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...near Oscillators 208sa b ma nd mb nc d c < ++ < ,s(7)sand x is approximated bysx ma nd mb nc = ++ ,s(8)swhere m and n are weighting factors.sThe proof of the inequality can be found insdetails in Ref.=-=[3]-=-, fascinating applications of thestechnology can be found in Refs.[4,5,6,7].sHe Chengtian (369?~447AD) is a famoussancient Chinese mathematics and astronomer, hesis an extremely important figure in de...

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...He’s Polynomials): This modified version of variational iteration method [33-39] is obtained by the elegant 481 x ⎛ 2 d y 1 3 n (x,s) − ⎞ y 2 2 n+ 1(x) = y 0(x) + λ(s) − x y 2 n ds ⎜ ds ⎟ 0 ⎝ ⎠ ∫ � . =-=(13)-=-World Appl. Sci. J., 4 (4): 479-486, 2008 Making the correction functional stationary, the Lagrange multiplier can be identified as λ(s) = (s-x); the following iterative scheme is obtained: y n x 2 d...

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...roximations that converge rapidly [6]. The variational iteration method (VIM) is a simple and yet powerful method for solving a wide class of nonlinear problems, first envisioned by He [11] (see also =-=[12, 13, 14, 15, 16]-=-). The VIM has successfully been applied to many situations. For example, He [12] solved the classical Blasius’ equation using VIM. He [13] used VIM to give approximate solutions for some well-known n...

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...1 , i = 1, 2, 3 n we have the following iterative schemes: x x � x ( k + 1) 1 ( k + 1) 2 ( k + 1) n i ,..., ( t) = x ( t) = x ( t) = x ( k ) 1 ( k ) 2 ( k ) n ( t) − ( t) − ( t) − 1 2 2 2 n n n 1 2 n =-=(6)-=- 1 denote the t ( k ) ( k) ( k ) ( k) � ( x′ 1 ( T ), f1( x1 ( T ), x2 ( T ),..., xn ( T )) − g1( T )) 0 t ( k ) ( k) ( k) ( k) � ( x′ 2 ( T ), f 2 ( x1 ( T ), x2 ( T ),..., xn ( T)) − g 2 ( T) ) 0 dT...

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... so properly chosen, the series (2.4) converges at p = 1, then we have u(τ) = u0(τ) + +∞∑ m=1 um(τ), (2.6) ****************************************************************************** Surveys in Mathematics and its Applications 5 (2010), 89 – 98 http://www.utgjiu.ro/math/sma HAM for KdV 91 which must be one of solutions of original nonlinear equation, as proved by[20]. As ~ = −1 and H(τ) = 1, Eq. (2.2) becomes (1− p)L[φ(τ ; p)− u0(τ)] + p N [φ(τ ; p)] = 0, (2.7) which is used mostly in the homotopy perturbation method[13], where as the solution obtained directly, without using Taylor series [14, 15]. According to the definition (2.5), the governing equation can be deduced from the zero-order deformation equation(2.2). Define the vector ~un = {u0(τ), u1(τ), . . . , un(τ)}. Differentiating equation (2.2) m times with respect to the embedding parameter p and then setting p = 0 and finally dividing them by m!, we have the so-called mth-order deformation equation L[um(τ)− χmum−1(τ)] = ~H(τ)<m(~um−1), (2.8) where <m(~um−1) = 1 (m− 1)! ∂m−1N [φ(τ ; p)] ∂pm−1 |p=0. (2.9) and χm = 0, m 6 1, 1, m > 1. (2.10) It should be emphasized that um(τ) for m > 1 is governed by the linear equation (2.8) ...

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...roblems have no small parameter at all. Many new methods, such as the variational method [14-16], variational iterations method [17-22], various modified Lindstedt-Poincare methods [23-26] and others =-=[27, 28]-=- are proposed to eliminate the shortcoming arising in the small parameter assumption. A review of recently developed nonlinear analysis methods can be found in [29]. Recently, the applications of homo...