Analytical Method for Solving the Fractional Order Generalized KdV Equation by a Beta-Fractional Derivative

Abstract

The present work is related to solving the fractional generalized Korteweg-de Vries (gKdV) equation in fractional time derivative form of order $\alpha$ . Some exact solutions of the fractional-order gKdV equation are attained by employing the new powerful expansion approach by using the beta-fractional derivative which is used to get many solitary wave solutions by changing various parameters. The obtained solutions include three classes of soliton wave solutions in terms of hyperbolic function, trigonometric function, and rational function solutions. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of nonlinearity. Some of the nonlinear equations arise in fluid dynamics and nonlinear phenomena.