New Iteration Methods for Time-Fractional Modified Nonlinear Kawahara Equation

Abstract

We put side by side the methodology of two comparatively new analytical techniques to get to the bottom of the system of nonlinear fractional modified Kawahara equation. The technique is described and exemplified with a numerical example. The dependability of both methods and the lessening in computations give these methods a wider applicability. In addition, the computations implicated are very simple and undemanding.

Enhanced Multistage Homotopy Perturbation Method: Approximate Solutions of Nonlinear Dynamic Systems

“…In this case, the homotopic representation of (12) is similar to that given by (9). By following the previous procedure and by regrouping the terms with the same order, we obtain that…”

“…Notice that, in this case, the homotopic representation of (8) is given by (9), together with the linear operator ( ) = ( / ) and the nonlinear operator ( ) = 2 − . Utilizing the third order expansion, we can write the set of first order linear equations %% Solution Definitions function dydt=numeric(t,y) % numerical solution dydt(1,1)=y(1)-exp(t).…”

“…The HPM decomposes a complex problem into a series of simple problems that simplifies the derivation of the approximate solutions. Therefore, the usage of the homotopy theory has been effective in deriving the approximate solutions of different types of problems, including nonlinear second order differential equations [7,8] or nonlinear fractional equations [9] that model the behavior of physical and engineering systems. Hojjati and Jafari compared the HPM and the ADM to obtain the distributions of stresses and displacements in a rotating annular elastic disk [10].…”

A New Iterational Method for Ordinary Equations Using Sumudu Transform

“…Recently, Sumudu transform is adopted in some famous analytical methods [16,17,18], the combination of Sumudu transform and homotopy perturbation method is used to simplify the solution process and improve the solution's accuracy.…”

Distributed Coordination of Fractional Dynamical Systems with Exogenous Disturbances

“…Recently, fractional calculus has attracted a significant number of researchers from physics and engineering. Many dynamical systems such as dielectric polarization, viscoelastic systems, and electromagnetic waves have turned out to be modeled by fractional-order differential equations, and many important results for fractional-order dynamical systems have been obtained [13][14][15][16][17]. Many multiple agents have emerged out the fractional cooperative behavior, for example: flocking and food searching of colony system by means of the individual secretions, submarine underwater robots in the undersea with a large number of microorganisms and viscous substances, and unmanned aerial vehicles running in the complex space environment [11,12].…”