An approximation solution of a nonlinear equation with Riemann–Liouville's fractional derivatives by He's variational iteration method

Numerical Solutions for Quadratic Integro-Differential Equations of Fractional Orders

“…The another method is HPM which defines as a coupling of the traditional perturbation method and homotopy in topology. Many authors successfully apply these methods to find the solutions of functional equations which arise in scientific and engineering problems [1] [5] [6] [7] [8] [9]. The Adomian decomposition method presents solution of functional equations but perhaps we find some difficulties that will arise during the computation of Adomian polynomials, the VIM and HPM overcome it is difficulties [7].…”

An Expansion Iterative Technique for Handling Fractional Differential Equations Using Fractional Power Series Scheme

“…As a result, the solution of such equations can be approximated with a large variety of analytical numerical methods. Among these, homotopy perturbation method Odibat and Momani, 2008), variational iteration method (Odibat and Momani, 2006;Abbasbandy, 2007;Molliq et al, 2009), homotopy analysis method (Song and Zhang, 2007;Hashim et al, 2009;El-Ajou et al, 2010;Arqup and El-Ajou, 2013), transform methods (Arıkoğlu and Özkol, 2007;Oturanç et al, 2008;Ertürk and Momani, 2008;Momani et al, 2014) and other methods (Schneider, 1996;Arqub et al, 2012;Al-Smadi et al, 2013;Arqub et al, 2013;Arqub et al, 2015).…”

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

“…Since most of the nonlinear fractional partial differential equations cannot be solved exactly, thus approximate and numerical methods must be used. Author of [5] found an approximate solution of a nonlinear equation with Riemann-Liouville's fractional derivatives by He's variational iterational method. Several methods have been used to solve fractional partial differential equations, such as Adomian's decomposition method (ADM) [6,7], Fourier transform method [8], Laplace transform method [2,3,9], and so on.…”

“…Several methods have been used to solve fractional partial differential equations, such as Adomian's decomposition method (ADM) [6,7], Fourier transform method [8], Laplace transform method [2,3,9], and so on. Some 0932-0784 / 10 / 1100-0935 $ 06.00 c 2010 Verlag der Zeitschrift für Naturforschung, Tübingen · http://znaturforsch.com fundamental works on various aspects of the fractional calculus are given by Abbasbandy [5], Al-Khaled and Momani [10], Caputo [11], Debnath [12], Diethelm et al [13], Jafari and Seifi [14,15], Hayat et al [16], Khan and Hayat [17], Kilbas and Trujillo [18], Kiryakova [19], Oldham and Spanier [20], Ray and Bera [21], Shawagfeh [22], Song and Zhang [23,24], Xu and Jie [25], Momani and his co-authors [26 -28], etc. The interested reader can see [29 -36] for more application of the method.…”