Analytic solutions in implicit form are derived for a nonlinear partial differential equation (PDE) with fractional derivative elements, which can model the dynamics of a deterministically excited Euler-Bernoulli beam resting on a viscoelastic foundation. Specifically, the initial-boundary value problem for the corresponding PDE is reduced to an initial value problem for a nonlinear ordinary differential equation in a Hilbert space. Next, by employing the cosine and sine families of operators, a variation of parameters representation of the solution map is introduced. Due to the presence of a nonlinear term, a local fixed point theorem is employed to prove the local existence and uniqueness of the solution. Relying on the regularity properties of cosine and sine families, taking into account the form of the nonlinear term, and considering the properties of the fractional derivative, the solution map of the abstract problem is cast into a derivative-free analytic solution in implicit-form for the initial boundary value problem. Results corresponding to the limiting purely elastic and purely viscous cases are also provided. The herein developed technique and derived implicit form solutions can be construed as generalizations of available results in the literature to account for fractional derivative elements. This is of significant importance given the vast utilization of fractional calculus modeling in modern engineering mechanics, and in viscoelastic material behavior in particular.

推导了含分数阶导数元素的非线性偏微分方程（PDE）的隐式解析解，该方程可对确定性激发的基于粘弹性基础的Euler-Bernoulli梁的动力学进行建模。具体而言，将对应的PDE的初边值问题简化为希尔伯特空间中非线性常微分方程的初值问题。接下来，通过使用算子的余弦和正弦族，引入解图的参数表示形式的变化。由于存在非线性项，因此采用局部不动点定理来证明该解的局部存在性和唯一性。依靠余弦和正弦族的正则性，并考虑到非线性项的形式，并考虑分数导数的性质，将抽象问题的解映射映射为初始边值问题的隐式无导数解析解。还提供了与有限的纯弹性和纯粘性情况相对应的结果。本文开发的技术和派生的隐式形式解决方案可以解释为文献中可用结果的概括，以说明分数导数元素。考虑到分数微积分建模在现代工程力学中，尤其是在粘弹性材料行为中的广泛应用，这一点非常重要。还提供了与有限的纯弹性和纯粘性情况相对应的结果。本文开发的技术和派生的隐式形式解决方案可以解释为文献中可用结果的概括，以说明分数导数元素。考虑到分数微积分建模在现代工程力学中，尤其是在粘弹性材料行为中的广泛应用，这一点非常重要。还提供了与有限的纯弹性和纯粘性情况相对应的结果。本文开发的技术和导出的隐式解决方案可以解释为文献中可用结果的概括，以说明分数导数元素。考虑到分数微积分建模在现代工程力学中，尤其是在粘弹性材料行为中的广泛应用，这一点非常重要。