A Novel Softmax Building Method for Finding Solutions of Benney-Luke Equation

Abstract

This paper firstly introduces a novel application of the softmax method to solve the Benney-Luke equation, a nonlinear partial differential equation widely used in studying water waves and fluid dynamics. The softmax function, traditionally used in classification problems within machine learning playing a main function in the hidden layer to analyze the data of neural network model, is applied here to obtain exact solutions regarding hyperbolic, trigonometric, rational and polynomial forms. The effectiveness of this method is to demonstrate the ability to provide diverse and explicit solutions with improved computational efficiency. The results are significant in expanding the tools for solving complex nonlinear equations, especially in physics, fluid mechanics, and applied mathematics.