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<p>In this paper, we investigate the strain wave equation, which governs the behavior of elastic wave propagation in different media of continuum mechanics. Due to the nonlinearity of the equation, it is often difficult to achieve exact solutions using certain traditional analytical and numerical approaches. In this work, we use the bilinear neural network method to provide exact solutions to the strain wave equation. This method effectively builds solutions by utilizing deep learning algorithms and bilinear transformations. The bilinear neural network method provides a potent substitute for traditional methods by methodically approximating nonlinear wave patterns and mapping them into a solvable framework. The strain wave equation provides an insight into the nonlinear long wave propagation behavior in shallow water by taking wave phase into account, and its versatility extends beyond fluid dynamics to various physical phenomena. By employing structured neural network models, the generalized lump, periodic, and other exact analytical solutions are obtained using the bilinear neural network approach. With the help of Mathematica, we extract these solutions and illustrate them in 3D, 2D, and contour plots. These visualizations helped illustrate the dynamics of the solutions obtained. The computational efficiency of the method was illustrated through its capacity to symbolically derive closed-form solutions without the need for iterative numerical methods and, in turn, diminish processing time and complexity significantly. The novelty of this work is the combination of bilinear transformation methods and neural network architectures to symbolically obtain exact analytical solutions as highlighted above for the nonlinear strain wave equation, representing a remarkable breakthrough compared to the conventional numerical and analytical methods. The validity of the solutions acquired was established by direct substitution in the bilinear equation as well as through checking their compliance with the equation of strain wave with the help of Mathematica.</p>

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On the analytical solution of the nonlinear strain wave equation through the bilinear neural network method

  • 期刊论文

  • 发表日期：

    2025年8月28日

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  • Humaira Muslim,
  • Nauman Ahmed,
  • Siegfried Macías,
  • Muhammad Ozair Ahmed,
  • Muhammad Z. Baber,
  • Baboucarr Ceesay,
  • Jorge E. Macías-Díaz

摘要

In this paper, we investigate the strain wave equation, which governs the behavior of elastic wave propagation in different media of continuum mechanics. Due to the nonlinearity of the equation, it is often difficult to achieve exact solutions using certain traditional analytical and numerical approaches. In this work, we use the bilinear neural network method to provide exact solutions to the strain wave equation. This method effectively builds solutions by utilizing deep learning algorithms and bilinear transformations. The bilinear neural network method provides a potent substitute for traditional methods by methodically approximating nonlinear wave patterns and mapping them into a solvable framework. The strain wave equation provides an insight into the nonlinear long wave propagation behavior in shallow water by taking wave phase into account, and its versatility extends beyond fluid dynamics to various physical phenomena. By employing structured neural network models, the generalized lump, periodic, and other exact analytical solutions are obtained using the bilinear neural network approach. With the help of Mathematica, we extract these solutions and illustrate them in 3D, 2D, and contour plots. These visualizations helped illustrate the dynamics of the solutions obtained. The computational efficiency of the method was illustrated through its capacity to symbolically derive closed-form solutions without the need for iterative numerical methods and, in turn, diminish processing time and complexity significantly. The novelty of this work is the combination of bilinear transformation methods and neural network architectures to symbolically obtain exact analytical solutions as highlighted above for the nonlinear strain wave equation, representing a remarkable breakthrough compared to the conventional numerical and analytical methods. The validity of the solutions acquired was established by direct substitution in the bilinear equation as well as through checking their compliance with the equation of strain wave with the help of Mathematica.