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Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation

  • Run-Fa Zhang
  • Sudao BiligeEmail author
Original Paper
  • 43 Downloads

Abstract

A new method named bilinear neural network is introduced in this paper, and the corresponding tensor formula is proposed to obtain the exact analytical solutions of nonlinear partial differential equations (PDEs). This is the first time that the neural network model is used to find the exact analytical solution, and this method covers almost all methods of constructing a function after bilinearization to solve nonlinear PDEs. Furthermore, this method is most likely a universal method to obtain the exact analytical solutions of nonlinear PDEs. Abundant arbitrary functions solutions of the reduced p-gBKP equation are obtained by using this method. Various beautiful plots of the presented solutions, which show diversity of exact solutions to PDEs, are made. By choosing appropriate values and functions, the fractal solitons waves are obtained and the self-similar characteristics of these waves are observed by reducing the observation range and magnifying local images. Via various three-dimensional plots, the evolution characteristics of these waves are exhibited.

Keywords

Bilinear neural network method Universal tensor formula Exact analytical solution Fractal soliton waves 

Notes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (11661060) and the Natural Science Foundation of Inner Mongolia Autonomous Region of China (2018LH01013).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsInner Mongolia University of TechnologyHohhotChina

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