By Shijun Liao

In contrast to different analytic ideas, the Homotopy research process (HAM) is self reliant of small/large actual parameters. along with, it presents nice freedom to settle on equation sort and resolution expression of similar linear high-order approximation equations. The HAM presents an easy option to warrantly the convergence of answer sequence. Such strong point differentiates the HAM from all different analytic approximation tools. moreover, the HAM could be utilized to resolve a few not easy issues of excessive nonlinearity.

This publication, edited by way of 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 study, the authors of every bankruptcy are best specialists within the HAM and its purposes.

Readership: Graduate scholars and researchers in utilized arithmetic, physics, nonlinear mechanics, engineering and finance.

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Now, as n → ∞ then Sn → u(r ) and γ n−M → 0. So, u(r)−UM (r ,δ, ) ≤ 1 M −k0 +1 γ uk0 (r ,δ) . 18) leads to occurrence of horizontal plateaus in -curve, in which they give valid region for the convergence controller parameter , where UM (r ,δ, ) converges. 2), for that we bring the below theorem. 6. 58) such that |f (x, y)| ≤ N, for all (x, y) in R. 59) for all (x, y) and (x, z) in R. Then there exists one, and only one, solution ϕ of y (n) = f x, y, y , . . 61) ϕ (x0 ) = α1 , ϕ (x0 ) = α2 , . .

1. Preliminaries Many of the mathematical modeling of the physical phenomena in science and engineering often lead to nonlinear differential equations. There are a lot of methods, from the past up to now, to give numerically approximate solutions of nonlinear differential equations such as Euler method, RungeKutta method, multistep method, Taylor series method, Hybrid methods, family of finite difference methods [1, 2], family of finite element methods [3], meshless methods, differential quadrature, spectral methods [4–6] etc.

25) n+1 β−x From the differentiability of s(x ) and β−α , it follows that U (x ) is differentiable on any subinterval of (a, b). 1 to U (x ) on the interval [α, β]. 26) which implies n − n (β−ξβ ) (n+1) (β−ξβ ) f (ξβ ) + (n+1) n+1 s(α) =0, n! 27) then, since β = ξβ , n+1 s(α) = (β−α) f (n+1) (ξβ ). (n + 1)! 28) Now, from Eq. 22), we have n+1 s(α) = f (β)−Gn (β, α) = (β−α) f (n+1) (ξβ ). (n + 1)! 29) Since the parameter β has been chosen arbitrary then ∀x ∈ [α, b], α ∈ [a, b] : f (x )−Gn (x ,α) = (x −α)n+1 (n+1) f (ξx ), (n + 1)!

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