On the energy decay of a nonlinear time-fractional Euler–Bernoulli beam problem including time-delay: theoretical treatment and numerical solution techniques

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.