In this work, an extended Euler–Bernoulli beam equation is addressed, where numerous phenomena are covered including damping, time-delay, and nonlinear source effects. A generalized fractional derivative is used to model dissipation of order less than one, which offers more flexibility for modeling tasks. Through a diffusive representation, the problem well-posedness is tackled and the exponential decay of the energy associated to global solutions is proved under some conditions. In order to validate our theoretical findings, we implement a finite difference scheme and we elucidate that the boundedness of the local propagation matrix may be inaccurate for the convergence evaluation in some situations. Furthermore, we show that deep neural networks are efficient alternatives to deal with computational and stability burdens resulting from the mesh refinement in standard numerical schemes.

在这项工作中，解决了扩展的欧拉-伯努利梁方程，其中涵盖了许多现象，包括阻尼、时滞和非线性源效应。广义分数阶导数用于对小于一阶的耗散进行建模，这为建模任务提供了更大的灵活性。通过扩散表示，解决了问题的适定性，并在某些条件下证明了与全局解相关的能量的指数衰减。为了验证我们的理论发现，我们实现了有限差分格式，并阐明了局部传播矩阵的有界性在某些情况下对于收敛评估可能不准确。此外，我们表明深度神经网络是处理标准数值方案中网格细化带来的计算和稳定性负担的有效替代方案。