Notes on the homotopy analysis method: Some definitions and theorems
نویسنده
چکیده
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 method more rigorously. Some lemmas and theorems about the homotopy-derivative and the deformation equation are proved. Besides, a few open questions are discussed, and a hypothesis is put forward for future studies. 2008 Elsevier B.V. All rights reserved.
منابع مشابه
Some notes on convergence of homotopy based methods for functional equations
Although homotopy-based methods, namely homotopy analysis method andhomotopy perturbation method, have largely been used to solve functionalequations, there are still serious questions on the convergence issue of thesemethods. Some authors have tried to prove convergence of these methods, butthe researchers in this article indicate that some of those discussions are faulty.Here, after criticizi...
متن کاملSome notes on convergence of homotopy based methods for functional equations
Although homotopy-based methods, namely homotopy analysis method andhomotopy perturbation method, have largely been used to solve functionalequations, there are still serious questions on the convergence issue of thesemethods. Some authors have tried to prove convergence of these methods, butthe researchers in this article indicate that some of those discussions are faulty.Here, after criticizi...
متن کاملSome notes concerning the convergence control parameter in homotopy analysis method
omotopy analysis method (HAM) is a promising method for handling func-tional equations. Recent publications proved the eectiveness of HAM in solvingwide variety of problems in dierent elds. HAM has a unique property whichmakes it superior to other analytic methods, this property is its ability to con-trol the convergence region of the solution series. In this work, we claried theadvantages and ...
متن کاملFixed point theorems for weakly contractive mappings on g-Metric spaces and a homotopy result
In this paper, we give some xed point theorems for '-weak contractivetype mappings on complete G-metric space, which was given by Zaed andSims [1]. Also a homotopy result is given.
متن کاملSolving infinite system of nonlinear integral equations by using F-generalized Meir-Keeler condensing operators, measure of noncompactness and modified homotopy perturbation.
In this article to prove existence of solution of infinite system of nonlinear integral equations, we consider the space of solution containing all convergence sequences with a finite limit, as with a suitable norm is a Banach space. By creating a generalization of Meir-Keeler condensing operators which is named as F-generalized Meir-Keeler condensing operators and measure of noncompactness, we...
متن کامل