Special Issue “Trends in Fractional Modelling in Science and Innovative Technologies”

Fractional calculus has played an important role in the fields of mathematics, physics, electronics, mechanics, and engineering in recent years. Modeling methods involving fractional operators have been continuously generalized and enhanced, especially during the last few decades.

Fractional calculus has an interesting history in the modeling of non-linear and anomalous problems in mathematics, physics, statistics, and engineering, involving a diversity of fractional-order integral and derivative operators, such as those named after Grunwald–Letnikov, Riemann–Liouville, Weyl, Caputo, Hadamard, Riesz, Erdelyi–Kober, etc. based on power-law memory. Beyond this positive classical basis, in recent years, new trends in fractional modeling involving operators with non-singular kernels have been created to model dissipative phenomena that cannot be adequately modeled by fractional differential operators based on singular kernels.

The goal of this Special Issue is to report the latest progress in fractional calculus oriented towards scientific and engineering problems in light of the classic (power-law) and modern trends (with non-singular kernels), thus demonstrating what can be done and how with this remarkable area of mathematical modeling.

The collection includes six articles, which will be briefly outlined.

The article Application of a Machine Learning Algorithm for Evaluation of Stiff Fractional Modeling of Polytropic Gas Spheres and Electric Circuits, by Fawaz Khaled Alarfaj et al. [1], demonstrates numerical solutions obtained by the designed technique contrasted with the multi-step reproducing kernel Hilbert space method (MS-RKM), Laplace transformation method (LTM), and Padé approximations.

The extensive graphical and statistical analysis of the designed technique showed that the DNN-LM algorithm is dependable and facilitates the examination of higher-order nonlinear complex problems due to the flexibility of the DNN architecture and the effectiveness of the optimization procedure.

Fractional Stefan Problem Solving by the Alternating Phase Truncation Method, by Chmielowska and Slota [2], addresses approximate temperature distributions in a domain with a moving boundary between the solid and the liquid phase. The method consists of reducing the whole considered domain to a liquid phase by adding sufficient heat at each point of the solid and then, after solving the heat equation transforming to the enthalpy form in the obtained region, subtracting the heat that has been added.

Zhao and Chen presented Unique Solutions for Sequential Fractional Differential Equations with Integral Multi-Point and Anti-Periodic Type Boundary Conditions [3], with an emphasis on the chaotic nature of a computational system related to the spread of disease into a specific environment involving a novel differential operator called the Atangana–Baleanu fractional derivative. To approximate the solutions of this fractional system, an efficient numerical method is adopted. The numerical method is an implicit approximate method that can provide very suitable numerical approximations for fractional problems due to symmetry. Symmetry is one of the distinguishing features of this technique compared to other methods in the literature.

The Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions, by Jocelyn Sabatier [4], presents an interesting point of view on the nature of fractional models described by fractional differential equations or pseudo-state space descriptions. Computation of the impulse response of a fractional model using the Cauchy method shows that they exhibit infinitely small and high time constants. This impulse response can be rewritten as a diffusive representation whose Fourier transform permits a representation of a fractional model by a diffusion equation in an infinite space domain. Fractional models can thus be viewed as doubly infinite dimensional models: infinite as distributed with distribution in an infinite domain. This infinite domain or the infinitely large time constants of the impulse response reveal a property intrinsic to fractional models: their infinite memory. Solutions to generate fractional behaviors without infinite memory are finally proposed.

Non-Local Kinetics: Revisiting and Updates Emphasizing Fractional Calculus Applications, by Jordan Hristov [5], encompass classical fractional kinetics based on the Continuum Time Random Walk diffusion model with the absence of stationary states, reviewing real-world problems from pharmacokinetics and modern material processing. Fractional allometry has been considered a potential area of application. The main focus in the analysis is on the memory functions in the convolution formulation, crossing from the classical power law to versions of the Mittag–Leffler function. The main focus is to revisit the non-local kinetic problems with an update on new issues relevant to new trends in fractional calculus.

Gunay et al., in A Fractional Approach to a Computational Eco-Epidemiological Model with Holling Type-II Functional Response [6], considered the significant combination of two research fields of computational biology and epidemiology. These problems mainly take ecological systems into account of the impact of epidemiological factors. In this paper, we examine the chaotic nature of a computational system related to the spread of disease into a specific environment involving a novel differential operator called the Atangana–Baleanu fractional derivative. To approximate the solutions of this fractional system, an efficient numerical method is adopted. The numerical method is an implicit approximate method that can provide very suitable numerical approximations for fractional problems due to symmetry.

We hope that this collection of articles will serve as a comprehensive source of information about modern trends in the development of fractional calculus and fractional modeling, as well as how the ideas concerning various phenomena are interpreted from a non-local point of view.