Xiao Hong, A. G. Davodi, S. M. Mirhosseini-Alizamini, M. M. A. Khater, Mustafa Inc

Important analytical methods such as the methods of exp-function, rational hyperbolic method (RHM) and sec–sech method are applied in this paper to solve fractional nonlinear partial differential equations (FNLPDEs) with a truncated $M$-fractional derivative (TMFD), which consist of exponential terms. A general modified fractional Degasperis–Procesi–Camassa–Holm equation (GM-FDP-CHE) is investigated with TMFD. The exp-function method is also applied to derive a variety of traveling wave solutions (TWSs) with distinct physical structures for this nonlinear evolution equation. The RHM is used to obtain single-soliton solutions for this equation. The sec–sech method is used to derive multiple-soliton solutions of the GM-FDP-CHE. These techniques can be implemented to find various differential equations exact solutions arising from problems in engineering. The analytical solution of the $M$-fractional heat equation is found. Graphical representations are also given.