Search Results (25)

A non-homogeneous fractional differential equation with a variable coefficient involving a Caputo fractional derivative is considered. The equation is first transformed into an integral equation and then solved using the method of successive approximations. The obtained general solution involves a generalized Mittag–Leffler-type function and Meijer G-functions. Example solutions corresponding to particular choices of the non-homogeneous term are presented. As an application of the considered non-homogeneous equation, direct and inverse source problems are studied. The solutions are expressed in the form of series expansions using an orthogonal basis obtained through separation of variables. Illustrative examples for the direct and inverse problems are also presented for specific choices of the initial and final time data and the source function. Full article

We extend the classical Euler-type integral representations for the Appell functions $F_1$, $F_2$, and $F_3$, to the appropriate Kampé de Fériet functions by using integration against the Meijer–Nørlund G-function. In particular, these representations provide analytic continuation of the corresponding Kampé de Fériet functions. We further focus on the following two applications. First, we obtain sufficient conditions for complete monotonicity on the positive quadrant for three families of the Kampé de Fériet functions. These conditions can be expressed directly in terms of parameters and imply, among other things, joint log-convexity and related inequalities for partial derivatives of the Kampé de Fériet functions. Second, we show how known reduction and transformation formulas for the Appell and the generalized hypergeometric functions can be lifted to Kampé de Fériet functions by concatenating parameter arrays via the integral representations. This yields several reduction formulas, including extensions of some classical and new product identities. Further combining integration against the Meijer–Nørlund G-function with Slater’s double series transformation we obtain several exotic identities for infinite sums of the generalized hypergeometric functions. Full article

Addition theorems have been indispensable tools for the reduction of quantum transition amplitudes. They are normally utilized at the start of the process to move the angular dependence within plane waves, Coulomb potentials, and the like, into a sum over spherical harmonics that allows the angular integration to be carried out. These have historically been “two-range” addition theorems, characterized by the two-fold notation $r_{>}=Max[r_1,r_2]$ and $r_{<}=Min[r_1,r_2]$ and comprising a single infinite series. More recently, “one-range” addition theorems have been created that have no such piecewise notation, but at the cost of the introduction of another infinite series. We use a very different approach to derive an infinite set of addition theorems for Slater orbitals, hydrogenic and Hylleraas wave functions, and so on, that retain the one-range variable dependence but have, at worst, a finite second series rather than an infinite one. In addition, unlike previous addition theorems, they are applicable to more than one coordinate system. One of these addition theorems may also be used for Yukawa-like functions that may appear late in the reduction of amplitude integrals, and we show its utility for an integral that has stubbornly defied reduction to analytic form for nearly sixty years. Finally, we craft indefinite integrals of 15 half-integer Macdonald functions multiplied by (inverse) powers and negative exponentials containing squares of the integration variable. Full article

The Riemann-Liuoville fractional integrals are the simplest and most popular operators of the classical fractional calculus. But their variants, the Erdélyi-Kober operators of fractional integration, have many more applications due to the freedom to choose the additional (three) parameters. We introduce and study a generalization of the Erdélyi-Kober and Riemann-Liuoville fractional integrals, where the elementary kernel function is replaced by a suitably chosen $I_{1,1}^{1,0}$-function. The I-functions introduced by Rathie in 1997 are generalized hypergeometric functions extending the Fox H-functions and the Meijer G-functions. Note that till recently this new class of special functions has not been popular because of their too complicated structure involving fractional powers of the Gamma functions and their multi-valued behavior. However, the I-functions happened to arise not only for the needs of statistical physics, but also since they included important special functions in mathematics that were not covered by the H- and G-functions. In our previous works, as Kiryakova and Paneva-Konovska, we have shown the relations of such functions, among which are the Mittag-Leffler and Le Roy type, their multi-index variants, and others related to fractional calculus, to the I-functions. Here, we propose a new theory of generalization of the Erdélyi-Kober fractional integrals, based on the use of an I-function as a kernel. This will serve next as a base to extend our generalized multi-order fractional calculus with operators involving $I_{m,m}^{m,0}$. In this paper, we also evaluate the images under these new generalized fractional integrals of special functions of very general form. Finally, in the Conclusion section, we comment on some earlier discussions on the relations between fractal geometry and fractional calculus, nowadays already without any doubts. Full article

In this paper, motivated by the contributions of several researchers, including (for example) Riemann, Kronecker, Lerch, Ramanujan, and Glaisher on some special integrals known as Mordell’s integrals, we consider the problem of the evaluation of some generalized Mordell integrals by applying the Mellin-Barnes-type contour integration. In particular, we have evaluated the following integrals: $I_1:={\int }_0^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t\mathrm{and}{I}_2:={\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t$ for real or complex parameters $a,b,c$, and g with $\Re \left(a\right)>0$ in terms of Meijer’s G-function and the generalized hypergeometric function _pF_q. Full article

This study investigates the performance of a mixed dual-hop free-space optical/underwater wireless optical communication (FSO-UWOC) system employing a decode-and-forward (DF) relay protocol, particularly under a comprehensive hybrid channel fading model. The FSO link is assumed to experience Gamma–Gamma atmospheric turbulence fading, combined with air path loss and pointing errors. Meanwhile, the UWOC link is modeled with generalized Gamma distribution (GGD) oceanic turbulence fading, along with underwater path loss and pointing errors. Based on the proposed hybrid channel fading model, closed-form expressions for the average outage probability (OP) and average bit error rate (BER) of the mixed dual-hop system are derived using the higher transcendental Meijer-G function. Similarly, the closed-form expression for the average ergodic capacity of the mixed relay system is obtained via the bivariate Fox-H function. Additionally, asymptotic performance analyses for the average outage probability and BER under high signal-to-noise ratio (SNR) conditions are provided. Finally, Monte Carlo simulations are conducted to validate the accuracy of the derived theoretical expressions and to illustrate the effects of key system parameters on the performance of the mixed relay FSO-UWOC system. Full article

On the one hand, convex functions are important to derive rigorous convergence rates, and on the other, synchronous functions are significant to solve statistical problems using Chebyshev inequalities. Therefore, fractional integral inequalities involving such functions play a crucial role in creating new models and methods. Although a large class of fractional operators have been used to establish inequalities, nevertheless, these operators having the Fox-H and the Meijer-G functions in their kernel have been applied to establish fractional integral inequalities for such important classes of functions. Taking motivation from these facts, the primary objective of this work is to develop fractional inequalities involving the Fox-H function for convex and synchronous functions. Since the Fox-H function generalizes several important special functions of fractional calculus, our results are significant to innovate the existing literature. The inventive features of these functions compel researchers to formulate deeper results involving them. Therefore, compared with the ongoing research in this field, our results are general enough to yield novel and inventive fractional inequalities. For instance, new inequalities involving the Meijer-G function are obtained as the special cases of these outcomes, and certain generalizations of Chebyshev inequality are also included in this article. Full article

The Low Earth Orbit (LEO) small satellites are extensively used for global connectivity to enable services in underpopulated, remote or underdeveloped areas. Their inherent broadcast nature exposes LEO–terrestrial communication links to severe security threats, which always reveal new challenges. The secrecy performance of the satellite-to-ground user link in the presence of a ground eavesdropper is studied in this paper. We observe both scenarios of the eavesdropper’s channel state information (CSI) being known or unknown to the satellite. Throughout the analysis, we consider that locations of the intended and unauthorized user are both arbitrary in the satellite’s footprint. On the other hand, we analyze the case when the user is in the center of the satellite’s central beam. In order to achieve realistic physical layer security features of the system, the satellite channels are assumed to undergo Gamma-shadowed Ricean fading, where both line-of-site and scattering components are influenced by shadowing effect. In addition, some practical effects, such as satellite multi-beam pattern and free space loss, are considered in the analysis. Capitalizing on the aforementioned scenarios, we derive the novel analytical expressions for the average secrecy capacity, secrecy outage probability, probability of non-zero secrecy capacity, and probability of intercept events in the form of Meijer’s G functions. In addition, novel asymptotic expressions are derived from previously mentioned metrics. Numerical results are presented to illustrate the effects of beam radius, satellite altitude, receivers’ position, as well as the interplay of the fading or/and shadowing impacts over main and wiretap channels on the system security. Analytical results are confirmed by Monte Carlo simulations. Full article

In this paper, we analyze the confidentiality of a hybrid radio frequency (RF)/free-space optical (FSO) system with regard to physical layer security (PLS). In this system, signals are transmitted between the source and destination using RF and FSO links, with the destination employing the maximal-ratio combining (MRC) scheme. A non-cooperative target (NCT) is assumed to have eavesdropping capabilities for RF and FSO signals in both collusion and non-collusion strategies. The Nakagami-m distribution models fading RF links, while FSO links are characterized by the Málaga ($\mathcal{M}$) distribution. Exact closed-form expressions for the system’s secrecy outage probability (SOP) and effective secrecy throughput (EST) are derived based on the generalized Meijer G-function with two variables. Asymptotic expressions for the SOP are also obtained under high-signal-to-noise-ratio (SNR) regimes. These conclusions are validated through Monte Carlo simulations. The superiority of the hybrid RF/FSO system in improving the communication security of a single link is confirmed in its comparison with conventional means of RF communication. Full article

Needed for cosmological and stellar nucleosynthesis, we are studying the closed-form analytic evaluation of thermonuclear reaction rates. In this context, we undertake a comprehensive analysis of three largely distinct velocity distributions, namely the Maxwell–Boltzmann distribution, the pathway distribution, and the Mittag-Leffler distribution. Moreover, a natural generalization of the Maxwell–Boltzmann velocity distribution is discussed. Furthermore, an explicit evaluation of the reaction rate integral in the high-energy cut-off case is carried out. Generalized special functions of mathematical physics like Meijer’s G-function and Fox’s H-functions and their utilization in mathematical physics are the prime focus of this paper. Full article

We apply known special functions from the literature (and these include the Fox $\mathbb{H}–$function, the exponential function, the Mittag-Leffler function, the Gauss Hypergeometric function, the Wright function, the $\mathbb{G}–$function, the Fox–Wright function and the Meijer $\mathbb{G}–$function) and fuzzy sets and distributions to construct a new class of control functions to consider a novel notion of stability to a fractional-order system and the qualified approximation of its solution. This new concept of stability facilitates the obtention of diverse approximations based on the various special functions that are initially chosen and also allows us to investigate maximal stability, so, as a result, enables us to obtain an optimal solution. In particular, in this paper, we use different tools and methods like the Gronwall inequality, the Laplace transform, the approximations of the Mittag-Leffler functions, delayed trigonometric matrices, the alternative fixed point method, and the variation of constants method to establish our results and theory. Full article

We extend previous research to derive three additional M-1-dimensional integral representations over the interval $[0,1]$. The prior version covered the interval $[0,\infty ]$. This extension applies to products of M Slater orbitals, since they (and wave functions derived from them) appear in quantum transition amplitudes. It enables the magnitudes of coordinate vector differences (square roots of polynomials) $|{\mathbf{x}}_1-{\mathbf{x}}_2|=\sqrt{x_1^2-2x_1{x}_{2}cos\theta +x_2^2}$ to be shifted from disjoint products of functions into a single quadratic form, allowing for the completion of its square. The M-1-dimensional integral representations of M Slater orbitals that both this extension and the prior version introduce provide alternatives to Fourier transforms and are much more compact. The latter introduce a 3M-dimensional momentum integral for M products of Slater orbitals (in M separate denominators), followed in many cases by another set of M-1-dimensional integral representations to combine those denominators into one denominator having a single (momentum) quadratic form. The current and prior methods are also slightly more compact than Gaussian transforms that introduce an M-dimensional integral for products of M Slater orbitals while simultaneously moving them into a single (spatial) quadratic form in a common exponential. One may also use addition theorems for extracting the angular variables or even direct integration at times. Each method has its strengths and weaknesses. We found that these M-1-dimensional integral representations over the interval $[0,1]$ are numerically stable, as was the prior version, having integrals running over the interval $[0,\infty ]$, and one does not need to test for a sufficiently large upper integration limit as one does for the latter approach. For analytical reductions of integrals arising from any of the three, however, there is the possible drawback for large M of there being fewer tabled integrals over $[0,1]$ than over $[0,\infty ]$. In particular, the results of both prior and current representations have integration variables residing within square roots asarguments of Macdonald functions. In a number of cases, these can be converted to Meijer G-functions whose arguments have the form $(a{x}^2+bx+c)/x$, for which a single tabled integral exists for the integrals from running over the interval $[0,\infty ]$ of the prior paper, and from which other forms can be found using the techniques given therein. This is not so for integral representations over the interval $[0,1]$. Finally, we introduce a fourth integral representation that is not easily generalizable to large M but may well provide a bridge for finding the requisite integrals for such Meijer G-functions over $[0,1]$. Full article

In the survey Kiryakova: “A Guide to Special Functions in Fractional Calculus” (published in this same journal in 2021) we proposed an overview of this huge class of special functions, including the Fox H-functions, the Fox–Wright generalized hypergeometric functions ${}_p{\Psi }_q$ and a large number of their representatives. Among these, the Mittag-Leffler-type functions are the most popular and frequently used in fractional calculus. Naturally, these also include all “Classical Special Functions” of the class of the Meijer’s G- and ${}_p{F}_q$-functions, orthogonal polynomials and many elementary functions. However, it so happened that almost simultaneously with the appearance of the Mittag-Leffler function, another “fractionalized” variant of the exponential function was introduced by Le Roy, and in recent years, several authors have extended this special function and mentioned its applications. Then, we introduced a general class of so-called (multi-index) Le Roy-type functions, and observed that they fall in an “Extended Class of SF of FC”. This includes the I-functions of Rathie and, in particular, the $\overline{H}$-functions of Inayat-Hussain, studied also by Buschman and Srivastava and by other authors. These functions initially arose in the theory of the Feynman integrals in statistical physics, but also include some important special functions that are well known in math, like the polylogarithms, Riemann Zeta functions, some famous polynomials and number sequences, etc. The I- and $\overline{H}$-functions are introduced by Mellin–Barnes-type integral representations involving multi-valued fractional order powers of $\Gamma$-functions with a lot of singularities that are branch points. Here, we present briefly some preliminaries on the theory of these functions, and then our ideas and results as to how the considered Le Roy-type functions can be presented in their terms. Next, we also introduce Gelfond–Leontiev generalized operators of differentiation and integration for which the Le Roy-type functions are eigenfunctions. As shown, these “generalized integrations” can be extended as kinds of generalized operators of fractional integration, and are also compositions of “Le Roy type” Erdélyi–Kober integrals. A close analogy appears with the Generalized Fractional Calculus with H- and G-kernel functions, thus leading the way to its further development. Since the theory of the I- and $\overline{H}$-functions still needs clarification of some details, we consider this work as a “Discussion Survey” and also provide a list of open problems. Full article

To improve the performance of fee-space optical communication systems, this paper analyzes the performance of a relay-aided hybrid fee-space optical (FSO)/radio frequency (RF) cooperation system based on a selective combination and decoding forward transmission scheme. In this system, the FSO sub-link experienced Málaga turbulence with pointing errors and the RF sub-link suffered Nakagami-m fading. Firstly, the probability density function (PDF) and cumulative distribution function (CDF) of the end-to-end output signal-to-noise ratio (SNR) of the relay-aided hybrid FSO/RF system are derived. Then, using the extended generalized bivariate Meijer’s G-function (EGBMGF) and the approximate analytical formula of the generalized Gauss–Laguerre integral, mathematical expressions of the end-to-end average bit error rate (ABER) and outage probability of the relay-aided hybrid FSO/RF system with different subcarrier intensity modulation and different detection schemes are derived. Through a simulation analysis of the system, the results show that compared with the other three modulation technologies, the hybrid FSO/RF direct link and relay-aided hybrid FSO/RF system with coherent binary phase shift keying (CBPSK) modulation have the best bit error performance. Compared with direct detection, the hybrid direct link and relay-aided hybrid system with coherent detection can significantly improve the communication performance. Increasing the RF fading parameter m can further improve the bit error and outage performance of the hybrid direct link and relay-aided hybrid system; the hybrid direct link can significantly mitigate the degradation of communication performance in the FSO system caused by pointing errors, and the relay-aided hybrid system can further improve the communication performance; under weak turbulence conditions, the impact of pointing errors on the performance of the relay-aided hybrid system can even be ignored. The greater the total number of paths in the relay-aided hybrid system, the better the communication performance of the system; however, the more hops, the worse the performance of the system. The outage probability of the hybrid direct link and relay-aided hybrid system are very sensitive to the decision threshold, and the larger the decision threshold, the worse the outage performance. The transmission distance of different hybrid direct links has little impact on the performance of hybrid direct links and relay-aided hybrid systems. Improving the signal-to-noise ratio of RF sub-links significantly improves the performance of hybrid direct links and relay-aided hybrid systems. Full article

This paper proposes a generalized space time block coded (GSTBC) enhanced fully optical generalized spatial modulation (EFOGSM) system based on Málaga (M) turbulent channel. GSTBC-EFOGSM adopts the hybrid concept of generalized space time block coded and optical spatial modulation to further utilize the high transmission rate of EFOGSM and the diversity advantage of GSTBC in free space optical (FSO) communication systems. Considering the combined effects of path loss, pointing error and atmospheric turbulence, the Meijer G function is used to derive the closed-form expression for the average bit error rate (ABER) of GSTBC-EFOGSM. Then, the ABER performance, data transmission rate, energy efficiency and computational complexity at the receiver of GSTBC-EFOGSM are compared with other optical spatial modulation schemes by simulation. In addition, the effects of key factors, such as data transmission rate, encoding ratio, number of photodetectors and modulation order, on the ABER performance of the system are also analyzed via simulation. Monte Carlo (MC) simulation is used to verify the correctness of the numerical simulation. The simulation results show that the GSTBC-EFOGSM system has better ABER performance and good performance gain. Full article