On the homotopy analysis method for nonlinear problems
TL;DR: 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.
About: This article is published in Applied Mathematics and Computation.The article was published on 2004-01-01. It has received 1589 citations till now. The article focuses on the topics: Homotopy analysis method & Homotopy.
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TL;DR: In this article, the basic ideas and current developments of the homotopy analysis method, an analytic approach to get convergent series solutions of strongly nonlinear problems, which recently attracts interests of more and more researchers, are described.
835 citations
TL;DR: In this paper, an optimal homotopy analysis approach is described by means of the nonlinear Blasius equation as an example, which can be used to get fast convergent series solutions of different types of equations with strong nonlinearity.
822 citations
Cites background from "On the homotopy analysis method for..."
...(5) can be further generalized, as shown by Liao [20,26,25]....
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TL;DR: In this paper, the homotopy analysis method (HAM) is compared with the numerical and HPM in the heat transfer file and the auxiliary parameter ℏ, which provides a simple way to adjust and control the convergence region of solution series.
643 citations
TL;DR: In this paper, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations (FPDE) with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives, and the results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.
Abstract: 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
554 citations
Cites methods from "On the homotopy analysis method for..."
...Hence we have [58–61] u(x, t) = u0(x, t)+ ∞ ∑ m=1 um(x, t), (3....
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...In this work, the homotopy analysis method (HAM) developed by Liao in [52–61] will be used to conduct an analytic study on the fractional KdV,K(2, 2), Burgers, BBM-Burgers, cubic Boussinesq, coupled KdV and Boussinesq-like B(m, n) equations....
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...In this article, we illustrate the validity of the HAM [57–61] for the nonlinear fractional partial differential equations....
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TL;DR: In this article, the authors examined the magnetohydrodynamic flow of non-Newtonian nanofluid in a pipe and derived explicit analytical expressions for the velocity field, the temperature distribution and nano concentration.
543 citations
References
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TL;DR: This website becomes a very available place to look for countless perturbation methods sources and sources about the books from countries in the world are provided.
Abstract: Following your need to always fulfil the inspiration to obtain everybody is now simple. Connecting to the internet is one of the short cuts to do. There are so many sources that offer and connect us to other world condition. As one of the products to see in internet, this website becomes a very available place to look for countless perturbation methods sources. Yeah, sources about the books from countries in the world are provided.
5,427 citations
"On the homotopy analysis method for..." refers background in this paper
...Perturbation techniques [1,2] are currently the main stream....
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"On the homotopy analysis method for..." refers methods in this paper
...the d-expansion method [4] and the Adomian s decomposition method [5], have been developed....
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2,395 citations
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TL;DR: In this article, the general problem of the stability of motion is considered and the authors propose a solution to the problem of stability in the stability-of-motion (SOM) problem.
Abstract: (1992) The general problem of the stability of motion International Journal of Control: Vol 55, No 3, pp 531-534
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TL;DR: In this article, limit process expansions applied to Ordinary Differential Equations (ODE) are applied to partial differential equations (PDE) in the context of Fluid Mechanics.
Abstract: 1 Introduction.- 2 Limit Process Expansions Applied to Ordinary Differential Equations.- 3 Multiple-Variable Expansion Procedures.- 4 Applications to Partial Differential Equations.- 5 Examples from Fluid Mechanics.- Author Index.
759 citations