Exact solutions of the space-time fractional equal width equation

Abstract

A class of fractional differential equations is investigated in this paper. By the use of modified Remann-Liouville derivative and the tanh-sech method, the exact bright soliton solutions for the space-time fractional equal width are obtained.

New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle

“…On a smaller scale, for example a scale of water molecule's size, water becomes discontinuous and all laws based on continuous space or continuous time become invalid. Generally we can use Mandelbrot's fractal theory [44] to model the discontinuous phenomena [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60]. Newton's calculus is established on an infinitesimal assumption and the function is differentiable, however, the molecule's motion in water at an infinitesimal interval of time or distance is not differentiable.…”

A periodic solution of the fractional sine-Gordon equation arising in architectural engineering

“…In one iteration process, one inserts equation (21) into the expansion (13) and setting q ! 1; we finally obtain…”

“…It has been proved already that discontinuous problems can be effectively modeled by fractional calculus and fractal calculus. [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] Ahmadinia and Safari 26 applied the local discontinuous Galerkin method to study the time-space fractional sine-Gordon equation; however, analytical approaches to equation ( 1) are rare and this paper will apply the homotopy perturbation method [27][28][29][30][31][32][33][34][35][36][37] to solve equation (1). Anjum and Ain 38 applied the He's fractional derivative for the time-fractional Camassa-Holm equation by employing the fractional complex transform to convert the time-fractional Camassa-Holm differential equation into its partial differential equation and then use HPM to find a fairly accurate solution.…”

The Soliton Solutions for Some Nonlinear Fractional Differential Equations with Beta-Derivative

“…Since it provides analytical solutions, this equation has inspired many scientists reading mathematical approaches for partial differential problems. Various approaches have been used to get accurate solutions for this sort of problem [39][40][41][42][43]. This study intends to construct soliton solutions to the time-fractional Korteweg-de Vries (KdV) equation [40,44], the time fractional equal width wave equation (EWE) and the time fractional modified fractional equal width equation (mEWE) [44] of the forms…”