Relaxation in filled polymers: A fractional calculus approach

Abstract

In recent years the fractional calculus approach to describing dynamic processes in disordered or complex systems such as relaxation or dielectric behavior in polymers or photo bleaching recovery in biologic membranes has proved to be an extraordinarily successful tool. In this paper we apply fractional relaxation to filled polymer networks and investigate the dependence of the decisive occurring parameters on the filler content. As a result, the dynamics of such complex systems may be well–described by our fractional model whereby the parameters agree with known phenomenological models.

On existence and uniqueness of positive solutions to a class of fractional boundary value problems

“…Obviously, (P, d) with d(x, y) = sup{|x(t) -y(t)|: t [0, 1]} is a complete metric space satisfying conditions (4) and (5).…”

“…Indeed, we can find numerous applications in viscoelasticity, electrochemistry control, porous media, electromagnetic, etc. [1][2][3][4][5][6].…”

Existence of three solutions for impulsive nonlinear fractional boundary value problems

“…Fractional differential equations (FDEs) have gained importance due to their numerous applications in various fields of science and engineering, such as physics, biophysics, blood flow phenomena, aerodynamics, electro magnetic, fluid flow, diffusive transport akin to diffusion, chemistry, electron-analytical chemistry, electro dynamics of complex medium, polymer rheology, viscoelasticity, control, porous media, probability, electrical networks, biology, etc. For details, see [13,17,18,28,31,33,37,39,44]. Many researchers have studied the existence of solutions for nonlinear FDEs with different tools such as fixed-point theorems, the topological degree theory, and the method of upper and lower solutions, for instance, see [5,6,20].…”

Monotone method for Riemann-Liouville multi-order fractional differential systems

“…Fractional differential equations have various applications in widespread fields of science, such as in engineering [5], chemistry [7,14,16], physics [1,2,9], and others [10,11]. Despite there being a number of existence theorems for nonlinear fractional differential equations, much as in the integer order case, this does not necessarily imply that calculating a solution explicitly will be routine, or even possible.…”