Homotopy Analysis Method for Solving Non-linear Various Problem of Partial Differential Equations (original) (raw)
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Homotopy Analysis Method for Solving System of Non-Linear Partial Differential Equations
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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.
APPLICATION OF HOMOTOPY ANALYSIS METHOD FOR SOLVING NONLINEAR PROBLEMS
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In this paper, the homotopy analysis method (HAM) is applied to obtain series solutions to linear and nonlinear systems of first-and second-order partial differential equations (PDEs). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. It is shown in particular that the solutions obtained by the variational iteration method (VIM) are only special cases of the HAM solutions.
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.