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In the paper, we consider the P H 2 -distribution, which is a particular case of the P H -distribution. In other words, The first service phase is exponentially distributed, and the service rate is μ. After the first service phase, the customer can to go away with probability p or continue the service with probability (1 − p) and service rate μ ′ . We study an analysis of an M / P H 2 / 1 queue model with catastrophes, which is regarded as a generalization of an M / M / 1 queue model with catastrophes. Whenever a catastrophe happens, all customers will be cleaned up immediately, and the queuing system is empty. The customers arrive at the queuing system based on a Poisson process, and the total service duration has two phases. Transient probabilities and steady-state probabilities of this queuing system are considered using practical applications of the modified Bessel function of the first kind, the Laplace transform, and probability-generating function techniques. Moreover, some important performance measures are obtained in the system. Finally, numerical illustrations are used to discuss the system's behavior, and conclusions and future directions of the model are given. [ABSTRACT FROM AUTHOR]

In this article, we obtain transformation formulas analogous to the identity of Ramanujan, Hardy and Littlewood in the setting of primitive Maass cusp form over the congruence subgroup $\Gamma _0(N)$ and also provide an equivalent criterion of the grand Riemann hypothesis for the $L$ -function associated with the primitive Maass cusp form over $\Gamma _0(N)$. [ABSTRACT FROM AUTHOR]

Future 6G communications are envisioned to enable a large catalogue of pioneering applications. These will range from networked Cyber-Physical Systems to edge computing devices, establishing real-time feedback control loops critical for managing Industry 5.0 deployments, digital agriculture systems, and essential infrastructures. The provision of extensive machine-type communications through 6G will render many of these innovative systems autonomous and unsupervised. While full automation will enhance industrial efficiency significantly, it concurrently introduces new cyber risks and vulnerabilities. In particular, unattended systems are highly susceptible to trust issues: malicious nodes and false information can be easily introduced into control loops. Additionally, Denial-of-Service attacks can be executed by inundating the network with valueless noise. Current anomaly detection schemes require the entire transformation of the control software to integrate new steps and can only mitigate anomalies that conform to predefined mathematical models. Solutions based on an exhaustive data collection to detect anomalies are precise but extremely slow. Standard models, with their limited understanding of mobile networks, can achieve precision rates no higher than 75%. Therefore, more general and transversal protection mechanisms are needed to detect malicious behaviors transparently. This paper introduces a probabilistic trust model and control algorithm designed to address this gap. The model determines the probability of any node to be trustworthy. Communication channels are pruned for those nodes whose probability is below a given threshold. The trust control algorithm comprises three primary phases, which feed the model with three different probabilities, which are weighted and combined. Initially, anomalous nodes are identified using Gaussian mixture models and clustering technologies. Next, traffic patterns are studied using digital Bessel functions and the functional scalar product. Finally, the information coherence and content are analyzed. The noise content and abnormal information sequences are detected using a Volterra filter and a bank of Finite Impulse Response filters. An experimental validation based on simulation tools and environments was carried out. Results show the proposed solution can successfully detect up to 92% of malicious data injection attacks. [ABSTRACT FROM AUTHOR]

This paper focuses on integrals and Borel transforms concerning λ$$ \lambda $$‐Bessel and λ$$ \lambda $$‐Tricomi functions, employing an umbral perspective as a central approach. Differintegral techniques, presently utilized in calculus, offer a rich reserve of tools that can be utilized across a broad spectrum of problems encompassing the realm of special functions and beyond. Employing integral transforms akin to the Borel type and the related formalism proves to be a potent approach, facilitating a connection between umbral and operational methodologies. By integrating these perspectives, we establish a novel and effective method for deriving integrals of hybrid special functions and achieving the summation of their corresponding generating functions. [ABSTRACT FROM AUTHOR]

Suppose that A 1 is a class of analytic functions f : D = { z ∈ C : | z | < 1 } → C with normalization f (0) = 1 . Consider two functions P l (z) = 1 + z and Φ N e (z) = 1 + z − z 3 / 3 , which map the boundary of D to a cusp of lemniscate and to a twi-cusped kidney-shaped nephroid curve in the right half plane, respectively. In this article, we aim to construct functions f ∈ A 0 for which (i) f (D) ⊂ P l (D) ∩ Φ N e (D) (ii) f (D) ⊂ P l (D) , but f (D) ⊄ Φ N e (D) (iii) f (D) ⊂ Φ N e (D) , but f (D) ⊄ P l (D) . We validate the results graphically and analytically. To prove the results analytically, we use the concept of subordination. In this process, we establish the connection lemniscate (and nephroid) domain and functions, including g α (z) : = 1 + α z 2 , | α | ≤ 1 , the polynomial g α , β (z) : = 1 + α z + β z 3 , α , β ∈ R , as well as Lerch's transcendent function, Incomplete gamma function, Bessel and Modified Bessel functions, and confluent and generalized hypergeometric functions. [ABSTRACT FROM AUTHOR]

The Bessel function and its various generalizations have extensively been studied in various branches of applied mathematics and theoretical physics, including the Geometric Function Theory. In this paper, we study basic characteristics of Bessel functions of order μ and degree ν . Among the results that we investigate are the results giving the characteristic properties of univalence, convexity and starlikeness. We further investigate the conditions under which the function L μ , ν are strongly convex and strongly starlike. Several corollaries are also mentioned depicting the usefulness of the main results, one of the Corollary providing improvement in a result for normalized Bessel function. [ABSTRACT FROM AUTHOR]

Using the Hadamard factorization, the exponential radii of starlikeness and convexity for various special functions like Wright function, Lommel function, Struve function, Ramanujan type entire function, cross product and product of Bessel function have been investigated. For certain ranges of the parameters appearing in these special functions, the precise values of the exponential radii of starlikeness and convexity are calculated as the solutions of transcendental equations. The interlacing property of the zeros of special functions and their derivatives is the fundamental technique utilized to demonstrate these results. [ABSTRACT FROM AUTHOR]

It is well-known that if one applies Kirchhoff migration (KM) to identify small objects when their values of magnetic permeabilities differ from those of the background (or transverse electric polarization), their location and outline shape cannot be satisfactorily retrieved because rings of large magnitudes centered at the location of objects appear in the imaging results. Fortunately, it is possible to recognize the existence and approximated location of objects in the 2D Fresnel dataset through the traditional KM, but no theoretical explanation for this phenomenon has been verified. Here we show that the imaging function of KM when tested on the Fresnel dataset can be expressed as squared zero-order and first-order Bessel functions and as an infinite series of Bessel functions of integer order greater than two. We also explain why the existence and approximate location of objects can be identified. This theoretical result is supported by numerical simulations on synthetic and experimental data.

In this paper, we propose a novel approach to the fractional power of the Laplace-Bessel operator $ \Delta_{\nu} $, defined as$ \Delta_{\nu} = \sum\limits_{i = 1}^{n}\frac{\partial^2}{\partial x_{i}^2} + \frac{\nu_i}{x_{i}}\frac{\partial}{\partial x_{i}}, \quad \nu_i\geq 0. $The fractional power of this operator is introduced as a pseudo-differential operator through the multi-dimensional Bessel transform. Our primary contributions encompass a normalized singular integral representation, Bochner subordination, and intertwining relations.

In this paper, taking into account the intriguing recent results of Rabotnov functions, Poisson functions, Bessel functions and Wright functions, we consider a new comprehensive subclass O μ (Δ 1 , Δ 2 , Δ 3 , Δ 4) of univalent functions defined in the unit disk Λ = { τ ∈ C : τ < 1 } . More specifically, we investigate some sufficient conditions for Rabotnov functions, Poisson functions, Bessel functions and Wright functions to be in this subclass. Some corollaries of our main results are given. The novelty and the advantage of this research could inspire researchers of further studies to find new sufficient conditions to be in the subclass O μ (Δ 1 , Δ 2 , Δ 3 , Δ 4) not only for the aforementioned special functions but for different types of special functions, especially for hypergeometric functions, Dini functions, Sturve functions and others. [ABSTRACT FROM AUTHOR]

We obtain an accurate analytic approximation for the Bessel function J 2 (x) using an improved multipoint quasirational approximation technique (MPQA). This new approximation is valid for all real values of the variable x, with a maximum absolute error of approximately 0.009. These errors have been analyzed in the interval from x = 0 to x = 1000 , and we have found that the absolute errors for large x decrease logarithmically. The values of x at which the zeros of the exact function J 2 (x) and the approximated function J ˜ 2 (x) occur are also provided, exhibiting very small relative errors. The largest relative error is for the second zero, with ε rel = 0.0004 , and the relative errors continuously decrease, reaching 0.0001 for the eleventh zero. The procedure to obtain this analytic approximation involves constructing a bridge function that connects the power series with the asymptotic approximation. This is achieved by using rational functions combined with other elementary functions, such as trigonometric and fractional power functions. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel approach to the fractional power of the Laplace-Bessel operator Δv, defined as... The fractional power of this operator is introduced as a pseudo-differential operator through the multidimensional Bessel transform. Our primary contributions encompass a normalized singular integral representation, Bochner subordination, and intertwining relations. [ABSTRACT FROM AUTHOR]

AbstractThis research reports an analytical and numerical study of thermo-solutal convection in pulsating flows between two coaxial cylinders under a pressure gradient and subject to Dirichlet-type boundary conditions for temperature and concentration along the vertical walls of the annulus. The numerical procedure used in this work is mainly based on the solution of the momentum equations and coupled with the energy and concentration equations. The finite difference method is adopted to solve governing equations using the implicit Crank-Nicolson time marching scheme in the fluid domain and the alternating direction implicit method inside the inner cylinder. The effect of the various parameters such as the fluid-solid conductivity ratio, thermal diffusivity ratio, buoyancy ratio, Lewis number, Richardson number, aspect ratio, and radius ratio on the heat and mass transfer characteristics and the flow structure is presented and discussed. The governing equations are solved analytically after separating into a steady case and an oscillatory case, where an analytical solution for velocity profile, temperature, and concentration distributions is obtained in terms of Bessel’s functions, while the analytical solutions of temperature and concentration profiles are keeping symmetry with the increase of the radius ratio. The results show that the analytical profile can reproduce the three fields obtained numerically and thus for the whole range of parameters. [ABSTRACT FROM AUTHOR]

In this work, a nonlocal boundary value problem for a degenerate equation of elliptic type with a singular coefficient and a spectral parameter is studied in a vertical half-strip. By applying the Hankel transformation and the method of separation of variables, a solution to the nonlocal boundary value problem is obtained in explicit form. [ABSTRACT FROM AUTHOR]

We study integrals containing gamma functions and derive a lot of new relations involving hypergeometric and other special functions. [ABSTRACT FROM AUTHOR]

Abstract. In this paper, we establish some analogues of the Paley-Wiener theorem for the Hankel type transform of half-integer order less than -1 in weighted L²-spaces. These Paley-Wiener-type theorems give a description of the class of even entire functions of exponential type σ ≤ 1 under this transformation in terms of solutions from the corresponding spaces of some differential equations. [ABSTRACT FROM AUTHOR]

In this article, we propose an efficient hybrid method to calculate the highly oscillatory Bessel integral ∫ 0 1 f (x) x - τ J m (ω x γ) d x with the Cauchy type singular point, where 0 < τ < 1 , m ≥ 0 , 2 γ ∈ N +. The hybrid method is established by combining the complex integration method with the Clenshaw– Curtis– Filon– type method. Based on the special transformation of the integrand and the additivity of the integration interval, we convert the integral into three integrals. The explicit formula of the first one is expressed in terms of the Meijer G function. The second is computed by using the complex integration method and the Gauss– Laguerre quadrature rule. For the third, we adopt the Clenshaw– Curtis– Filon– type method to obtain the quadrature formula. In particular, the important recursive relationship of the required modified moments is derived by utilizing the Bessel equation and the properties of Chebyshev polynomials. Importantly, the strict error analysis is performed by a large amount of theoretical analysis. Our proposed methods only require a few nodes and interpolation multiplicities to achieve very high accuracy. Finally, numerical examples are provided to verify the validity of our theoretical analysis and the accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

The problem of the real-time microwave imaging of small, moving objects from a scattering matrix without diagonal elements, whose elements are measured scattering parameters, is considered herein. An imaging algorithm based on a Kirchhoff migration operated at single frequency is designed, and its mathematical structure is investigated by establishing a relationship with an infinite series of Bessel functions of integer order and antenna configuration. This is based on the application of the Born approximation to the scattering parameters of small objects. The structure explains the reason for the detection of moving objects via a designed imaging function and supplies some of its properties. To demonstrate the strengths and weaknesses of the proposed algorithm, various simulations with realdata are conducted. [ABSTRACT FROM AUTHOR]

Under study is the Tricomi–Neumann problem for a three-dimensional mixed-type equation with three singular coefficients in a mixed domain consisting of a quarter of a cylinder and a triangular straight prism. We prove the unique solvability of the problem in the class of regular solutions by using the separation of variables in the hyperbolic part of the mixed domain, which yields the eigenvalue problems for one-dimensional and two-dimensional equations. Finding the eigenfunctions of the problems, we use the formula of the solution of the Cauchy–Goursat problem to construct a solution to the two-dimensional problem. In result, we find the solutions to eigenvalue problems for the three-dimensional equation in the hyperbolic part. Using the eigenfunctions and the gluing condition, we derive a nonlocal problem in the elliptic part of the mixed domain. To solve the problem in the elliptic part, we reformulate the problem in the cylindrical coordinate system and separating the variables leads to the eigenvalue problems for two ordinary differential equations. We prove a uniqueness theorem by using the completeness property of the systems of eigenfunctions of these problems and construct the solution to the problem as the sum of a double series. Justifying the uniform convergence of the series relies on some asymptotic estimates for the Bessel functions of the real and imaginary arguments. These estimates for each summand of the series made it possible to prove the convergence of the series and its derivatives up to the second order, as well as establish the existence theorem in the class of regular solutions. [ABSTRACT FROM AUTHOR]

Sums of the form are considered where are special functions of hypergeometric type. Such sums involving Bessel, Struve, incomplete gamma functions, and Laguerre, Hermite, Jacobi, Legendre, and Chebyshev polynomials can be represented in terms of the Kampé de Fériet and generalized Horn hypergeometric functions of two variables. [ABSTRACT FROM AUTHOR]

The partial sums of the series expansion of the Bessel function Jα of the first kind provide lower and upper bounds on the positive real axis for order α ≥-1=2. [ABSTRACT FROM AUTHOR]

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The Dirichlet problem for a multidimensional elliptic system in the half-space is considered. With the help of the Fourier transform, the problem of the solvability of the problem is reduced to the study of a second-order partial differential equation. [ABSTRACT FROM AUTHOR]

The deterioration of winding defects is one of the important causes of power transformer fires and even explosion failures. The change of leakage magnetic field distribution is the most direct response to winding defects. Currently there are few sensors suitable for online measurement of the internal magnetic field of transformers. Based on the Faraday magneto-optical effect, a magnetic field sensor with wide range and high sensitivity is proposed in this paper, which is suitable for the interior use of transformers. The straight-through optical structure with interior polarizer is adopted, and the sensor has a measurement range of 1.5 T and a sensitivity of 1 mT. It also possesses a small size, with a length of about 30 mm after encapsulation. The influence mechanism of vibration and temperature is revealed through theoretical analysis and numerical simulation. It is proposed to filter out the interference of vibration by characteristic frequency analysis and to compensate for temperature by a two-probe structure. An anti-interference test verifies the effectiveness of this method, and it can reduce the error from 80.56% to 2.63% under the combined interference of vibration and temperature.

In the paper, we consider the PH2-distribution, which is a particular case of the PH-distribution. In other words, The first service phase is exponentially distributed, and the service rate is μ. After the first service phase, the customer can to go away with probability p or continue the service with probability (1−p) and service rate μ′. We study an analysis of an M/PH2/1 queue model with catastrophes, which is regarded as a generalization of an M/M/1 queue model with catastrophes. Whenever a catastrophe happens, all customers will be cleaned up immediately, and the queuing system is empty. The customers arrive at the queuing system based on a Poisson process, and the total service duration has two phases. Transient probabilities and steady-state probabilities of this queuing system are considered using practical applications of the modified Bessel function of the first kind, the Laplace transform, and probability-generating function techniques. Moreover, some important performance measures are obtained in the system. Finally, numerical illustrations are used to discuss the system’s behavior, and conclusions and future directions of the model are given.

Suppose that A1 is a class of analytic functions f:D={z∈C:|z|<1}→C with normalization f(0)=1. Consider two functions Pl(z)=1+z and ΦNe(z)=1+z−z3/3, which map the boundary of D to a cusp of lemniscate and to a twi-cusped kidney-shaped nephroid curve in the right half plane, respectively. In this article, we aim to construct functions f∈A0 for which (i) f(D)⊂Pl(D)∩ΦNe(D) (ii) f(D)⊂Pl(D), but f(D)⊄ΦNe(D) (iii) f(D)⊂ΦNe(D), but f(D)⊄Pl(D). We validate the results graphically and analytically. To prove the results analytically, we use the concept of subordination. In this process, we establish the connection lemniscate (and nephroid) domain and functions, including gα(z):=1+αz2, |α|≤1, the polynomial gα,β(z):=1+αz+βz3, α,β∈R, as well as Lerch’s transcendent function, Incomplete gamma function, Bessel and Modified Bessel functions, and confluent and generalized hypergeometric functions.

We consider Bernstein's inequality for the Riesz derivative of order of entire functions of exponential type in the uniform norm on the real line. For this operator, the corresponding interpolation formula is obtained; this formula has nonequidistant nodes. Using this formula, the sharp Bernstein inequality is obtained for all ; namely, the extremal entire function and the sharp constant are written out. [ABSTRACT FROM AUTHOR]

The two-parameter Wright special function is an interesting mathematical object that arises in the theory of the space and time-fractional diffusion equations. Moreover, many other special functions are particular instantiations of the Wright function. The article demonstrates finite representations of the Wright function in terms of sums of generalized hypergeometric functions, which in turn provide connections with the theory of the Gaussian, Airy, Bessel, and Error functions, etc. The main application of the presented results is envisioned in computer algebra for testing numerical algorithms for the evaluation of the Wright function. [ABSTRACT FROM AUTHOR]

The model of horizontal layered structure in elastic solids has a wide range of applications in our real-world scientific research. To explore the propagation characteristics of cylindrical Rayleigh waves in a horizontally layered structure, we decompose the displacement using the Helmholtz decomposition method in the cylindrical coordinate system in order to use potential functions to describe the displacement and stress of the medium. Following this, the paper employs the transfer matrix method to derive the dynamic equations of axisymmetric Rayleigh waves in a horizontally layered elastic solid. After that, the equations are solved to obtain the Rayleigh wave dispersion curves and displacement expressions with Bessel function under these conditions. Lastly, through the calculations of four representative numerical examples, this paper verifies the consistency of the algorithm in calculating cylindrical Rayleigh wave dispersion curves and its agreement with experimental measurements, while demonstrating the characteristics of cylindrical Rayleigh wave displacement. [ABSTRACT FROM AUTHOR]

Let L² ((0; 1); x4 dx) be the weighted Lebesgue space of all measurable functions f : (0; 1) → C, satisfying ∫0¹ t4 |f(t)|² dt < +∞. Let J−5/2 be the Bessel function of the first kind of index −5/2 and (ρk)k∈N be a sequence of distinct nonzero complex numbers. Necessary and sufficient conditions for the completeness of the system {ρk² √xρk J−5/2(xρk) : k ∈ N} in the space L² ((0; 1); x4 dx) are found in terms of an entire function with the set of zeros coinciding with the sequence (ρk)k∈N. In this case, we study an integral representation of some class E4,+ of even entire functions of exponential type σ ≤ 1. This complements similar results on approximation properties of the systems of Bessel functions of negative half-integer index less than −1, due to B. Vynnyts’kyi, V. Dilnyi, O. Shavala and the author. [ABSTRACT FROM AUTHOR]

In this paper, we consider the perturbed Hill equation on the entire axis. Using a transformation operator that preserves asymptotics at infinity, triangular representations of special solutions of this equation are found. Estimates for the representation kernels are obtained. [ABSTRACT FROM AUTHOR]

We obtain an accurate analytic approximation for the Bessel function J2(x) using an improved multipoint quasirational approximation technique (MPQA). This new approximation is valid for all real values of the variable x, with a maximum absolute error of approximately 0.009. These errors have been analyzed in the interval from x=0 to x=1000, and we have found that the absolute errors for large x decrease logarithmically. The values of x at which the zeros of the exact function J2(x) and the approximated function J˜2(x) occur are also provided, exhibiting very small relative errors. The largest relative error is for the second zero, with εrel=0.0004, and the relative errors continuously decrease, reaching 0.0001 for the eleventh zero. The procedure to obtain this analytic approximation involves constructing a bridge function that connects the power series with the asymptotic approximation. This is achieved by using rational functions combined with other elementary functions, such as trigonometric and fractional power functions.

In this paper, taking into account the intriguing recent results of Rabotnov functions, Poisson functions, Bessel functions and Wright functions, we consider a new comprehensive subclass Oμ(Δ1,Δ2,Δ3,Δ4) of univalent functions defined in the unit disk Λ={τ∈C:τ<1}. More specifically, we investigate some sufficient conditions for Rabotnov functions, Poisson functions, Bessel functions and Wright functions to be in this subclass. Some corollaries of our main results are given. The novelty and the advantage of this research could inspire researchers of further studies to find new sufficient conditions to be in the subclass Oμ(Δ1,Δ2,Δ3,Δ4) not only for the aforementioned special functions but for different types of special functions, especially for hypergeometric functions, Dini functions, Sturve functions and others.

The two-parameter Wright special function is an interesting mathematical object that arises in the theory of the space and time-fractional diffusion equations. Moreover, many other special functions are particular instantiations of the Wright function. The article demonstrates finite representations of the Wright function in terms of sums of generalized hypergeometric functions, which in turn provide connections with the theory of the Gaussian, Airy, Bessel, and Error functions, etc. The main application of the presented results is envisioned in computer algebra for testing numerical algorithms for the evaluation of the Wright function.

Mathematical functions, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a foundation of current science. In this paper, we find a new integral representation of the Whittaker function of the first kind and show a relevant summation formula for Kummer's confluent hypergeometric functions. We also perform the specifications of our identities to link to known and new results. [ABSTRACT FROM AUTHOR]