The Efficient Method to Solve the Conformable Time Fractional Benney Equation

Abstract

The current study employed the innovative conformable fractional method to analyze the nonlinear Benney equations involving the conformable fractional derivative. Conformable fractional Benney equations have been examined by the conformable q‐Shehu analysis transform method. By including nonlinear factors, it offers a more precise depiction of wave propagation compared to linear models. Various natural phenomena, including ocean waves, plasma waves, and some forms of solitons, display nonlinear behavior that cannot be precisely explained by linear equations. The fractional Benney equation is important because it extends the classical Benney equation, which describes the evolution of weakly nonlinear and weakly dispersive long waves in shallow water. By incorporating fractional calculus operators, the fractional Benney equation provides a more accurate description of wave propagation phenomena in certain physical systems characterized by nonlocal or memory‐dependent behavior. The utilization of the Benney equation enables researchers to simulate these occurrences with greater realism. This study investigates the convergence and inaccuracy of the future scheme. The conformable q‐Shehu homotopy analysis transform method (Cq‐SHATM) generates h‐curves that demonstrate the convergence interval of the series solution obtained. In order to determine the effectiveness and suitability of the Cq‐SHATM, uniqueness and convergence theorems have been proven. This study presents an application that showcases the potential advantages and efficacy of the suggested method. Moreover, an error analysis is conducted to validate the precision of the scheme. Computational simulations are performed to verify the accuracy of the upcoming method. This study presents the results gained from the numerical and graphical analysis. The method presented in this work demonstrates a high level of computational accuracy and simplicity in analyzing and solving complex phenomena associated with conformable fractional nonlinear partial differential equations in the fields of science and technology.