APPLICATION OF HOMOTOPY ANALYSIS METHOD FOR SOLVING NONLINEAR PROBLEMS (original) (raw)
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Homotopy analysis method: A new analytical technique for nonlinear problems
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In this paper, the basic ideas of a new kind of analytical technique, namely the Homotopy Analysis Method (HAM), are briefly described. Different from perturbation techniques, the HAM does not depend on whether or not there exist small parameters in nonlinear equations under consideration. Therefore, it provides us with a powerful tool to analyse strongly nonlinear problems. A simple but typical example is used to illustrate the validity and the great potential of the HAM. Moreover, a pure mathematical theorem, namely the General Taylor Theorem, is given in appendix, which provides us with some rational knowledge for the validity of this new analytical technique.
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In this paper, new powerful modification of homotopy analysis technique (NMHAM) was submitted to create an approximate solution of nonhomogeneous nonlinear ordinary and partial differential equations. The NMHAM is a combination of the new technique of homotopy analysis method(NHAM) [4] and the new technique of homotopy analysis method(nHAM) [7].Three illustrative examples are employed to illustrate the accuracy and computational proficiency of this approach. The outcomes uncover that the NMHAM is more accurate than the NHAM and nHAM.
HOMOTOPY ANALYSIS METHOD: A NEW ANALYTIC METHOD FOR NONLINEAR PROBLEMS
In this paper, the basic ideas of a new (malytic techniq,te, ntmwly the llomotopy Analysis:Method (HAM). are described. Diff'erent from perturl~ttioJt methods, the ralidit.r of the tlAM is huh'l)ende, t o, whether or not there exist sin,l/pammwters h~ considered notdinear equations. Therefore, it provides tls with a poweJ]'zd alttdytic tool Jbr strongly nonlhtear problems. A typical no,lhwar problem is used as an example to ver(])' the validity and the great potential of the HAM.
Journal of Applied Mathematics, 2013
A modifiedq-homotopy analysis method (mq-HAM) was proposed for solvingnth-order nonlinear differential equations. This method improves the convergence of the series solution in thenHAM which was proposed in (see Hassan and El-Tawil 2011, 2012). The proposed method provides an approximate solution by rewriting thenth-order nonlinear differential equation in the form ofnfirst-order differential equations. The solution of thesendifferential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.
The exact solutions of nonlinear problems by Homotopy Analysis Method (HAM
The present paper presents the comparison of analytical techniques. We establish the existence of the phenomena of the noise terms in the perturbation series solution and find the exact solution of the nonlinear problems. If the noise terms exist, the Homotopy Analysis method gives the same series solution as in Adomian Decomposition Method as well as homotopy Perturbation Method (Wahab et al, 2015) and we get the exact solution using the initial guess in Homotopy Analysis Method using the results obtained by Adomian Decomposition Method.
Homotopy Analysis Method for Solving Non-linear Various Problem of Partial Differential Equations
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In this paper, solve several important equations such as korteweg-devries (kdv) problem, Boussinesq equation of non-homogeneous problem and non-homogeneous system Hirota-Satsuma problem of partial differential equation by Homotopy analysis method (HAM). Studied comparison exact solution with numerical results , this method have shown that is very effective and convenient and gives numerical solutions in the form of convergent series with easily computable components for solving non-linear various problem of partial differential equation .
Numerically solving non-linear problems by the homotopy analysis method
Computational Mechanics, 1997
In this paper, the Homotopy Analysis Method (HAM) proposed by Liao (is greatly improved by introducing a nonzero variable " h. Based on the HAM, a new numerical approach for strongly non-linear problems is proposed and applied to solve, as an example, a non-linear heat transfer problem, i.e. microwave heating of an unit plate, so as to verify its validity and great potential. Our numerical experiments show that, by the proposed approach, iteration is not absolutely necessary for solving non-linear problems. This fact may deepen our understanding about numerical techniques for non-linear problems and widen our ®eld of vision. Moreover, the basic ideas proposed in this paper may afford us a great possibility to greatly improve our current numerical techniques.
Modified Homotopy Perturbation Technique for the Approximate Solution of Nonlinear Equations
Chinese Journal of Mathematics, 2014
We use a new modified homotopy perturbation method to suggest and analyze some new iterative methods for solving nonlinear equations. This new modification of the homotopy method is quite flexible. Various numerical examples are given to illustrate the efficiency and performance of the new methods. These new iterative methods may be viewed as an addition and generalization of the existing methods for solving nonlinear equations.