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The fractional exponential function is considered. General expansions in fractional powers are used to solve fractional population dynamics problems. Laguerre-type exponentials are also considered, and an application to Laguerre-type fractional logistic equation is shown.

We extended the classical Bernoulli and Euler numbers and polynomials to introduce the Laguerre-type Bernoulli and Euler numbers and related fractional polynomials. The case of fractional Bernoulli and Euler polynomials and numbers has already been considered in a previous paper of which this article is a further generalization. Furthermore, we exploited the Laguerre-type fractional exponentials to define a generalized form of the classical Laplace transform.

Fractional derivative operators are finding applications in a wide variety of fields with their ability to better model certain phenomena exhibiting spatial and temporal nonlocality. One area in which these operators are applicable is in the field of electromagnetism, thereby modelling transient wave propagation in complex media. To apply fractional derivative operators to electromagnetic problems, the operator must adhere to certain principles, like the trigonometric functions invariance property. The Grünwald–Letnikov and Marchaud fractional derivative operators comply with these principles and therefore could be applied. The fractional derivative arises when modelling frequency-dispersive dielectric media. The time-domain convolution integral in the relation between the electric displacement and the polarisation density, containing an empirical extension of the Debye model, is approximated directly. A common approach is to recursively update the convolution integral by approximating the time series by a truncated sum of decaying exponentials, with the coefficients found through means of optimisation or fitting. The finite-difference time-domain schemes using this approach have shown to be more computationally efficient compared to other approaches using auxiliary differential equation methods.

We review dielectric resonator antenna (DRA) designs. This review examines recent advancements across several categories, specifically focusing on their applicability in array configurations for millimeter-wave (mmW) bands, particularly in the context of 5G and beyond 5G applications. Notably, the off-chip DRA designs, including in-substrate and compact DRAs, have gained prominence in recent years. This surge in popularity can be attributed to the rapid development of cost-effective multilayer laminate manufacturing techniques, such as printed circuit boards (PCBs) and low-temperature co-fired ceramic (LTCC). Furthermore, there is a growing demand for DRAs with beam-steering, dual-band functions, and on-chip alignment availability, as they offer versatile alternatives to traditional lossy printed antennas. DRAs exhibit distinct advantages of lower conductive losses and greater flexibility in shapes and materials. We discuss and compare the performances of different DRA designs, considering their material usage, manufacturing feasibility, overall performance, and applications. By exploring the pros and cons of these diverse DRA designs, this review provides valuable insights for researchers in the field.

We use power series with rational exponents to find exact solutions to initial value problems for fractional differential equations. Certain problems that have been previously studied in the literature can be solved in a closed form, and approximate solutions are derived by constructing recursions for the relevant expansion coefficients.

Bernoulli and Euler numbers and polynomials are well known and find applications in various areas of mathematics, such as number theory, combinatorial mathematics, series expansions, and the theory of special functions. Using fractional exponential functions, we extend the classical Bernoulli and Euler numbers and polynomials to introduce their fractional-index-based types. This reveals a symmetry in relation to the classical numbers and polynomials. We demonstrate some examples of these generalized mathematical entities, which we derive using the computer algebra system Mathematica©.

An approach to compensate scan losses displayed by full-Azimuth multifacet phased arrays is proposed in this article. This approach is based on a constructive superposition of scanned beams from adjacent array panels resulting in enhanced composed coverage. Said constructive superposition is achieved by coordinating the insertion phase shifts applied at the array element level. Special attention is given to panel arrays arranged in equilateral triangle and square prisms. The benefits of the proposed beamforming approach are validated by full-wave simulations, as well as passive antenna measurements taken on physical prototypes. The achieved improvement in equivalent isotropically radiated power (EIRP) excessively compensates for the experienced scan loss and reaches 6 dB at the maximal scan angles. An additional advantage of the developed methodology consists in the mitigation of the composed beam squint at wide scan angles.

We approximate the solution of a generalized form of the Bagley–Torvik equation using Taylor’s expansions in fractional powers. Then, we study the fractional Laguerre-type logistic equation by considering the fractional exponential function and its Laguerre-type form. To verify our findings, we conduct numerical tests using the computer algebra program Mathematica.

Bell’s polynomials are used in many different fields of mathematics, ranging from number theory to operator theory. This paper shows a relevant application in probability theory aimed at computing the moments of generalized Gaussian distributions. To this end, a table containing the first values of the complete Bell’s polynomials is provided. Furthermore, a dedicated code for approximating the moments of the general distributions in terms of complete Bell’s polynomials is detailed. Several test cases concerning different nested functions are discussed.

Bell's polynomials have been used in many different fields, ranging from number theory to operators theoryIn this article we show a method to compute the Laplace Transform (LT) of nested analytic functions. To this aim, we provide a table of the first few values of the complete Bell's polynomials, which are then used to evaluate the LT of composite exponential functions. Furthermore a code for approximating the Laplace Transform of general analytic composite functions is created and presented. A graphical verification of the proposed technique is illustrated in the last section.

A wideband high-gain circularly polarized (CP) shaped dielectric horn-lens antenna (SDHLA) operating in the frequency band between 6.7 and 18.2 GHz [fractional impedance bandwidth (FIBW) of 92.4%] with a 3-dB axial-ratio in the frequency range from 8.1 to 16.3 GHz [fractional axial-ratio bandwidth (FARBW) of 67.2%], is presented. The antenna, composed of a suitably shaped dielectric horn, integrated with a super-ellipsoidal-axicon dielectric lens made out of stacked thin dielectric disks, is mounted on a printed circuit board (PCB) where a microstrip line terminated with a wideband radial stub is used to excite a S-shaped slot through which the circular polarization is achieved. Parameterized 3D Lamé curves, describing the horn and lens profile, are used to optimize the antenna design. The antenna features a peak realized gain exceeding 13.1 dBi that is beneficial in a variety of applications, such as digital video broadcasting (DVB), remote sensing, weather monitoring, satellite communications, and air traffic control. The full-wave electromagnetic solver CST Studio Suite™, based on a locally conformal finite integration technique (FIT), was employed to design and characterize the antenna whose performances were found to be in good agreement with the experimental measurements.

Conformal array pattern synthesis represents a challenging problem for current state-of-the-art design techniques. To their detriment, these techniques typically suffer from a significant computational burden. A deterministic method for the synthesis of singly curved conformal antenna arrays based on the superformula and the auxiliary array pattern (AAP) formulation is presented. This method, being semianalytical in nature, is able to evaluate the location and excitation coefficients of each radiating element subject to different design constraints in a deterministic manner while utilizing multiple embedded element patterns. Therefore, this method is more computationally efficient than the state-of-the-art alternatives. The proposed synthesis technique has been validated through full-wave electromagnetic simulations and experimental measurements taken on physical demonstrators relevant to different applications within radar and communication systems. The obtained results show accurate agreement between simulations and measurements, requiring only seconds of computation on conventional hardware.