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The well‐known Drinfeld–Sokolov–Wilson system is widely used to describe the flow of the shallow water. This paper considers the time–space fractional Drinfeld–Sokolov–Wilson system, where He's variational method together with the two‐scale transform is applied to find its solitary solutions. The numerical results have been shown through graphs to demonstrate the applicability and efficiency of our approach. The main advantage of variational approach is that it can reduce the order of differential equation and make the equation more simple. The obtained results in this work are expected to shed a bright light on the study of fractional nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper suggests a fractal mathematical model for Fangzhu's water collection by taking into account its nanoscale surface morphology. A fractal variational principle is established by the semi‐inverse method, which elucidates the conservation laws in the fractal space. Additionally, the variational formulation can also reveal the structure of the solution. An approximate analytical solution is obtained via the two‐scale transform method and He's frequency formula. Some special cases of the model are discussed numerically, showing the method is remarkably simple and capable to solve nonlinear oscillators with fractal derivatives. [ABSTRACT FROM AUTHOR]