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In this article, we construct a family of closed form traveling wave solutions to the space-time fractional Equal Width equation (EW) and the space-time fractional modified Equal Width equation (mEW) by using newly proposed modified rational fractional (Dξ G/G2)-expansion method and the exp-function method. The considered equations are turned into fractional order ordinary differential equations with the help of a complex fractional transformation along with conformable fractional derivative and then the methods are used to investigate their solutions. The achieved solutions are in terms of trigonometric function, hyperbolic function and rational function which might be used to analyze deeply the physical complex phenomena of real world as they are new and bear much more generality. Two more well-established methods, the (G/G) -expansion method and the rational (G/G) expansion method, are also taken into account to unravel the suggested equations which do not provide any solution. The results reveal that the proposed method is efficient, straightforward and concise which might further be useful tool to examine any other nonlinear evolution equations of fractional order arising in various physical problems.

EXACT ANALYTIC WAVE SOLUTIONS TO SOME NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS FOR THE SHALLOW WATER WAVE ARISE IN PHYSICS AND ENGINEERING