Volume 19, Issue 5

Discussions under this title, inspired by the famous Gauguin painting, were held during a Round Table in frames of the International Conference “ Fractional Differentiation and Applications ” (ICFDA ’16) held in Novi Sad (Serbia), 18-20 July 2016, http://www.icfda16.com/public/; see details at Announcements (21 September Closure): http://icfda16.com/public/announcements.php; and Report on ICFDA ’16 in Editorial Note in this journal’s issue, pp. 1059–1060. Along with the presentations made during this Round Table, we include here some contributions by the participants sent afterwards and also by few colleagues planning but failed to attend. The intention of these discussions was to continue the useful traditions from the first conferences on Fractional Calculus (FC) topics, to pose open problems, challenging hypotheses and questions “where to go”, “how to save and improve the prestige of FC”, to share opinions and try to find ways to resolve them.

The relaxation properties of dielectric materials are described, in the frequency domain, according to one of the several models proposed over the years: Kohlrausch-Williams-Watts, Cole-Cole, Cole-Davidson, Havriliak-Negami (with its modified version) and Excess wing model are among the most famous. Their description in the time domain involves some mathematical functions whose knowledge is of fundamental importance for a full understanding of the models. In this work, we survey the main dielectric models and we illustrate the corresponding time-domain functions. In particular, we stress the attention on the completely monotone character of the relaxation and response functions. We also provide a characterization of the models in terms of differential operators of fractional order.

Fractional (in time and in space) evolution equations defined on Dirichlet regular bounded open domains, driven by fractional integrated in time Gaussian spatiotemporal white noise, are considered here. Sufficient conditions for the definition of a weak-sense Gaussian solution, in the mean-square sense, are derived. The temporal, spatial and spatiotemporal Hölder continuity, in the mean-square sense, of the formulated solution is obtained, under suitable conditions, from the asymptotic properties of the Mittag-Leffler function, and the asymptotic order of the eigenvalues of a fractional polynomial of the Dirichlet negative Laplacian operator on such bounded open domains.

A new geometric interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders is proposed. The suggested geometric interpretation of the fractional derivatives is based on modern differential geometry and the geometry of jet bundles. We formulate a geometric interpretation of the fractional-order derivatives by using the concept of the infinite jets of functions. For this interpretation, we use a representation of the fractional-order derivatives by infinite series with integer-order derivatives. We demonstrate that the derivatives of non-integer orders connected with infinite jets of special type. The suggested infinite jets are considered as a reconstruction from standard jets with respect to order.

Over the last decade, it has been demonstrated that many systems in science and engineering can be modeled more accurately by fractional-order than integer-order derivatives, and many methods are developed to solve the problem of fractional systems. Due to the extra free parameter order α , fractional-order based methods provide additional degree of freedom in optimization performance. Not surprisingly, many fractional-order based methods have been used in image processing field. Herein recent studies are reviewed in ten sub-fields, which include image enhancement, image denoising, image edge detection, image segmentation, image registration, image recognition, image fusion, image encryption, image compression and image restoration. In sum, it is well proved that as a fundamental mathematic tool, fractional-order derivative shows great success in image processing.

This paper proposes a novel structural derivative approach to tackle the perplexing modeling problem of ultraslow diffusion. The structural function plays a central role in this new strategy as a kernel transform of underlying time-space fabric of physical systems. Ultraslow diffusion has been observed in numerous lab experiments and field observations, whose behaviors deviate dramatically from the standard anomalous diffusion models characterizing power function of time. The logarithmic diffusion model has since been used to describe bizarre process of ultraslow diffusion but with very limited success. This study applies the inverse Mittag-Leffler function as the structural function in the structural derivative modeling ultraslow diffusion of a random system of two interacting particles. It is observed that the dynamics of two interacting particles are respectively the ballistic motion at the short time scale and the Sinai ultraslow diffusion at the long time scale. Compared with the logarithmic diffusion model, the inverse Mittag-Leffler diffusion model has higher accuracy and manifests clearer physical mechanism. Numerical experiments show that the structural derivative is a feasible mathematical tool to model the ultraslow diffusion using the inverse Mittag-Leffler function as its structural function.

This paper is devoted to the investigation of regional controllability of the fractional order sub-diffusion process. We first derive the equivalent integral equations of the abstract sub-diffusion systems with Caputo and Riemann-Liouville fractional derivatives by utilizing the Laplace transform. The new definitions of regional controllability of the system studied are introduced by extending the existence contributions. Then we analyze the regional controllability of the fractional order sub-diffusion system with minimum energy control in two different cases: B ∈ L$\mathcal{L}$ ( R m , L 2 (Ω)). T he adjoint system of fractional order sub-diffusion system is also presented at the same time. Two applications are worked out in the end to verify the effectiveness of our results.

The only purpose of this paper is to show how fractional calculus can contribute to the development of the very interesting and promising field of micro and nano machines, from the deeper understanding of the world in which they work, to their design and control. The essentials of Brownian motors and untethered swimming microrobots are briefly presented, and two naive examples on how fractional calculus can contribute to open new perspectives are given.

We study a boundary value problem associated to a fractional differential inclusion with “maxima”. Several existence results are obtained by using suitable fixed point theorems when the right hand side has convex or non convex values.

The article concerns dynamics of an open system (OS) considered as a subsystem of some closed Hamiltonian system. Description of such OS is carried out in terms of integro-differential generalization of the Liouville equation. The Mellin transform allows to present this equation in terms of fractional operators. In frame of these concepts the response problem is formulated and applied to the supercapacitor dynamics. A good agreement with experimental data is found.

Editors’ notes The Editors acknowledge the permission of the “Prof. Marin Drinov Academic Publishing House”, where the English translation by P. Rusev of this paper was published, to reprint it now in FCAA journal, from: NIKOLA OBRECHKOFF, SELECTED PAPERS , Vol. II, 474 pp, ISBN 10 954-322-121-9(II), ISBN 13 978-954-322-121-9(II)), Sofia, 2006. : In this paper N. Obrechkoff defined a general method of summation or, more precisely, a method of Borel (B) type for analytical continuation of complex function f defined by convergent series of the kindc0+c1z+c2z2+… $c_0+c_1z+c_2z^2+\dots$ . The main goal is the summation of such series at boundary points of the corresponding region of summability under conditions of the kind B for the singular points of the function f , as they have been introduced in his earlier paper [8], 1928.Eα p(x)=∑ n=0∞ xnΓ (α n+p+1),0<α ≤ 2,integerp≥ 0,asx→ ∞ , $$E_{\alpha}^p(x)=\sum_{n=0}^{\infty}{x^n \over\Gamma(\alpha{n}+p+1)}, \ \ 0 \lt \alpha \leq 2, \,\,\, \mbox{integer}\,\, \ p \geq 0, \, \ \mbox{as} \ x \to \infty,$$ (*) which for p = 0 reduces to the well-known Mittag-Leffler function E α ( x ). He obtained thatEα p(x)=βx− pβexβ+η (x),β=1α ,for|argx|≤ α π 2$E_{\alpha}^p(x) =\beta{x^{-p\beta}}e^{x^{\beta}}+\eta(x),\, \beta=\displaystyle{1\over{\alpha}},\,\,\, \text{for} \,\,\, |\arg{x}|\leq\alpha\displaystyle{\pi \over{2}}$, where η ( x ) → 0 as x → ∞. Note that in contemporary denotations, (*) is the 2-parameter Mittag-Leffler function E α , μ ( x ) with μ = p + 1, known now as the Queen Function of Fractional Calculus (Gorenflo and Mainardi, 1997). Further, the already proved theorems are specified for the Borel as well as for the Mittag-Leffler type summations . To this end, Obrechkoff studied the asymptotics of the function (see § 11)