The variational iteration method for Whitham-Broer-Kaup system with local fractional derivatives

Abstract

The Whitham-Broer-Kaup equations are modified using local fractional derivatives, and the equations are then solved by the variational iteration method. Yang-Laplace transform method is adopted to make the solution process simpler.

Fractional Soliton and Semirational Solutions of a Fractional Two-Component Generalized Hirota Equation

“…There are many effective methods for constructing exact solutions of nonlinear evolution systems, for example, the inverse scattering transform [22], Hirota bilinear method [23], homogeneous balance method [24], and exp-function method [25]. Because this paper focuses on solving fractional integrable systems, we also introduce some methods such as Adomian decomposition method [26], variational iteration method [27], and homotopy perturbation method [28] used in recent literature to handle fractional-order nonlinear equations. Traditional Darboux transformation (DT) [29] can construct the well-known soliton solutions, while the generalized DT (GDT) [30] based on DT [29] was originally designed to obtain rational solutions and has recently been applied to the construction of rogue wave solutions [31] and semirational solutions [32].…”

Modified Homotopy Perturbation Method and Approximate Solutions to a Class of Local Fractional Integrodifferential Equations

“…The local fractional calculus [1] developed in recent years provides a powerful mathematical tool to handling with such type of nondifferentiable functions. Fractional calculus, which is widely believed to have originated more than 300 years ago, has attracted much attention [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. It is of theoretical and practical value to solve fractional differential equations (DEs) directly connecting with fractional dynamical processes in a great many fields.…”

“…With the development of fractional calculus, many numerical and analytical methods for fractional DEs have been developed, such as integral transform method [1], series expansion method [3], Adomian decomposition method [4], Fan subequation method [5], variational iteration method (VIM) [6], variable separation method [7], finite difference method [8], homotopy perturbation method (HPM) [9], combined the HPM with Laplace transform [10], exp-function method [11], and Hirota bilinear method [12]. The HPM proposed by He [18] couples the homotopy method and the perturbation technique, which needs no the small parameters embedded in differential equations.…”