Numerical Fractional Calculus Framework for Nonlocal Euler–Bernoulli Beam Deflection Analysis

In this study, the bending behavior of beams is investigated using the fractional Euler–Bernoulli beam model. This model is developed based on fractional calculus, particularly employing the Riesz–Caputo derivatives, and is capable of accurately accounting for nonlocal and size-dependent effects in structural beams. Unlike classical models that rely on integer-order derivatives, the present model uses fractional-order derivatives, which offer greater precision in analyzing beam behavior at small scales such as micro and nano levels. In this work, various beams with different boundary conditions and loading types are analyzed. To solve the governing fractional equations, a numerical algorithm based on the finite difference method is developed, which also allows for the incorporation of a variable characteristic length function along the beam. The numerical simulation results demonstrate that the order of the fractional derivative and the characteristic length have a direct impact on the amount of beam deflection. These findings indicate that the Euler–Bernoulli model based on Riesz–Caputo derivatives has high potential for realistic simulation of beam bending behavior at small scales, making it an effective tool for accurate analysis of microscale structures.