Topic
Homotopy analysis method
About: Homotopy analysis method is a research topic. Over the lifetime, 5031 publications have been published within this topic receiving 129601 citations.
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TL;DR: In this paper, homotopy perturbation method (HPM), which does not need small parameters in the equations, is compared with the perturbations and numerical methods in the heat transfer field.
496 citations
TL;DR: In this article, a guided tour through the mathematics needed for a proper understanding of homotopy perturbation method as applied to various nonlinear problems is presented, and a new interpretation of the concept of constant expansion is given.
Abstract: The present work constitutes a guided tour through the mathematics needed for a proper understanding of homotopy perturbation method as applied to various nonlinear problems. It gives a new interpretation of the concept of constant expansion in the homotopy perturbation method.
483 citations
Journal Article•
473 citations
TL;DR: In this paper, the boundary-layer flows over a stretched impermeable wall are solved by means of an analytic technique, namely the homotopy analysis method, and two branches of solutions are found.
450 citations
TL;DR: Based on homotopy, which is a basic concept in topology, a general analytic method was proposed to obtain series solutions of nonlinear differential equations in this article, where the authors showed that the second-order nonlinear oscillation equation can be replaced by an infinite number of (2κ)thorder linear differential equations, where κ≥ 1 can be any a positive integer.
Abstract: Based on homotopy, which is a basic concept in topology, a general analytic method (namely the homotopy analysis method) is proposed to obtain series solutions of nonlinear differential equations. Different from perturbation techniques, this approach is independent of small/large physical parameters. Besides, different from all previous analytic methods, it provides us with a simple way to adjust and control the convergence of solution series. Especially, it provides us with great freedom to replace a nonlinear differential equation of order n into an infinite number of linear differential equations of order k, where the order k is even unnecessary to be equal to the order n. In this paper, a nonlinear oscillation problem is used as example to describe the basic ideas of the homotopy analysis method. We illustrate that the second-order nonlinear oscillation equation can be replaced by an infinite number of (2κ)th-order linear differential equations, where κ≥ 1 can be any a positive integer. Then, the homotopy analysis method is further applied to solve a high-dimensional nonlinear differential equation with strong nonlinearity, i.e., the Gelfand equation. We illustrate that the second-order two or three-dimensional nonlinear Gelfand equation can be replaced by an infinite number of the fourth or sixth-order linear differential equations, respectively. In this way, it might be greatly simplified to solve some nonlinear problems, as illustrated in this paper. All of our series solutions agree well with numerical results. This paper illustrates that we might have much larger freedom and flexibility to solve nonlinear problems than we thought traditionally. It may keep us an open mind when solving nonlinear problems, and might bring forward some new and interesting mathematical problems to study.
432 citations