Analytical study of the nonlinear dynamical systems: Application of the neural networks method

In this paper, we study the numerous complex dynamics of the nonlinear partial differential equations, namely the nonlinear Murray equation and nano-ionic currents along microtubules dynamical equations. Research has focused on solitary wave solutions because they provide important insights into nonlinear processes and have a variety of practical applications. Their exceptional behaviours and reliability represent creative nonlinear models across numerous fields, including physical, biological, and medical modeling. This research introduces Riccati subequation neural networks to derive exact solutions for space-time partial differential equations. The suggested technique integrates the solutions of the Riccati problem into neural networks. Neural networks are multi-layer computational representations consisting of activation and weights functions connecting neurons across input, hidden, and output layers. In this method, each neuron in the first hidden layer is allocated to the solutions of the Riccati equation. Thus, the new trial functions are derived. The suggested approach provides exact solutions of space-time partial differential equations. To validate the mathematical framework of this technique, we examine the proposed equations, resulting in the derivation of generalized hyperbolic function solutions, generalized trigonometric function solutions, and generalized rational solutions. This research presents novel solutions, as the presented approach is applied to the neural networks model for the first time. The dynamic properties of some solutions related to waves are shown using various graphics. This study advances knowledge of nonlinear dynamics in specific systems by demonstrating the method's efficacy.