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Analytical evaluations using neural network-based method for wave solutions of combined Kairat-II-X differential equation in fluid mechanics
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  • Published: 07 February 2026

Analytical evaluations using neural network-based method for wave solutions of combined Kairat-II-X differential equation in fluid mechanics

  • Ping Zhou1,
  • Jalil Manafian2,3,
  • Mehrdad Lakestani2,4,
  • Onur Alp Ilhan5,
  • Nayier Jangi Bahador2,3,
  • A. A. Fattah6,
  • K. H. Mahmoud6,
  • Rzayeva Nuray7 &
  • …
  • Nigar Nagiyeva7 

Scientific Reports , Article number:  (2026) Cite this article

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We are providing an unedited version of this manuscript to give early access to its findings. Before final publication, the manuscript will undergo further editing. Please note there may be errors present which affect the content, and all legal disclaimers apply.

Subjects

  • Engineering
  • Mathematics and computing
  • Physics

Abstract

In this paper, a symbolic computation method based on a neural network architecture, the improved neural network-based method, for obtaining novel exact solutions to combined Kairat-II-X differential equation is proposed. We secure various types of soliton solutions and periodic waves through the considered approach. Furthermore, similar to existing neural network-based schemes, this improved technique also applies the output of neural networks obtained via feedforward computation as a trial function. By introducing various activation functions, novel trial functions are extracted. These functions incorporate the neural networks’ weights and biases, in that connection transforming the solution of the combined Kairat-II-X differential equation into a problem of determining these parameters. Using neural network-based technique and the improved variant, we derive a number of exact solutions including dark solitons, singular solitons, combined hyperbolic function solutions for Kairat-II-X equation. The proposed method is compared in detail with physics-informed neural networks in terms of computational theory. The physical relevance of the driven solutions is carefully examined by providing a range of graphs that show how the solutions behave for particular parameter values. Our findings suggest that they could be applied in the future to determine novel and diverse solutions to nonlinear evolution equations that arise in mathematical physics and engineering.

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The author would like to acknowledge the Deanship of Graduate Studies and Scientific Research, Taif University for funding this work.

Funding

Training Program for Young and Middle aged Research Backbone Teachers of Wenshan University, No. 2023GG06. Fund Project of Yunnan Provincial Department of Education, No. 2025J1009.

Author information

Authors and Affiliations

  1. School of Artificial Intelligence, Wenshan University, Wenshan, Yunnan, China

    Ping Zhou

  2. Department of Applied Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, University of Tabriz, Tabriz, Iran

    Jalil Manafian, Mehrdad Lakestani & Nayier Jangi Bahador

  3. Natural Sciences Faculty, Lankaran State University, 50, H. Aslanov str., Lankaran, Azerbaijan

    Jalil Manafian & Nayier Jangi Bahador

  4. Research Center of Performance and Productivity Analysis, Istinye University, Istanbul, Türkiye

    Mehrdad Lakestani

  5. Department of Mathematics, Faculty of Education, Erciyes University, 38039, Melikgazi, Kayseri, Turkey

    Onur Alp Ilhan

  6. Department of Physics, College of Khurma University College, Taif University, 21944, Taif, Saudi Arabia

    A. A. Fattah & K. H. Mahmoud

  7. Faculty of Physics and Mathematics, Department of Informatics, Nakhchivan State University, Nakhchivan, Azerbaijan

    Rzayeva Nuray & Nigar Nagiyeva

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  1. Ping Zhou
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  8. Rzayeva Nuray
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Contributions

“All author’s wrote the main manuscript text and they prepared Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. All authors reviewed the manuscript.”

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Correspondence to Jalil Manafian or Nayier Jangi Bahador.

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Zhou, P., Manafian, J., Lakestani, M. et al. Analytical evaluations using neural network-based method for wave solutions of combined Kairat-II-X differential equation in fluid mechanics. Sci Rep (2026). https://doi.org/10.1038/s41598-026-38761-8

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  • Received: 12 December 2025

  • Accepted: 30 January 2026

  • Published: 07 February 2026

  • DOI: https://doi.org/10.1038/s41598-026-38761-8

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Keywords

  • Neural networks
  • Dark solitons
  • Singular solitons
  • Combined hyperbolic function solutions
  • Soliton solutions
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