Implicit analytic solutions for a nonlinear fractional partial differential beam equation

Abstract

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.