© 2020 Elsevier GmbHIn this work, two different schemes, the extended hyperbolic auxiliary and the simplest equation method, are employed to construct the exact solutions involving parameters of the Biswas–Arshed model (BAM) with truncated M-fractional derivative. Moreover, semi-inverse variational principle are applied to underlying equation to acquire analytical solution. These methods have a broad applicability to many other nonlinear evolution equations in mathematical physics. Different traveling wave solutions have been investigated by invoking these methods. 3D graphic representations are given to explain the principal effect of parameter μ on dynamical properties of the soliton solutions. The stability property of the obtained solutions is tested to show the ability of our obtained solutions through the physical experiments. Moreover, the general solution of nonlinear ordinary differential equation corresponding to underlying equation is found using traveling wave reduction.