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Under the current research, the local fractional (3+1)-dimensional modified Zakharov–Kuznetsov equation (MZKE) is explored. With the Mittag–Leffler function (MLF) defined on the Cantor sets (CS), four special functions, namely the SHχ(ℑχ), CHχ(ℑχ), SEχ(ℑχ) and CSχ(ℑχ) are extracted to construct an auxiliary function. Then the auxiliary function, along with Yang’s non-differentiable (ND) transformation, is manipulated to explore the ND exact solutions (ESs). By means of the proposed method, four different sets of the ND ESs are found in just one step. The nonlinear dynamics of the ND exact solutions on the CS are illustrated graphically. Furthermore, the ND exact solutions for χ =ln 2/ln3 and the classic exact solutions for χ = 1 are also compared and discussed in detail via the 2-D curves. The attained results reveal that the method is a simple but effective tool to deal with local fractional PDEs arising in physics and maths. [ABSTRACT FROM AUTHOR]

In this work, the (2 + 1) -dimensional Kadomtsev–Petviashvili model is investigated. A novel variational scheme, namely, the variational transform wave method (VTWM), is successfully established to seek the solitary wave solution of the Kadomtsev–Petviashvili model. Furthermore, the fractal solitary solution of fractal Kadomtsev–Petviashvili model is also studied based on the local fractional derivative. Numerical examples are given to fully demonstrate that the VTWM is straightforward, efficient and attractive. Finally, the physical properties of solitary wave solutions are demonstrated by some 3D graphs. [ABSTRACT FROM AUTHOR]

The fractional derivatives have been widely applied in many fields and has attracted widespread attention. This paper extracts a new fractional (3+1)-dimensional modified Zakharov–Kuznetsov equation (MZKe) with the local fractional derivative (LFD) for the first time. Two special functions, namely, the LT δ (Ξ δ) and LC δ (Ξ δ) functions that are derived on the basis of the Mittag-Leffler function (MLF) defined on the Cantor set (CS), are employed to construct the auxiliary trial function to look into the exact solutions (ESs). Aided by Yang's non-differentiable (ND) transformation, six groups of the ND ESs are found. The ND ESs on the CS for δ = ln 2 / ln 3 are depicted graphically. Additionally, as a comparison, the ESs of the classic (3+1)-dimensional MZKe for δ = 1 are also illustrated. The outcomes reveal that the derived method is powerful and effective, and can be used to deal with the other local fractional PDEs. [ABSTRACT FROM AUTHOR]