Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems

In this paper, Homotopy Analysis Method (HAM) is applied to numerically approximate the eigenvalues of the fractional Sturm-Liouville problems. The eigenvalues are not unique. These multiple solutions, i.e., eigenvalues, can be calculated by starting the HAM algorithm with one and the same initial guess and linear operator L\mathcal{L}. It can be seen in this paper that the auxiliary parameter (^h/_2p),\hbar, which controls the convergence of the HAM approximate series solutions, has another important application. This important application is predicting and calculating multiple solutions.     

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.     

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.     

Approximate traveling wave solution for shallow water wave equation

Reconstruction of Variational Iteration Method (RVIM) is used for computing the coupled Whitham-Broer-Kaup shallow water. Then RVIM solution is verified against exact one and is compared with powerful approximate solutions, the Homotopy Perturbation Method (HPM) and Homotopy Analysis Method (HAM). The existent error of the methods is computed and convergence of the RVIM solution has been presented. Results obtained expose effectiveness and capability of this method to solve the nonlinear systems in mechanics, analytically.     

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.     

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.     

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.     

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.     

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 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.     

Determination of optimal convergence-control parameter value in homotopy analysis method

In the framework of the Homotopy Analysis Method (HAM) the so-called convergence-control parameter $c_{0}$ (Liao (Int J Non-Linear Mech 32:815–822, 1997) originally used the symbol $\hbar $ to denote the auxiliary parameter. But, $\hbar $ is well-known as Planck’s constant in quantum mechanics. To avoid misunderstanding, Liao (Commun Nonlinear Sci Numer Simulat 15:2003–2016, 2010) suggest to use the symbol $c_0$ to denote the basic convergence-control parameter.) has a key role in convergence of obtained series solution of solving non-linear equations. In this paper a modified approach in the determining of the convergence-control parameter value $c_{0}$ is proposed. The purpose of this paper is to find a proper convergence-control parameter $c_0$ to get a convergent series solution, or a faster convergent one. This modified approach minimizes the norm of a discrete residual function, systematically, in which seeks to find an optimal value of the convergence-control parameter $c_0$ at each order of HAM approximation, instead of the so-called $c_0$ -curve process. The proved theorems and numerical results demonstrate the validity, efficiency, and performance of the proposed approach.     

Homotopy analysis method with a non-homogeneous term in the auxiliary linear operator

We demonstrate the efficiency of a modification of the normal homotopy analysis method (HAM) proposed by Liao [2] by including a non-homogeneous term in the auxiliary linear operator (this can be considered as a special case of “further generalization” of HAM given by Liao in [2]). We then apply the modified method to a few examples. It is observed that including a non-homogeneous term gives faster convergence in comparison to normal HAM. We also prove a convergence theorem, which shows that our technique yields the convergent solution.     

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.     

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.     

Superlinear Convergence of a General Algorithm for the Generalized Foley–Sammon Discriminant Analysis

Linear Discriminant Analysis (LDA) is one of the most efficient statistical approaches for feature extraction and dimension reduction. The generalized Foley–Sammon transform and the trace ratio model are very important in LDA and have received increasing interest. An efficient iterative method has been proposed for the resulting trace ratio optimization problem, which, under a mild assumption, is proved to enjoy both the local quadratic convergence and the global convergence to the global optimal solution (Zhang, L.-H., Liao, L.-Z., Ng, M.K.: SIAM J. Matrix Anal. Appl. 31:1584, 2010). The present paper further investigates the convergence behavior of this iterative method under no assumption. In particular, we prove that the iteration converges superlinearly when the mild assumption is removed. All possible limit points are characterized as a special subset of the global optimal solutions. An illustrative numerical example is also presented.     

Nano boundary layers over stretching surfaces

In this paper, we present similarity solutions for the nano boundary layer flows with Navier boundary condition. We consider viscous flows over a two-dimensional stretching surface and an axisymmetric stretching surface. The resulting nonlinear ordinary differential equations are solved analytically by the Homotopy Analysis Method. Numerical solutions are obtained by using a boundary value problem solver, and are shown to agree well with the analytical solutions. The effects of the slip parameter K and the suction parameter s on the fluid velocity and on the tangential stress are investigated and discussed. As expected, we find that for such fluid flows at nano scales, the shear stress at the wall decreases (in an absolute sense) with an increase in the slip parameter K.     

A method of directly defining the inverse mapping for solutions of coupled systems of nonlinear differential equations

Recently, Liao introduced a new method for finding analytical solutions to nonlinear differential equations. In this paper, we extend this idea to nonlinear systems. We study the system of nonlinear differential equations that governs nonlinear convective heat transfer at a porous flat plate and find functions that approximate the solutions by extending Liao’s Method of Directly Defining the Inverse Mapping (MDDiM).     

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.     

Solution of the Thomas–Fermi equation with a convergent approach

The explicit analytic solution of the Thomas–Fermi equation thorough a new kind of analytic technique, namely the homotopy analysis method, was employed by Liao [1]. However, the base functions and the auxiliary linear differential operator chosen were such that the convergence to the exact solution was fairly slow. New base functions and auxiliary linear operator to form a better homotopy are the main concern of the present paper. Optimum convergence control parameter concept is used together with a mathematical proof of the convergence.