Analytical solution of the fractional-order Euler–Bernoulli beam equation

This study investigates the dynamic response and vibration characteristics of an Euler-Bernoulli beam resting on a Winkler foundation under the influence of a concentrated distributed moving load.A fractional-order framework is employed to model the viscoelastic behavior of the beam.The governing fourth-order fractional partial differential equation is solved analytically using the Fourier sine transform, which reduces the problem to a second-order fractional differential equation.The reduced equation is further treated using Jumarie's Laplace transform and its convolution theorem to obtain closed-form solutions for the deflection and bending moment.The influence of the fractional derivative order on the beam's dynamic response is explored, revealing strong memory effects at lower orders.Moreover, as varies from 0.5 to 1.0, the beam transitions from highly viscoelastic-like behavior to classical elastic behavior.At lower values, deflection decays slowly, exhibiting prolonged oscillations and significant memory retention.In contrast, at higher values, the system shows rapid energy dissipation and transient responses.Numerical simulations demonstrate the evolution of the transverse deflection across space and time, highlighting how fractional-order damping influences vibrational decay and system stability.