Abstract
The extension of the fractional order derivative to the distributed order fractional derivative (DOFD) is somewhat simple from a formal point of view, but it does not yet have a simple, obvious analytic form that allows its fast numerical calculation, which is necessary when solving differential equations with DOFD. In this paper, we supply a simple analytic kernel for the Caputo DOFD and the Caputo-Fabrizio DOFD, which may be used for numerical calculation in cases where the weight function is unity. This, in turn, could potentially allow faster solution of differential equations containing DOFD. Utilizing an analytical formulation of simple physical systems with phenomenological equations that include a DOFD, we show the relevant differences between the Caputo DOFD and the Caputo-Fabrizio DOFD. Finally, we propose a model based on DOFD for modeling composed materials that comprise different constituents, and show its compatibility with thermodynamics.
Highlights
The distributed order fractional derivative (DOFD) was introduced in 1967 [1] and, rightly, received no attention from the scientific community, since the simpler derivative of fractional order was being successfully used in several different fields of science, such as as a filter for studying spectral properties, in diffusion, and in Maxwell equations [2,3]; Bagley and Torvik, in [4], developed a theory to show the existence of the order domain and the solution of distributed order equations
Jiao et al [11] treated the problems of stability, simulation, applications and perspectives of dynamic systems modeled with the use of this new operator, and Gorenflo et al [12] found a fundamental solution of a distributed order time-fractional diffusion equations
The original definition of the fractional derivative of distributed order includes cases where the derivative is affected by a weight function, which adds more difficulties to those that already exist in the numerical computation of the derivative, including that of the computer time required for its numerical estimation
Summary
The distributed order fractional derivative (DOFD) was introduced in 1967 [1] and, rightly, received no attention from the scientific community, since the simpler derivative of fractional order was being successfully used in several different fields of science, such as as a filter for studying spectral properties, in diffusion, and in Maxwell equations [2,3]; Bagley and Torvik, in [4], developed a theory to show the existence of the order domain and the solution of distributed order equations. We find the time domain expression of h(t) appearing in Equations (5) and (6) required for the computation of cDdvw(t). We find the time domain expression of h(t) appearing in Equations (5) and (6) required for the computation of cDdvw(t) by integrating along the Bromwich-Hankel path h(t) = −(1/2iπ){ exp(−rt)dr/[rd exp(iπd) − rc exp(iπc)]r(ln r + iπ)+. We find the time domain expression of h(t) appearing in Equations (27) and (28) required for the computation of the cDdvw(t) by integrating along the path of the complex plane with s = r exp(iθ) and setting θ = 3π/2 and θ = π/2, which is integrated over the interval [−∞, ∞] of the imaginary axis h(t) = (1/2iπ){ exp(−rt)dr[log{(d − ir(1 − d))/(c − ri(1 − d))}]/(1 + ir). A comparison between the kernels of the Caputo DOFD and of Caputo-Fabrizio DOFD in the ranges 0.1–0.2, 0.45–0.55 and 0.8–0.9, as shown in the Figures 3–5, indicates that, for values of the narrow intervals closer to unity, the two DOFDs are almost identical, which is not the case when the interval is close to zero
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