Comparison between homotopy analysis method and optimal homotopy asymptotic method for nth‐order integro‐differential equation

This paper presents general framework for solving the nth‐order integro‐differential equation using homotopy analysis method (HAM) and optimal homotopy asymptotic method (OHAM). OHAM is parameter free and can provide better accuracy over the HAM at the same order of approximation. Furthermore, in OHAM the convergence region can be easily adjusted and controlled. Comparison, via two examples, between our solution using HAM and OHAM and the exact solution shows that the HAM and the OHAM are effective and accurate in solving the nth‐order integro‐differential equation. Copyright © 2011 John Wiley & Sons, Ltd.     

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     

A new spectral-homotopy analysis method for solving a nonlinear second order BVP

A modification of the homotopy analysis method (HAM) for solving nonlinear second-order boundary value problems (BVPs) is proposed. The implementation of the new approach is demonstrated by solving the Darcy–Brinkman–Forchheimer equation for steady fully developed fluid flow in a horizontal channel filled with a porous medium. The model equation is solved concurrently using the standard HAM approach and numerically using a shooting method based on the fourth order Runge–Kutta scheme. The results demonstrate that the new spectral homotopy analysis method is more efficient and converges faster than the standard homotopy analysis method.     

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.     

Explicit series solutions of some linear and nonlinear Schrodinger equations via the homotopy analysis method

In this paper, by means of the homotopy analysis method (HAM), the solutions of some Schrodinger equations are exactly obtained in the form of convergent Taylor series. The HAM contains the auxiliary parameter ?, that provides a convenient way of controlling the convergent region of series solutions. This analytical method is employed to solve linear and nonlinear examples to obtain the exact solutions. HAM is a powerful and easy-to-use analytic tool for nonlinear problems.     

Solving nonlinear fractional partial differential equations using the homotopy analysis method

In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional KdV, K(2,2), Burgers, BBM‐Burgers, cubic Boussinesq, coupled KdV, and Boussinesq‐like B(m,n) equations with initial conditions, which are introduced by replacing some integer‐order time derivatives by fractional derivatives. The homotopy analysis method for partial differential equations of integer‐order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions of the studied models are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Analytical approximations for a population growth model with fractional order

In this paper, we apply the homotopy analysis method (HAM) to solve the fractional Volterra’s model for population growth of a species in a closed system. This technique is extended to give solutions for nonlinear fractional integro–differential equations. The whole HAM solution procedure for nonlinear fractional differential equations is established. Further, the accurate analytical approximations are obtained for the first time, which are valid and convergent for all time t. This indicates the validity and great potential of the homotopy analysis method for solving nonlinear fractional integro–differential equations.     

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.     

A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations

In this paper, based on the homotopy analysis method (HAM), a powerful algorithm is developed for the solution of nonlinear ordinary differential equations of fractional order. The proposed algorithm presents the procedure of constructing the set of base functions and gives the high-order deformation equation in a simple form. 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 ?$?$. The analysis is accompanied by numerical examples. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.     

New explicit approximate solution of MHD viscoelastic boundary layer flow over stretching sheet

In this paper, the study the momentum and heat transfer characteristics in an incompressible electrically conducting non‐Newtonian boundary layer flow of a viscoelastic fluid over a stretching sheet. The partial differential equations governing the flow and heat transfer characteristics are converted into highly nonlinear coupled ordinary differential equations by similarity transformations. The resultant coupled highly nonlinear ordinary differential equations are solved by means of, homotopy analysis method (HAM) for constructing an approximate solution of heat transfer in magnetohydrodynamic (MHD) viscoelastic boundary layer flow over a stretching sheet with non‐uniform heat source. The proposed method is a strong and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The HAM solutions contain an auxiry parameter, which provides a convenient way of controlling the convergence region of series solutions. The results obtained here reveal that the proposed method is very effective and simple for solving nonlinear evolution equations. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in physics. Copyright © 2012 John Wiley & Sons, Ltd.     

An analytical approach for solving nonlinear boundary value problems in finite domains

Based on the homotopy analysis method (HAM), a general analytical approach for obtaining approximate series solutions to nonlinear two-point boundary value problems in finite domains is proposed. To demonstrate its effectiveness, this approach is applied to solve three nonlinear problems, and the analytical solutions obtained are more accurate than the numerical solutions obtained via the shooting method and the sinc-Galerkin method.     

A one-step optimal homotopy analysis method for nonlinear differential equations

In this paper, a one-step optimal approach is proposed to improve the computational efficiency of the homotopy analysis method (HAM) for nonlinear problems. A generalized homotopy equation is first expressed by means of a unknown embedding function in Taylor series, whose coefficient is then determined one by one by minimizing the square residual error of the governing equation. Since at each order of approximation, only one algebraic equation with one unknown variable is solved, the computational efficiency is significantly improved, especially for high-order approximations. Some examples are used to illustrate the validity of this one-step optimal approach, which indicate that convergent series solution can be obtained by the optimal homotopy analysis method with much less CPU time. Using this one-step optimal approach, the homotopy analysis method might be applied to solve rather complicated differential equations with strong nonlinearity.     

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.     

Analytical solution for large deflections of a cantilever beam under nonconservative load based on homotopy analysis method

In this article, large deflection and rotation of a nonlinear beam subjected to a coplanar follower static loading is studied. It is assumed that the angle of inclination of the force with respect to the deformed axis of the beam remains unchanged during deformation. The governing equation of this problem is solved analytically for the first time using a new kind of analytic technique for nonlinear problems, namely, the homotopy analysis method (HAM). The present solution can be used in wide range of load and length for beams under large deformations. The results obtained from HAM are compared with those results obtained by fourth order Range Kutta method. Finally, the load‐displacement characteristics of a uniform cantilever under a follower force normal to the deformed beam axis are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27:541–553, 2011     

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.     

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.     

Solution of Delay Differential Equation by Means of Homotopy Analysis Method

The algorithm of approximate analytical solution for delay differential equations (DDE) is obtained via homotopy analysis method (HAM) and modified homotopy analysis method (MHAM). Various examples of linear, nonlinear and system of initial value problems of DDE are solved and the results obtained show that these algorithms are accurate and efficient for the DDE. The convergence of this algorithm is also proved.     

Effects of thermal radiation on the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface

In the present investigation we have analyzed the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface. The effects of thermal radiation are carried out for two cases of heat transfer analysis known as (1) Prescribed exponential order surface temperature (PEST) and (2) Prescribed exponential order heat flux (PEHF). The highly nonlinear coupled partial differential equations of Jeffrey fluid flow along with the energy equation are simplified by using similarity transformation techniques based on boundary layer assumptions. The reduced similarity equations are then solved analytically by the homotopy analysis method (HAM). The convergence of the HAM series solution is obtained by plotting (^h/_2p)\hbar-curves for velocity and temperature. The effects of physical parameters on the velocity and temperature profiles are examined by plotting graphs.     

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.