The homotopy perturbation method for nonlinear oscillators with discontinuities

Comparison of homotopy perturbation method and homotopy analysis method

A new approximate analytical technique to address for non-linear problems, namely Optimal Homotopy Asymptotic Method (OHAM) is proposed and has been applied to thin film flow of a fourth grade fluid down a vertical cylinder. This approach however, does not depend upon any small/large parameters in comparison to other perturbation method. This method provides a convenient way to control the convergence of approximation series and allows adjustment of convergence regions where necessary. The series solution has been developed and the recurrence relations are given explicitly. The results reveal that the proposed method is very accurate, effective and easy to use.

https://www.degruyter.com/downloadpdf/j/phys.2008.6.issue-3/s11534-008-0061-x/s11534-008-0061-x.xml

Optimal homotopy asymptotic method with application to thin film flow

Analytic solution for axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet

In this paper, Adomian’s decomposition method is proposed to solve the well-known Blasius equation. Comparison with homotopy perturbation method and Howarth’s numerical solution reveals that the Adomian’s decomposition method is of high accuracy.

A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method

In this paper, the basic ideas of a new analytic technique, namely the Homotopy Analysis Method (HAM), are described. Different from perturbation methods, the validity of the HAM is independent on whether or not there exist small parameters in considered nonlinear equations. Therefore, it provides us with a powerful analytic tool for strongly nonlinear problems. A typical nonlinear problem is used as an example to verify the validity and the great potential of the HAM.

Homotopy analysis method: a new analytic method for nonlinear problems

A generalized Taylor series of a complex function was derived and some related theorems about its convergence region were given. The generalized Taylor theorem can he applied to greatly enlarge convergence regions of approximation series given by other traditional techniques. The rigorous proof of the generalized Taylor theorem also provides us with a rational base of the validity of a new kind of powerful analytic technique for nonlinear problems, namely the homotopy analysis method.

廖世俊

Papers

Homotopy analysis method: a new analytic method for nonlinear problems

On a generalized taylor theorem: a rational proof of the validity of the homotopy analysis method