This paper is about the investigation of exact solutions of important economic model; Ivancevic option pricing model (IOPM) with M-fractional derivative. To achieve this aim, three different methods; expa function method, extended Sinh-Gordon equation expansion method (EShGEEM) and extended (G′/G)-expansion method are used. Obtained solutions consisting of trigonometric, hyperbolic trigonometric, rational and exponential. The obtained solutions are new than the existing solutions in the literature. The got solutions are also verified by using Mathematica tool. Graphically justification are also done by plotting 2-D,3-D and contour graphs. The importance of this paper is that M-fractional derivative is first time use for this model. On the bases of achieved results it is suggested that these methods are simple, reliable and fruitful than the other methods.

The goal of the present paper is to obtain and analyze new exact travelling wave solutions and bifurcation behavior of modified Zakharov-Kuznetsov (mZK) equation with higher order dispersion term. For this purpose, first and second simple methods are used to build soliton solutions of travelling wave solutions. Furthermore, bifurcation behavior of traveling waves including new type of quasiperiodic and multi-periodic traveling wave motions have been examined depending on the physical parameters. Multistability for the nonlinear mZK equation has been investigated depending on fixed values of physical parameters with various initial conditions. The suggested methods for the analytical solutions are powerful and benefical tools to obtain the exact travelling wave solutions of nonlinear evolution equations (NLEEs). Two and three-dimensional plots are also provided to illustrate the new solutions. Bifurcation and multistability behaviors of traveling wave solution of the nonlinear mZK equation with higher order dispersion will add some value in the literature of mathematical and plasma physics.