DYNAMICS OF THE NEW EXACT WAVE SOLUTIONS TO THE LOCAL FRACTIONAL VAKHNENKO–PARKES EQUATION
Yan-Hong Liang, Kang‐Jia Wang
Abstract
The fractional derivative is a research hotspot since it can better describe and model the complex physical phenomena compared to the integer derivatives. In this work, a new fractional Vakhnenko–Parkes equation with the local fractional derivative for the fractal relaxation medium is proposed and explored. Upon defining the Mittag-Leffler function on the Cantor sets, the Mittag-Leffler function-based method, combined with Yang’s fractal travelling wave transformation, is utilized to construct the exact fractal wave solutions. Correspondingly, the wave structures of the fractal wave solutions on the Cantor sets for [Formula: see text] are presented graphically. It is found that the fractal wave solutions reduce to the exact solutions of the classic Vakhnenko–Parkes equation for [Formula: see text], which strongly proves the correctness of the obtained solutions. The MLFBM adopted in this work can be employed to investigate other nonlinear partial differential equations with the local fractional derivative.