On the homotopy analysis method for nonlinear problems

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Abstract

A powerful, easy-to-use analytic tool for nonlinear problems in general, namely the homotopy analysis method, is further improved and systematically described through a typical nonlinear problem, i.e. the algebraically decaying viscous boundary layer flow due to a moving sheet. Two rules, the rule of solution expression and the rule of coefficient ergodicity, are proposed, which play important roles in the frame of the homotopy analysis method and simplify its applications in science and engineering. An explicit analytic solution is given for the first time, with recursive formulas for coefficients. This analytic solution agrees well with numerical results and can be regarded as a definition of the solution of the considered nonlinear problem.

Introduction

In most cases it is difficult to solve nonlinear problems, especially analytically. Perturbation techniques [1], [2] are currently the main stream. Perturbation techniques are based on the existence of small/large parameters, the so-called perturbation quantity. Unfortunately, many nonlinear problems in science and engineering do not contain such kind of perturbation quantities at all. Some nonperturbative techniques, such as the artificial small parameter method [3], the δ-expansion method [4] and the Adomian’s decomposition method [5], have been developed. Different from perturbation techniques, these nonperturbative methods are independent upon small parameters. However, both of the perturbation techniques and the nonperturbative methods themselves can not provide us with a simple way to adjust or control the convergence region and rate of given approximate series.

Liao [6] proposed a powerful analytic method for nonlinear problems, namely the homotopy analysis method [7], [8], [9], [10], [11], [12], [13]. Different from all reported perturbation and nonperturbative techniques mentioned above, the homotopy analysis method itself provides us with a convenient way to control and adjust the convergence region and rate of approximation series, when necessary. Briefly speaking, the homotopy analysis method has the following advantages

  • it is valid even if a given nonlinear problem does not contain any small/large parameters at all;

  • it itself can provide us with a convenient way to adjust and control the convergence region and rate of approximation series when necessary;

  • it can be employed to efficiently approximate a nonlinear problem by choosing different sets of base functions.


To systematically describe the basic ideas of the homotopy analysis method and to show its validity, let us consider a viscous boundary layer flow due to a moving sheet occupying the negative x-axis and moving continuously in the positive x-direction at a velocityus=u0x0|x|κ,0<κ<1,where (x,y) denotes the coordinate in Cartesian system. The boundary layer flow is governed byux+vy=0,uux+vuy2uy2,where u and v are the velocity components in the x- and y-directions, respectively. The corresponding boundary conditions areu=us,v=0aty=0,u→0asy→+∞.Under the similar transformationψ=F(ξ)2νus|x|,ξ=yus2ν|x|,where ψ is the stream function defined by u=∂ψ/∂y and v=−∂ψ/∂x, the Eqs. , becomeF′′′(ξ)+(κ−1)F(ξ)F′′(ξ)−2κ[F(ξ)]2=0andF(0)=0,F(0)=1,F(+∞)=0,where the prime denotes differentiation with respect to ξ. For details, please refer to Kuiken [14].

Kuiken [14] gave such an asymptotic expressionf∼(ξ−ξ0)αi=0Nc01−ici(ξ−ξ0)−i(1+α),whereα=1−κ1+κand the coefficients ci are given by recursive formulas and the coefficients c0,ξ0 are determined by an iterative numerical approach. Thus, rigorously speaking, Kuiken’s solution is semi-analytic and semi-numerical one. Besides, the above expression is valid only for ξξ0≫1, because it is singular at ξ=ξ0. To the best of our knowledge, no one has reported an explicit, purely analytic solution of , , valid in the whole region 0⩽ξ⩽+∞.

In this paper the homotopy analysis method is further improved and systematically described in a usual procedure through a typical example mentioned above. Two rules are described, which play important roles in the frame of the homotopy analysis method and simplify its applications in science and engineering. An explicit analytic solution of above nonlinear problem is given for the first time.

Section snippets

Homotopy analysis method

In this section the homotopy analysis method is further improved and systematically described to give an explicit analytic solution of the nonlinear problem mentioned above. A usual procedure of the homotopy analysis method is proposed for the first time.

Homotopy-Padé approach

As verified in our previous publications [7], [9], [10], [11], [12], [13], it is the auxiliary parameter ℏ which provides us with a simple way to adjust or control the convergence rate and region of approximations given by the homotopy analysis method. Alternatively, in many (but not all) cases the convergence rate and/or region of approximations given by the homotopy analysis method can be greatly enlarged by the so-called Homotopy-Padé approach proposed by Liao and Cheung [13]. To explain it,

Conclusions

In this paper a powerful, easy-to-use analytic technique for nonlinear problems in general, namely the homotopy analysis method, is further improved and systematically described through a typical nonlinear problem, i.e. the viscous boundary layer flow due to a moving sheet, governed by , . A usual procedure of the homotopy analysis method is proposed for the first time. Two rules, the rule of solution expression and the rule of coefficient ergodicity, are proposed, which play important roles in

Acknowledgements

Thanks to “National Science Fund for Distinguished Young Scholars” (Approval no. 50125923) of Natural Science Foundation of China for the financial support.

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