Development of a Predictor-Corrector Algorithm for the Numerical Solution of the Time-Fractional Vibration Equation

Abstract

This study employs a predictor–corrector approach to solve the time-fractional vibration equation governing the transverse deflection of a cable of length L fixed at both ends. The model incorporates the Riemann–Liouville time-fractional derivative to accurately represent memory effects and damping behavior characteristic of composite and viscoelastic materials. Spatial discretization is performed using a finite difference method, while the temporal fractional derivative is approximated through a carefully formulated predictor–corrector scheme. This technique effectively addresses the initial conditions and captures the nonlocal temporal dynamics introduced by the fractional derivative. Numerical experiments demonstrate that the proposed method is accurate, stable, and computationally efficient in simulating damped vibrations in elastic and composite cables. By offering a reliable numerical framework, the approach enables more precise analysis of vibrating systems with memory effects and material heterogeneity, thereby contributing to improved modeling and design in engineering applications involving time-dependent mechanical behavior.