveling wave solutions of Korteweg-de Vries (KdV) equation. The work emphasizes the power of the method that can be used in problems of identical nonlinearity.
TL;DR: In this article, the unsteady boundary-layer flow and heat transfer in an incompressible viscous electrically conducting fluid, caused by an impulsive stretching of the surface in two lateral directions and by suddenly increasing the surface temperature from that of surrounding fluid are studied analytically.
Abstract: In this paper, the unsteady boundary-layer flow and heat transfer in an incompressible viscous electrically conducting fluid, caused by an impulsive stretching of the surface in two lateral directions and by suddenly increasing the surface temperature from that of surrounding fluid are studied analytically. By using the homotopy analysis method, the accurate series solutions are obtained which are uniformly valid for all dimensionless time in the whole spatial region 0 ⩽ η ∞ . To the best of our knowledge, such kind of solutions have not been reported.