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In this article, the nonlinear partial fractional differential equation, namely the KdV equation is renewed with the help of modified Riemann- Liouville fractional derivative. The equation is transformed into the nonlinear ordinary differential equation by using the fractional complex transformation. The goal of this paper is to construct new analytical solutions of the space and time fractional nonlinear KdV equation through the extended  -expansion method. The work produces abundant exact solutions in terms of hyperbolic, trigonometric, rational, exponential, and complex forms, which are new and more general than existing results in literature. The newly generated solutions show that the executed method is a well-organized and competent mathematical tool to investigate a class of nonlinear evolution fractional order equations.

A Comparative Study of Space and Time Fractional KdV Equation through Analytical Approach with Nonlinear Auxiliary Equation