Abstract
We implement relatively analytical techniques, the homotopy perturbation method, and variational iteration method to find the approximate solutions for time and space fractional Benjamin-Bona Mahony equation. The fractional derivatives are described in the Caputo sense. These methods are used in applied mathematics to obtain the analytic approximate solutions for the nonlinear Bejamin-Bona Mahoney (BBM) partial fractional differential equation. We compare between the approximate solutions obtained by these methods. Also, we present the figures to compare between the approximate solutions. Also, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. We use the improved <svg style="vertical-align:-2.3205pt;width:47.275002px;" id="M1" height="23.612499" version="1.1" viewBox="0 0 47.275002 23.612499" width="47.275002" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,20.662)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,5.944,20.662)"><path id="x1D43A" d="M713 296l-5 -25q-47 -7 -59 -20t-23 -72l-15 -79q-9 -48 -3 -74q-15 -3 -55 -13t-63 -15t-59.5 -10t-70.5 -5q-149 0 -243 80t-94 220q0 169 127.5 276.5t336.5 107.5q91 0 206 -36l-10 -165l-29 -1q1 85 -47.5 126t-146.5 41q-153 0 -243.5 -97t-90.5 -242
q0 -122 68.5 -198.5t188.5 -76.5q121 0 139 75l20 86q13 58 -1.5 70.5t-99.5 20.5l5 26h267z" /></g> <g transform="matrix(.008,-0,0,-.008,18.25,6.8)"><path id="x2032" d="M227 744l-123 -338l-31 15l73 368q12 3 41.5 -8t36.5 -20z" /></g> <g transform="matrix(.017,-0,0,-.017,22.025,20.662)"><path id="x2F" d="M368 703l-264 -866h-60l265 866h59z" /></g><g transform="matrix(.017,-0,0,-.017,29.028,20.662)"><use xlink:href="#x1D43A"/></g><g transform="matrix(.017,-0,0,-.017,41.336,20.662)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg>-expansion function method to find exact solutions of nonlinear fractional BBM equation.
Highlights
In recent years, there has been a great deal of interest in fractional differential equations
We present the figures to compare between the approximate solutions
Not much work has been done for nonlinear problems, and only a few numerical schemes have been proposed to solve nonlinear fractional differential equations
Summary
There has been a great deal of interest in fractional differential equations. Applications have included classes of nonlinear equation with multiorder fractional derivative, and this motivates us to develop a numerical scheme for their solutions [13]. Numerical and analytical methods have included Adomian decomposition method (ADM) [14,15,16,17], variational iteration method (VIM) [18,19,20], homotopy perturbation method [21,22,23,24], homotopy analysis method [25,26,27], and the fractional complex transformation [28, 29] to get some special exact solutions for nonlinear partial fractional differential equation. The main objective of this paper is to use two different methods such as homotopy perturbation method and Journal of Applied Mathematics variational iteration method for calculating the analytic approximate solutions of the Bejamin-Bona Mahoney equation. We find some exact solutions to time and space fractional Bejamin-Bona Mahoney equation by using the improved (G/G)-expansion function method
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