The comparison between Homotopy Analysis Method and Optimal Homotopy Asymptotic Method for nonlinear age-structured population models

This paper presents comparison between Homotopy Analysis Method (HAM) and Optimal Homotopy Asymptotic Method (OHAM) for the solution of nonlinear age-structured population models. Three examples have been presented to illustrate and compare these methods. In OHAM the convergence region can be easily adjusted and controlled. Comparison between our solution and the exact solution shows that the both methods are effective and accurate in solving nonlinear age-structured population models with HAM being the more accurate for the same number of terms. It was also found that OHAM require more CPU time.     

Homotopy perturbation method for (2+1) -dimensional coupled Burgers system

In this paper, a numerical solution of the (2+1)-dimensional coupled Burgers system is studied by using the Homotopy Perturbation Method (HPM). For this purpose, the available analytical solutions obtained by tanh method will be compared to show the validity and accuracy of the proposed numerical algorithm. The results approve the convergence and accuracy of the Homotopy Perturbation Method for numerically analyzed (2+1) coupled Burgers system.     

Jacobi elliptic function solutions of the (1 + 1)‐dimensional dispersive long wave equation by Homotopy Perturbation Method

In this article, we try to obtain approximate Jacobi elliptic function solutions of the (1 + 1)‐dimensional long wave equation using Homotopy Perturbation Method. This method deforms a difficult problem into a simple problem which can be easily solved. In comparison with HPM, numerical methods leads to inaccurate results when the equation intensively depends on time, while He's method overcome the above shortcomings completely and can therefore be widely applicable in engineering. As a result, we obtain the approximate solution of the (1 + 1)‐dimensional long wave equation with initial conditions. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

An Analytical Study of Boundary Layer Flows on a Continuous Stretching Surface

In this paper the momentum and heat transfer characteristics for a self-similarity boundary layer on exponentially stretching surface modeled by a system of nonlinear differential equations is studied. The system is solved using the Homotopy Analysis Method (HAM), which yields an analytic solution in the form of a rapidly convergent infinite series with easily computable terms. 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. By suitable choice of the auxiliary parameter ℏ, reasonable solutions for large modulus can be obtained.     

Three analytical methods applied to Jeffery–Hamel flow

Three new analytical approximate techniques for addressing nonlinear problems are applied to Jeffery–Hamel flow. Homotopy Analysis Method (HAM), Homotopy Perturbation Method (HPM) and Differential Transformation Method (DTM) are proposed and used in this research. These methods are very useful and applicable for solving nonlinear problems. Then, the results are compared with numerical results and the validity of these methods is shown. Comparison between obtained results showed that HAM is more acceptable and accurate than two other methods. Ultimately, the effects of Reynolds number and divergent and convergent model of the channel on features of the flow are discussed.     

Homotopy perturbation method for coupled Schrödinger–KdV equation

In this paper, numerical analysis of the coupled Schrödinger–KdV equation is studied by using the Homotopy Perturbation Method (HPM). The available analytical solutions of the coupled Schrödinger–KdV equation obtained by multiple traveling wave method are compared with HPM to examine the accuracy of the method. The numerical results validate the convergence and accuracy of the Homotopy Perturbation Method for the analyzed coupled Schrödinger–KdV equation.     

An Approximation of the Analytical Solution of the Linear and Nonlinear Integro-Differential Equations by Homotopy Perturbation Method

This paper aims to introduce an analytic technique, namely the Homotopy perturbation method (HPM) for the solution of integro-differential equations. From the computational viewpoint, the comparison shows that the homotopy perturbation method is efficient and easy to use.     

On the selection of auxiliary functions, operators, and convergence control parameters in the application of the Homotopy Analysis Method to nonlinear differential equations: A general approach

The Homotopy Analysis Method of Liao [Liao SJ. Beyond perturbation: introduction to the Homotopy Analysis Method. Boca Raton: Chapman & Hall/CRC Press; 2003] has proven useful in obtaining analytical solutions to various nonlinear differential equations. In this method, one has great freedom to select auxiliary functions, operators, and parameters in order to ensure the convergence of the approximate solutions and to increase both the rate and region of convergence. We discuss in this paper 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. Further, we discuss various convergence requirements on solutions.     

Analytical substrate flux approximation for the Monod boundary value problem

We present an analytical approximation for the diffusive flux of a substrate into a reactive layer, in which the substrate is degraded according to Monod kinetics. This problem is described by a nonlinear two-point boundary value problem. The approximation is derived based on a Homotopy Analysis Method idea and verified computationally, by comparison against a numerical solution of the problem. The analytical approximation is easy to evaluate and depends only on model parameters.     

Homotopy Analysis Method for Solving(2+1)-dimensional Navier-Stokes Equations with Perturbation Terms

In this paper Homotopy Analysis Method (HAM) is implemented for obtaining approximate solutions of (2+1)-dimensional Navier-Stokes equations with perturbation terms. The initial approximations are obtained using linear systems of the Navier-Stokes equations;by the iterations formula of HAM,the first approxima-tion solutions and the second approximation solutions are successively obtained and Homotopy Perturbation Method(HPM)is also used to solve these equations;finally, approximate solutions by HAM of (2+1)-dimensional Navier-Stokes equations with-out perturbation terms and with perturbation terms are compared. Because of the freedom of choice the auxiliary parameter of HAM,the results demonstrate that the rapid convergence and the high accuracy of the HAM in solving Navier-Stokes equa-tions;due to the effects of perturbation terms,the 3rd-order approximation solutions by HAM and HPM have great fluctuation.     

Solutions of the SIR models of epidemics using HAM

In this paper, we investigate the accuracy of the Homotopy Analysis Method (HAM) for solving the problem of the spread of a non-fatal disease in a population. The advantage of this method is that it provides a direct scheme for solving the problem, i.e., without the need for linearization, perturbation, massive computation and any transformation. Mathematical modeling of the problem leads to a system of nonlinear ODEs. MATLAB 7 is used to carry out the computations. Graphical results are presented and discussed quantitatively to illustrate the solution.     

An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate

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. This approach does not depend upon any small/large parameters. This method provides us with a convenient way to control the convergence of approximation series and adjust convergence regions when necessary. The series solution is developed and the recurrence relations are given explicitly. The results reveal that the proposed method is effective and easy to use.     

Homotopy Analysis Method for the heat transfer of a non-Newtonian fluid flow in an axisymmetric channel with a porous wall

In this article, a powerful analytical method, called the Homotopy Analysis Method (HAM) is introduced to obtain the exact solutions of heat transfer equation of a non-Newtonian fluid flow in an axisymmetric channel with a porous wall for turbine cooling applications. The HAM is employed to obtain the expressions for velocity and temperature fields. Tables are presented for various parameters on the velocity and temperature fields. These results are compared with the solutions which are obtained by Numerical Methods (NM). Also the convergence of the obtained HAM solution is discussed explicitly. These comparisons show that this analytical method is strongly powerful to solve nonlinear problems arising in heat transfer.     

On the coupling of the homotopy perturbation method and Laplace transformation

In this paper, a Laplace homotopy perturbation method is employed for solving one-dimensional non-homogeneous partial differential equations with a variable coefficient. This method is a combination of the Laplace transform and the Homotopy Perturbation Method (LHPM). LHPM presents an accurate methodology to solve non-homogeneous partial differential equations with a variable coefficient. The aim of using the Laplace transform is to overcome the deficiency that is mainly caused by unsatisfied conditions in other semi-analytical methods such as HPM, VIM, and ADM. The approximate solutions obtained by means of LHPM in a wide range of the problem’s domain were compared with those results obtained from the actual solutions, the Homotopy Perturbation Method (HPM) and the finite element method. The comparison shows a precise agreement between the results, and introduces this new method as an applicable one which it needs fewer computations and is much easier and more convenient than others, so it can be widely used in engineering too.     

Some solutions of the linear and nonlinear Klein-Gordon equations using the optimal homotopy asymptotic method

We investigate the effectiveness of the Optimal Homotopy Asymptotic Method (OHAM) in solving time dependent partial differential equations. To this effect we consider the homogeneous, non-homogeneous, linear and nonlinear Klein-Gordon equations with boundary conditions. The results reveal that the method is explicit, effective, and easy to use.     

Solving the one-loop soliton solution of the Vakhnenko equation by means of the Homotopy analysis method

A powerful, easy-to-use analytic technique for nonlinear problems, namely the Homotopy analysis method, is applied to solve the Vakhnenko equation, a nonlinear equation with loop soliton solutions governing the propagation of high-frequency waves in a relaxing medium. By means of the transformation of independent variables, an analysis one-loop soliton solution expressed by a series of exponential functions is obtained, which agrees well with the exact solution. This indicates the validity and great potential of the Homotopy analysis method in solving complicated solitary wave problems.     

An easy trick to a periodic solution of relativistic harmonic oscillator

In this paper, the relativistic harmonic oscillator equation which is a nonlinear ordinary differential equation is investigated by Homotopy perturbation method. Selection of a linear operator, which is a part of the main operator, is one of the main steps in HPM. If the aim is to obtain a periodic solution, this choice does not work here. To overcome this lack, a linear operator is imposed, and Fourier series of sines will be used in solving the linear equations arise in the HPM. Comparison of the results, with those of resulted by Differential Transformation and Harmonic Balance Method, shows an excellent agreement.     

He's homotopy perturbation method for solving Korteweg‐de Vries Burgers equation with initial condition

In this article, we present the Homotopy Perturbation Method (Shortly HPM) for obtaining the numerical solutions of the Korteweg‐de Vries Burgers (KdVB) equation. The series solutions are developed and the reccurance relations are given explicity. The initial approximation can be freely chosen with possibly unknown constants which can be determined by imposing the boundary and initial conditions. The results reveal that HPM is very simple and effective. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Analytical solutions of a coupled nonlinear system arising in a flow between stretching disks

In this paper, we consider the axi-symmetric flow between two infinite stretching disks. By using a similarity transformation, we reduce the governing Navier-Stokes equations to a system of nonlinear ordinary differential equations. We first obtain analytical solutions via a four-term perturbation method for small and large values of the Reynolds number R. Also, we apply the Homotopy Analysis Method (which may be used for all values of R) to obtain analytical solutions. These solutions converge over a larger range of values of the Reynolds number than the perturbation solutions. Our results agree well with the numerical results of Fang and Zhang [22]. Furthermore, we obtain the analytical solutions valid for moderate values of R by use of Homotopy Analysis.     

The study of Heat and Mass Transfer in a Visco elastic fluid due to a continuous stretching surface using Homotopy Analysis Method

In this paper, an approximate analytical solution is derived for the flow velocity and temperature due to the laminar, two-dimensional flow of non-Newtonian incompressible visco elastic fluid due to a continuous stretching surface. The surface is stretched with a velocity proportional to the distance $x$ along the surface. The surface is assumed to have either power-law heat flux or power-law temperature distribution. The presence of source/sink and the effect of uniform suction and injection on the flow are considered for analysis. An approximate analytical solution has been obtained using Homotopy Analysis Method(HAM) for various values of visco elastic parameter, suction and injection rates. Optimal values of the convergence control parameters are computed for the flow variables. It was found that the computational time required for averaged residual error calculation is very very small compared to the computation time of exact squared residual errors. The effect of mass transfer parameter, visco elastic parameter, source/sink parameter and the power law index on flow variables such as velocity, temperature profiles, shear stress, heat and mass transfer rates are discussed.