On the non‐differentiable exact solutions of the (2 + 1)‐dimensional local fractional breaking soliton equation on Cantor sets

Abstract

In this article, a new (2 + 1)‐dimensional local fractional breaking soliton equation is derived with the local fractional derivative. Applying the traveling wave transform of the non‐differentiable type, the (2 + 1)‐dimensional local fractional breaking soliton equation is converted into a nonlinear local fractional ordinary differential equation. By defining a set of elementary functions on Cantor sets, a novel analytical technique namely the Mittag–Leffler function‐based method is employed for the first time ever to construct the exact solutions. The solutions on the Cantor sets are presented via the 3‐D contours. It reveals that the proposed method is effective and powerful and is expected to give some inspiration for the study of the local fractional PDEs.

A new fractal model of the convective-radiative fins with temperature-dependent thermal conductivity

“…2), it becomes invalid, so we need to give a modification of it. Recently, the fractal and fractional calculus are adopted to model many complex phenomenon arising in the extreme conditions such as the un-smooth boundary [16][17][18][19][20][21], microgravity space [22,23], fractal media [24], porous media [25] and so on [26][27][28]. Inspired by these research results, here we apply the fractal calculus to Eq.…”

Abundant Soliton Structures to the ( 2 + 1 )-Dimensional Heisenberg Ferromagnetic Spin Chain Dynamical Model

“…In recent years, the fractal and fractional derivatives [50][51][52][53][54][55][56][57][58][59][60][61] have attracted extensive attention in different fields. How to apply the proposed methods to study the fractal and fractional PDEs will be the focus of our future research.…”

Variational approach for the fractional exothermic reactions model with constant heat source in porous medium

“…It is well known that integer order derivatives are local in nature, so these derivatives do not accurately describe the problem, especially for processes with historical memory. Recently, the fractal and fractional derivatives have drawn wide attention, and has been used widely to describe many complex phenomenon arising in different fields such as the bioscience [6][7][8], optics [9,10], cold plasma [11], vibration [12][13][14], circuits [15,16], unsmooth boundary [17][18][19][20][21][22] and so on [23][24][25][26][27][28][29]. Due to the nonlocal and nonsingular properties of the fractional derivatives, the fractional derivatives are more suitable for modelling the complex processes with historical memory than integer derivatives.…”