This paper is devoted to the solution and stability of a one-dimensional model depicting Rao--Nakra sandwich beams, incorporating damping terms characterized by fractional derivative types within the domain, specifically a generalized Caputo derivative with exponential weight. To address existence, uniqueness, stability, and numerical results, fractional derivatives are substituted by diffusion equations relative to a new independent variable, $\xi$, resulting in an augmented model with a dissipative semigroup operator. Polynomial decay of energy is achieved, with a decay rate depending on the fractional derivative parameters. Both the polynomial decay and its dependency on the parameters of the generalized Caputo derivative are numerically validated. To this end, an energy-conserving finite difference numerical scheme is employed.

翻译：本文致力于求解和分析描述Rao-Nakra夹层梁的一维模型，该模型在域内引入了分数阶导数型阻尼项，具体为带指数权重的广义Caputo导数。为处理解的存在性、唯一性、稳定性及数值结果，我们将分数阶导数替换为关于新自变量$\xi$的扩散方程，从而得到一个具有耗散半群算子的增广模型。能量呈现多项式衰减，其衰减速率取决于分数阶导数参数。广义Caputo导数的参数对多项式衰减的影响及其衰减特性均通过数值模拟得到验证。为此，我们采用了能量守恒的有限差分数值格式。