Homotopy analysis method for quadratic Riccati differential equation

In this paper, the quadratic Riccati differential equation is solved by means of an analytic technique, namely the homotopy analysis method (HAM). Comparisons are made between Adomian’s decomposition method (ADM), homotopy perturbation method (HPM) and the exact solution and the homotopy analysis method. The results reveal that the proposed method is very effective and simple.     

Comparison of homotopy analysis method and homotopy-perturbation method for purely nonlinear fin-type problems

In this paper, the homotopy analysis method (HAM) is compared with the homotopy-perturbation method (HPM) and the Adomian decomposition method (ADM) to determine the temperature distribution of a straight rectangular fin with power-law temperature dependent surface heat flux. Comparisons of the results obtained by the HAM with that obtained by the ADM and HPM suggest that both the HPM and ADM are special case of the HAM.     

Some issues on HPM and HAM methods: A convergence scheme

The homotopy method for the solution of nonlinear equations is revisited in the present study. An analytic method is proposed for determining the valid region of convergence of control parameter of the homotopy series, as an alternative to the classical way of adjusting the region through graphical analysis. Illustrative examples are presented to exhibit a vivid comparison between the homotopy perturbation method (HPM) and the homotopy analysis method (HAM). For special choices of the initial guesses it is shown that the convergence-control parameter does not cover the HPM. In such cases, blindly using the HPM yields a non convergence series to the sought solution. In addition to this, HPM is shown not always to generate a continuous family of solutions in terms of the homotopy parameter. By the convergence-control parameter this can however be prevented to occur in the HAM.     

On comparison of the solutions for an axisymmetric flow

Recently, Ariel (Comput Math Appl, 54 (2007), 1169–1183) explored the axially stretching flow of a viscous fluid in the presence of a velocity slip. He computed the solutions by noniterative technique, the homotopy perturbation method (HPM), and the perturbation and asymptotic methods (for small and large values of the slip parameter, respectively). Through comparison between these solutions, he claimed that HPM solution is the best solution showing close agreement with an exact solution. Here, we recomputed the flow problem considered in Ariel's work for the series solution by homotopy analysis method (HAM). It is found that HAM solution is identical with the presented exact solution in Ariel's work. Furthermore, the HAM solution is better than the HPM solution. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009     

Homotopy analysis method for singular IVPs of Emden–Fowler type

In this paper, approximate and/or exact analytical solutions of singular initial value problems (IVPs) of the Emden–Fowler type in the second-order ordinary differential equations (ODEs) are obtained by the homotopy analysis method (HAM). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions. It is shown that the solutions obtained by the Adomian decomposition method (ADM) and the homotopy-perturbation method (HPM) are only special cases of the HAM solutions.     

A series solution of the fin problem with a temperature-dependent thermal conductivity

In this work, we apply the homotopy analysis method to solve fin problems with temperature-dependent thermal conductivity. The results are compared with the solutions obtained by the Adomian decomposition method (ADM), homotopy perturbation method (HPM) and with the Taylor series expansion method as well. The results of the comparisons have shown that the ADM and HPM are just an approximation of the present study and the accuracy of the solution obtained by the present study is much better than the latter algorithms.     

Comparison between the homotopy analysis method and homotopy perturbation method to solve coupled Schrodinger-KdV equation

In this paper, we apply the homotopy analysis method (HAM) and the homotopy perturbation method (HPM) to obtain approximate analytical solutions of the coupled Schrodinger-KdV equation. The results show that HAM is a very efficient method and that HPM is a special case of HAM.     

A new method for homoclinic solutions of ordinary differential equations

Consideration is given to the homoclinic solutions of ordinary differential equations. We first review the Melnikov analysis to obtain Melnikov function, when the perturbation parameter is zero and when the differential equation has a hyperbolic equilibrium. Since Melnikov analysis fails, using Homotopy Analysis Method (HAM, see [Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press; 2003; Liao SJ. An explicit, totally analytic approximation of Blasius’ viscous flow problems. Int J Non-Linear Mech 1999;34(4):759–78; Liao SJ. On the homotopy analysis method for nonlinear problems. Appl Math Comput 2004;147(2):499–513] and others [Abbasbandy S. The application of the homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 2006;360:109–13; Hayat T, Sajid M. On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder. Phys Lett A, in press; Sajid M, Hayat T, Asghar S. Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dyn, in press]), we obtain homoclinic solution for a differential equation with zero perturbation parameter and with hyperbolic equilibrium. Then we show that the Melnikov type function can be obtained as a special case of this homotopy analysis method. Finally, homoclinic solutions are obtained (for nontrivial examples) analytically by HAM, and are presented through graphs.     

Soliton solutions for the Fitzhugh–Nagumo equation with the homotopy analysis method

An analytic technique, the homotopy analysis method (HAM), is applied to obtain the soliton solution of the Fitzhugh–Nagumo equation. The homotopy analysis method (HAM) is one of the most effective method to obtain the exact solution and provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region of series solution.     

Series Solutions of Systems of Nonlinear Fractional Differential Equations

Differential equations of fractional order appear in many applications in physics, chemistry and engineering. An effective and easy-to-use method for solving such equations is needed. In this paper, series solutions of the FDEs are presented using the homotopy analysis method (HAM). The HAM provides a convenient way of controlling the convergence region and rate of the series solution. It is confirmed that the HAM series solutions contain the Adomian decomposition method (ADM) solution as special cases.      

Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation

In this paper, the homotopy analysis method (HAM) proposed by Liao in 1992 and the homotopy perturbation method (HPM) proposed by He in 1998 are compared through an evolution equation used as the second example in a recent paper by Ganji et al. [D.D. Ganji, H. Tari, M.B. Jooybari, Variational iteration method and homotopy perturbation method for nonlinear evolution equations. Comput. Math. Appl. 54 (2007) 1018–1027]. It is found that the HPM is a special case of the HAM when =-1. However, the HPM solution is divergent for all x and t except t=0. It is also found that the solution given by the variational iteration method (VIM) is divergent too. On the other hand, using the HAM, one obtains convergent series solutions which agree well with the exact solution. This example illustrates that it is very important to investigate the convergence of approximation series. Otherwise, one might get useless results.     

The homotopy analysis method for solving higher dimensional initial boundary value problems of variable coefficients

In this article, higher dimensional initial boundary value problems of variable coefficients are solved by means of an analytic technique, namely the Homotopy analysis method (HAM). Comparisons are made between the Adomian decomposition method (ADM), the exact solution and the homotopy analysis method. The results reveal that the proposed method is very effective and simple. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Application of semi‐analytic methods for the Fitzhugh–Nagumo equation,which models the transmission of nerve impulses

In this work, the homotopy perturbation method (HPM), the variational iteration method (VIM) and the Adomian decomposition method (ADM) are applied to solve the Fitzhugh–Nagumo equation. Numerical solutions obtained by these methods when compared with the exact solutions reveal that the obtained solutions produce high accurate results. The results show that the HPM, the VIM and the ADM are of high accuracy and are efficient for solving the Fitzhugh–Nagumo equation. Also the results demonstrate that the introduced methods are powerful tools for solving the nonlinear partial differential equations. Copyright © 2010 John Wiley & Sons, Ltd.     

The Optimal Homotopy Analysis Method for solving linear optimal control problems

In this paper, an Optimal Homotopy Analysis Method (Optimal HAM) is applied to solve the linear optimal control problems (OCPs), which have a quadratic performance index. This approach contains at most two convergence-control parameters which depend on the control system and is computationally rather efficient. A squared residual error for the system is defined, which can be used to find the unknown optimal convergence-control parameters by using Mathematica package BVPh (version 2.0). The results of comparisons among the proposed method, the homotopy perturbation method (HPM), the Adomian decomposition method (ADM), the differential transform method (DTM) and the homotopy analysis method (HAM) provide verification for the validity of the proposed approach. Moreover, numerical results are presented by several examples involving scalar and 2nd-order systems to clarify the efficiency and high accuracy of the proposed approach.     

Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method

In this work, the homotopy analysis method (HAM), one of the most effective method, is implemented for finding approximate solutions of the Burger and regularized long wave (RLW) equations. Comparisons are made between the results of the proposed method and homotopy perturbation method (HPM). It illustrates the validity and the great potential of the homotopy analysis method in solving nonlinear partial differential equations.     

Homotopy analysis method for generalized Benjamin–Bona–lMahony equation

An analytic technique, the homotopy analysis method (HAM), is applied to solve the generalized Benjamin–Bona–Mahony (BBM) equation. An explicit series solution is given, different from traditional analytic techniques, our approach is independent of knowing some parameters. This analytic method provides us with a new way to obtain series solutions of such problems. The homotopy analysis method contains the auxiliary parameter ħ, which provides us with a simple way to adjust and control the convergence region of series solution.     

Solitary wave solutions for a time-fraction generalized Hirota–Satsuma coupled KdV equation by an analytical technique

In this work, we implement a relatively analytical technique, the homotopy perturbation method (HPM), for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo derivatives. This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations which applied in engineering mathematics. The corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations. He’s homotopy perturbation method (HPM) which does not need small parameter is implemented for solving the differential equations. It is predicted that HPM can be found widely applicable in engineering.     

Solutions of Emden–Fowler equations by homotopy-perturbation method

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.     

An approximation of the analytic solution of some nonlinear heat transfer in fin and 3D diffusion equations using HAM

In this article, the approximate solution of nonlinear heat diffusion and heat transfer equation are developed via homotopy analysis method (HAM). This method is a strong and easy‐to‐use analytic tool for investigating nonlinear problems, which does not need small parameters. HAM contains the auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region of solution series. By suitable choice of the auxiliary parameter ?, we can obtain reasonable solutions for large modulus. In this study, we compare HAM results, with those of homotopy perturbation method and the exact solutions. The first differential equation to be solved is a straight fin with a temperature‐dependent thermal conductivity and the second one is the two‐ and three‐dimensional unsteady diffusion problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010