Fractal rheological models and fractional differential equations for viscoelastic behavior

Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives

“…A suitable hierarchical arrangement of springs and dashpots gives rise to spring-pot behaviour (described below), either exactly at all timescales for an infinite tree (Heymans and Bauwens 1994), or in the longterm (or low-frequency) limit for an infinite ladder or infinite Sierpinski gasket (Heymans and Bauwens 1994, Schiessel and Blumen 1993, 1995. In the latter case short-term behaviour is similar to a Maxwell model with one element replaced by a spring-pot.…”

Strain-dependent fractional molecular diffusion in humid spider silk fibres

“…It is formulated by the generalized diffusion equation (1.2), which is solved by a generalized exponential function known as the Mittag-Leffler function. Moreover, it has been found that the same fractional calculus that provides the basis for the fractional molecular diffusion also describes the macroscopic mechanical properties of silkworm silk fibres [18], thus providing a link between molecular dynamical properties and the macroscopic mechanical behaviour inspired by fractal concepts [20]. However, a comprehensive study of the function K a ¼ K a (s) relating the generalized fractional molecular diffusion coefficient to the applied strain s on the fibres is still missing.…”

Nonlocal boundary value problems for fractional differential inclusions with Erdélyi-Kober fractional integral boundary conditions

“…is a solution of (8), then x is a solution to (1). To transform (8) into a fixed point problem, we define the operator¯ bȳ…”