We describe, very briefly, 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. Definitions of some new concepts such as the homotopy-derivative, the convergence-control parameter and so on, are given to redescribe the metho...

Homotopy Analysis Transform Method (HATM) is applied to tackle timespace fractional Gas dynamics equation. The proposed HATM is an elegant coupling of homotopy analysis method (HAM) and Laplace transform. The method gives an analytical solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. The numerical solutions obtained...

The purpose of this article to provide an explicit example where continuation based on the homotopy method fails. The example is a one-parameter homotopy for periodic orbits between two well-known nonlinear systems, the normal form of the Hopf bifurcation and the van der Pol system. Our analysis shows that various types of obstructions can make approximation over the whole range of the homotopy...

in the present work, a hybrid of fourier transform and homotopy perturbation method is developed for solving the non-homogeneous partial differential equations with variable coefficients. the fourier transform is employed with combination of homotopy perturbation method (hpm), the so called fourier transform homotopy perturbation method (fthpm) to solve the partial differential equations. the c...

In this paper, we propose a new numerical algorithm, namely q-homotopy analysis Sumudu transform method (q-HASTM), to obtain the approximate solution for the nonlinear fractional dynamical model of interpersonal and romantic relationships. The suggested algorithm examines the dynamics of love affairs between couples. The q-HASTM is a creative combination of Sumudu transform technique, q-homotop...

We have used the homotopy analysis method (HAM) to obtain solution of Davey-Stewartson equations of fractional order. The fractional derivative is described in the Caputo sense. The results obtained by this method have been compared with the exact solutions. Stability and convergence of the proposed approach is investigated. The effects of fractional derivatives for the systems under considerat...