Investigation of the exact solutions via sub-equation neural network method to the nonlinear systems in fluid and nuclear physics
This paper aims to explore the nonlinear dynamics of the well-known nonlinear partial differential equations, namely, Estevez–Mansfield–Clarkson (EMC) and Sharma–Taso–Olver (STO) equations. The presented models have useful applications in various fields. The EMC equation clarifies the complex dynamics of waves in shallow water and fluid physics. In nuclear physics, the STO model is pertinent to particle fission and fusion processes. This work offers Riccati sub-equation neural networks to provide exact solutions for space–time partial differential equations. The proposed method incorporates the solutions of the Riccati problem into neural networks. Neural networks are multi-layer computer models with activation functions and weight functions that connect neurons across the input, hidden, and output layers. In this approach, each neuron in the first hidden layer is assigned to the solutions of the Riccati equation. Consequently, the new trial functions are established. The proposed method provides exact solutions to the studied models in the forms of bright, dark, singular, combined, and complex solitons. Moreover, generalized hyperbolic function solutions, trigonometric function solutions, and generalized rational solutions are also recovered. This study introduces innovative solutions as the proposed methodology is used in the neural network model. A variety of graphs have been sketched for the physical behavior of the obtained solutions. By establishing the dependability of the method used, this research’s outcomes could advance our grasp of nonlinear behavior in targeted systems.