By Shijun Liao
In contrast to different analytic ideas, the Homotopy research technique (HAM) is self sustaining of small/large actual parameters. along with, it presents nice freedom to settle on equation variety and resolution expression of comparable linear high-order approximation equations. The HAM presents an easy strategy to warrantly the convergence of answer sequence. Such forte differentiates the HAM from all different analytic approximation tools. furthermore, the HAM will be utilized to resolve a few hard issues of excessive nonlinearity.
This publication, edited through the pioneer and founding father of the HAM, describes the present advances of this robust analytic approximation approach for hugely nonlinear difficulties. Coming from assorted nations and fields of analysis, the authors of every bankruptcy are most sensible specialists within the HAM and its functions.
Readership: Graduate scholars and researchers in utilized arithmetic, physics, nonlinear mechanics, engineering and finance.
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Extra resources for Advances in the Homotopy Analysis Method
Can we employ the HAM to gain the above-mentioned unstable periodic solutions of Lorenz equation found by Viswanath ? More importantly, it would be very interesting if the HAM could be employed to find some new periodic solutions of Lorenz equation with physical parameters leading to chaos! This is mainly because Lorenz equation is one of the most famous ones in nonlinear dynamics and nonlinear science. 2. Periodic orbits of Newtonian three-body problem Let us consider one of the most famous problem in mechanics and applied mathematics: the Newtonian three-body problem, say, the motion of three celestial bodies under their mutual gravitational attraction.
Magyari, Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones, Z. angew. Math. Phys. 57, 777–792 (2006). J. Liao, On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves, Commun. Nonlinear Sci. Numer. Simulat. 16: 1274 – 1303 (2011). L. L. J. Liao and M. Stiassnie, On the steady-state fully resonant progressive waves in water of finite depth, J. Fluid Mech. 710: 379–418 (2012). H. H. Kim, Homotopy analysis method for option pricing under stochastic volatility, Appl.
Elipe and A. Riaguas, On the figure-8 periodic solutions in the three-body problem, Chaos, Solitons Fractals. 30: 513–520 (2006).  M. Nauenberg, Continuity and stability of families of figure eight orbits with finite angular momentum, Celest. Mech. 97: 1–15 (2007). ˇ  M. Suvakov and V. Dmitraˇsinovi´c, Three classes of newtonian three-body planar periodic orbits, Phys. Rev. Lett. 110: 114301 (2013). rs/3body/. G. Stokes, On the effect of the internal friction of fluids on the motion of pendulums, Trans.
Advances in the Homotopy Analysis Method by Shijun Liao