Analysis of a Fractional Variational Problem Associated with Cantilever Beams Subjected to a Uniformly Distributed Load

Abstract

In this paper, we investigate the existence and uniqueness of minimizers of a fractional variational problem generalized from the energy functional associated with a cantilever beam under a uniformly distributed load. We apply the fractional Euler–Lagrange condition to formulate the minimization problem as a boundary value problem and obtain existence and uniqueness results in both L2 and L∞ settings. Additionally, we characterize the continuous dependence of the minimizers on varying loads in the energy functional. Moreover, an approximate solution is derived via the homotopy perturbation method, which is numerically demonstrated in various examples. The results show that the deformations are larger for smaller orders of the fractional derivative.

Keywords

• cantilever beam
• existence and uniqueness of minimizers
• fractional boundary value problem
• Euler–Lagrange theorem
• homotopy perturbation method