Vol. 6, No. 2 (2017): Regular Issue

Fractional calculus represents the generalization of integration and differentiation to an arbitrary order. Since the very first occurrence of fractional differentiation more than 300 years ago, fractional calculus and research related to its possible application have deserved ever-growing attention and interest. The research community has managed to bring forward ideas and concepts that justify the importance of fractional calculus for future engineering and science discoveries. What has begun as a means to describe abnormal behaviours in viscoelasticity or diffusion, power law phenomena, long range processes or fractal structures has spread to almost all engineering fields and applied sciences. Nowadays, its use in control engineering has been gaining more and more popularity in both modeling and identification, as well as in the controller tuning.

Experimental study of the mass transfer kinetics for methanol in mesoporous silica is presented. Analysis of the experimental data shows that there is no good correspondence between them and corresponding solutions found according to the second Fick’s law for various pores geometries of the silica. Contrary, we show a good fit of the experimental data by a solution of the time-fractional diffusion equation with proper boundary conditions that correspond to experiment. Our results support that mass transfer in silica, which is a geometrically restricted media, may exhibit anomalous features, due to the geometrical constraints associated with randomly porous structure of a solid.

A model-reduction methodology for fractional-order systems based on the Hankel-norm is presented. The methodology involves the truncation of a Laurent series associated with the fractional-order system in a transformed domain. The truncated Laurent series coefficients are used to construct a finite-order transfer function to approximate the original system. Standard model-reduction techniques are then applied to obtain a final low-order approximation. The Hankel norm of the approximation error can be specified a priori. The approximation method is applied to several fractional-order and other infinite-order systems. It is shown to be more generally applicable than standard finite-order modeling techniques.

We study the existence of solutions for fractional integro-differential inclusions with nonlocal boundary conditions and with multi-order fractional integral conditions. We establish Filippov type existence results in the case of nonconvex set-valued maps.

To perform a robot-assisted surgery of a prosthesis implantation on a patient’s femur, we may need to get the femoral head-neck orientation for the application. We can extract that information from Computed Tomography scans, using image processing. In image processing, edge detection often makes use of integer-order differentiation operators (e.g. Canny and LoG operators). This paper shows that introducing non-integer (fractional) differentiation to edge detectors (Fractional Canny, Fractional LoG, Fractional Derivative operators) can improve automatic edge detection results.

In this paper, the balancing of a collection of electric energy storage devices is considered. Specific devices discussed are capacitors, supercapacitors, and batteries. The balancing problem is formulated as a complex network with dynamic nodes. The connection problem is approached from a graph theory viewpoint and solved by using homogeneous or entangled graphs. The utility of this approach is demonstrated with several examples

The space fractional advection equation is a linear partial pseudodifferential equation with spatial fractional derivatives in space and is used to model transport at the earth surface. This equation arises when velocity variations are heavy tailed. Space fractional diffusion equation mathematically models the solutes that move through fractal media. In this paper, we are interested in finding the approximation solution of an intermediate fractional advection diffusion equation by using the quadratic spline function. The approximation solution is proved to be conditionally stable. Finally, some numerical examples are given based on this method.

In this paper we introduce a nonlinear fractional differential equation of Lane-Emden type. We establish a solution which satisfies the Müntz-Szász theorem conditions in terms of power series. Particular solutions are established for different values of the parameters. A validation of our method is based on a case verified with the aid of a Maple program.

In this paper we introduce and establish the solutions for the fractional Hermite, Legendre and Bessel equations. The construction of the solution is established on the basis of Müntz - Szász theorem [19]. These new equations open new applications in the field of fractional quantum models, or to new applications in engineering.

To perform a robot-assisted surgery of a prosthesis implantation on a patient’s femur, we may need to get the femoral head-neck orientation for the application. We can extract that information from Computed Tomography scans, using image processing. In image processing, edge detection often makes use of integer-order differentiation operators (e.g. Canny and LoG operators). This paper shows that introducing non-integer (fractional) differentiation to edge detectors (Fractional Canny, Fractional LoG, Fractional Derivative operators) can improve automatic edge detection results

The positive and minimal realisation problem for fractional continuous-time linear single-input and single-output (SISO) systems is formulated. Method based on the one-dimensional digraph for finding a positive and minimal realisation of a given proper transfer function is proposed. Two special cases of the digraph structure are given. Sufficient conditions for the existence of a positive minimal realisation of a given proper transfer function systems are established. The algorithm for computation of a positive minimal realisation is proposed and illustrated with a numerical example. The algorithm is based on a parallel computing method to gain needed speed and computational power for such a solution.

The authors have previously reported the existence of a morphism between the Riemann zeta function and the “Cole and Cole” canonical transfer functions observed in dielectric relaxation, electrochemistry, mechanics and electromagnetism. The link with self-similar structures has been addressed for a long time and likewise the discovered of the incompleteness which may be attached to any dynamics controlled by non-integer derivative operators. Furthermore it was already shown that the Riemann Hypothesis can be associated with a transition of an order parameter given by the geometric phase attached to the fractional operators. The aim of this note is to show that all these properties have a generic basis in category theory. The highlighting of the incompleteness of non-integer operators considered as critical by some authors is relevant, but the use of the morphism with zeta function reduces the operational impact of this issue without limited its epistemological consequences.

There is considerable interest in the study of fractional order derivative integrator but obtaining analytical impulse and step responses is a difficult problem. Therefore all methods reported on to date use approximations for the fractional derivative/integrator both for analytical based computations and more relevantly in simulation studies. In this paper, an analytical formula is first derived for the inverse Laplace transform of fractional order integrator, 1/sα where α∈R and 0<α<1 using Stirling’s formula and Gamma function. Then, the analytical step response of fractional integrator has been computed from the derived impulse response of 1/sα. The obtained analytical formulas for impulse and step responses of fractional order integrator are exact results except the very small error due to the neglected terms of Stirling’s series. The results are compared with some well known integer order approximation methods and Grünwald-Letnikov (GL) approximation technique. It has been shown via numerical examples that the presented method is very successful according to other methods.