On the Appearance of the Fractional Derivative in the Behavior of Real Materials

Abstract

Generalized constitutive relationships for viscoelastic materials are suggested in which the customary time derivatives of integer order are replaced by derivatives of fractional order. To this point, the justification for such models has resided in the fact that they are effective in describing the behavior of real materials. In this work, the fractional derivative is shown to arise naturally in the description of certain motions of a Newtonian fluid. We claim this provides some justification for the use of ad hoc relationships which include the fractional derivative. An application of such a constitutive relationship to the prediction of the transient response of a frequency-dependent material is included.

Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions

“…Many authors have worked thoroughly on their numerical solution [15][16][17][18][19][20][21][22]. We restrict our attention to the linear case that includes important models, such as the Bagley-Torvik equation [23], the fractional oscillation equation [24] and many others.…”

Impulse response of a generalized fractional second order filter

“…On one hand, using the notion c 2012 Diogenes Co., Sofia pp. 97-116 , DOI: 10.2478/s13540-012-0007-2 of fractional calculus equations may be a closer characterization to the real world than using non-fractional calculus ones, [5]. On the other hand, fractional calculus provides a powerful tool for the description of memory and hereditary effects in various substances and in modeling dynamical processes in fractal media [6].…”

Optimal Approximation, Simulation and Analog Realization of the Fundamental Fractional Order Transfer Function

“…In the last decades a considerable focus on fractional calculus has been stimulated by applications of these concepts in various areas of physics and engineering. Many systems are known to reveal fractional order system dynamics, such as viscoelastic systems (Torvik and Bagley, 1984), electrode-electrolyte polarization (Ichise et al, 1971), interface polarization (Sun and Onaral, 1983), cardiac behaviour (Goldberger et al, 1985), dielectric relaxation (Cole and Cole, 1941;Davidson and Cole, 1950). Because of their representation by irrational transfer functions, fractional order systems were studied marginally in theory and practice.…”