Elastic curves with fractional derivative

This thesis aims to extend classical elastic curve theories by incorporating fractional derivative concepts. Current needs in engineering and materials science require more flexible and realistic approaches for cases where traditional integer-order derivative-based models fall short. In particular, fractional derivatives can more accurately reflect the memory and hereditary properties of materials exhibiting viscoelastic behavior or possessing microstructural complexity in modeling their deformations. The thesis first examines common fractional derivative definitions, such as Riemann-Liouville and Caputo, along with their fundamental properties in detail. Subsequently, the classical Euler-Bernoulli beam theory and the modeling of elastic curves through variational principles are reviewed. Building upon this foundation, fractional derivative-based elastic curve models are derived by substituting classical derivative terms in the potential energy functional with fractional derivatives. Generalized Euler-Lagrange equations are formulated for these new models, and appropriate boundary conditions are established. This study provides significant contributions to the development and understanding of research on fractional derivative-based elastic curves. Future work may focus on fractional derivative-based elastic curves defined by energy functionals associated with curves on surfaces.