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<p>In this research we mainly introduced a method called M-truncated bilinear neural network method (M<sub>t</sub>-BNNM) to explore and construct analytical exact and approximate solutions of nonlinear partial differential equations (PDEs) with the sense of M-truncated fractional order derivative. This proposed method is the improved version of bilinear neural network method (BNNM). This is the first time we combined the Hirota bilinear method to the M-truncated fractional order derivative to solve the PDEs. To establish the sovereignty, generalisation and flexibility of the proposed method, we considered a M-truncated fractional nonlinear PDE, specifically the (1 + 1) dimensional Boussinesq equation (BE). We have obtained new exact and approximate solutions of our considered equation can be developed by our newly proposed technique M<sub>t</sub>-BNNM. For physical visualization, we have drawn 3-D plots, 2-D line plots, contour plots and density plots to detect the self-similar wave propagation and profuse dynamical behaviour of these gained solutions. Moreover, we have added the qualitative analysis of the higher order Boussinesq type equation like the (3 + 1)-dimensional Wazwaz Kaur Boussinesq (WKB) equation. Qualitative investigation of PDEs is utilized to understand the overall behaviour of solutions. It exposes important features such as bifurcation, phase portrait, and multi-stability analysis helping us predict how systems respond to parameter variations and interpret complex real-world phenomena.</p>

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M-truncated bilinear neural network method with qualitative analysis for nonlinear partial differential equations

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月30日

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  • Nur Hasan Mahmud Shahen,
  • Md. Al Amin,
  • M. M. Rahman

摘要

In this research we mainly introduced a method called M-truncated bilinear neural network method (Mt-BNNM) to explore and construct analytical exact and approximate solutions of nonlinear partial differential equations (PDEs) with the sense of M-truncated fractional order derivative. This proposed method is the improved version of bilinear neural network method (BNNM). This is the first time we combined the Hirota bilinear method to the M-truncated fractional order derivative to solve the PDEs. To establish the sovereignty, generalisation and flexibility of the proposed method, we considered a M-truncated fractional nonlinear PDE, specifically the (1 + 1) dimensional Boussinesq equation (BE). We have obtained new exact and approximate solutions of our considered equation can be developed by our newly proposed technique Mt-BNNM. For physical visualization, we have drawn 3-D plots, 2-D line plots, contour plots and density plots to detect the self-similar wave propagation and profuse dynamical behaviour of these gained solutions. Moreover, we have added the qualitative analysis of the higher order Boussinesq type equation like the (3 + 1)-dimensional Wazwaz Kaur Boussinesq (WKB) equation. Qualitative investigation of PDEs is utilized to understand the overall behaviour of solutions. It exposes important features such as bifurcation, phase portrait, and multi-stability analysis helping us predict how systems respond to parameter variations and interpret complex real-world phenomena.