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

Jan Muhammad; Ghulam Hussain Tipu; Yasser Alrashedi; Mofareh Alhazmi; Usman Younas; Jan Muhammad; Ghulam Hussain Tipu; Yasser Alrashedi; Mofareh Alhazmi; Usman Younas

doi:10.3934/math.2025657

AIMS Mathematics

2025, Volume 10, Issue 6: 14596-14616. doi: 10.3934/math.2025657

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

1.
Department of Mathematics, Shanghai University, Shanghai 200444, China
2.
Department of Mathematics, College of Science, Taibah University, P.O. Box 344, Madinah 42353, Saudi Arabia
3.
Mathematics Department, College of Science, Jouf University, P.O. Box 2014, Sakaka, Saudi Arabia

Received: 01 May 2025 Revised: 12 June 2025 Accepted: 16 June 2025 Published: 26 June 2025
MSC : 35C07, 35C08, 35C15

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.
- Riccati subequation neural networks,
- nonlinear equations,
- solitons
Citation: Jan Muhammad, Ghulam Hussain Tipu, Yasser Alrashedi, Mofareh Alhazmi, Usman Younas. Analytical study of the nonlinear dynamical systems: Application of the neural networks method[J]. AIMS Mathematics, 2025, 10(6): 14596-14616. doi: 10.3934/math.2025657

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Abstract

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.

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