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Using the bilinear neural network method and the neural network-based symbolic calculation approach to find exact solutions of the (3+1)-dimensional Kairat-X extended equation

Crossref DOI link: https://doi.org/10.1007/s11071-025-11522-4

Published Online: 2025-06-30

Published Print: 2025-10

Update policy: https://doi.org/10.1007/springer_crossmark_policy

Authors

Hai, Ni

Bilige, Sudao
Funding

Funding for this research was provided by:

Natural Science Foundation of Inner Mongolia Autonomous Region of China (2024MS01009)

Natural Science Foundation of Inner Mongolia Autonomous Region of China (2024MS01009)

Basic research funds for universities directly under the Inner Mongolia Autonomous Region (JY20230100)

Basic research funds for universities directly under the Inner Mongolia Autonomous Region (JY20230100)

NationalNatural Science Foundation of China (12061054)

NationalNatural Science Foundation of China (12061054)
License Information

Text and Data Mining valid from 2025-06-30

Version of Record valid from 2025-06-30
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Article History

Received: 2 May 2025

Accepted: 19 June 2025

First Online: 30 June 2025

Declarations

:

: The authors declare that they have no Conflict of interest.

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