On new abundant exact traveling wave solutions to the local fractional Gardner equation defined on Cantor sets

Abstract

In this article, we derive a new local fractional Gardner equation based on the local fractional derivative. A novel traveling wave transform of the non‐differentiable type is utilized to convert the local fractional Gardner equation into a nonlinear local fractional ordinary differential equation. Then a new method called the Mittag–Leffler function based method is proposed for the first time ever to construct the abundant traveling wave solutions. By this method, three families (six sets) of the traveling wave solutions on Cantor set are obtained. Finally, the performances of the various solutions on Cantor set are presented through the three‐dimensional contours. It shows that the proposed method is straightforward, powerful and can provide more forms of the traveling wave solutions for the local fractional equations, which is expected to be helpful for the study of the traveling wave theory for local fractional equations.