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.

This paper is about the study of space-time fractional Fokas-Lenells equation that describes nonlinear wave propagation in optical fibers. Three prominent schemes are employed for extracting different types of exact soliton solutions. In particular, the expa function method, the hyperbolic function method and the simplest Riccati equation scheme are investigated for the said model. As a sequela, a series of soliton solutions are obtained and verified through Mathematica. The obtained solutions are significant additions in some specific fields of physics and engineering. Furthermore, the 3D graphical descriptions are left to analyze the pulse propagation for the reader.