Confluent Hypergeometric Functions Research Articles

Approximations of the extended Graetz problem to quantify particulate matter transport and deposition in tubes

Two different analytical solutions of the extended Graetz problem are analyzed in order to quantify the dynamics of particulate matter (PM) in a cylindrical element. Advection, diffusion and deposition of particles are calculated in a symmetrical circular tube with fully developed laminar flow applying the theory of confluent hypergeometric functions. The following in silico experiments are proposed: (i) the simulation of a thin tube deposition, acting as a filtering element for PM and (ii) the modeling of isokinetic sampling in the inner region of the tube. Two mathematical formulations are compared to obtain the general analytical solution of the advection-diffusion equation in cylindrical coordinates. After applying the variable separation method, a second order ordinary differential equation (ODE) is obtained, this ODE is solved with two distinct forms: (a) applying a transformation of variables to Whittaker’s function and (b) using a new variable transformation proposition. No studies were found that compare these analytical solutions, as well as the use of this new variable transformation, although both cases are essentially different ways of applying the Frobenius method. In this work, 75 eigenvalues and their constants are presented for the first time using five decimal places. The new series solution has high accuracy compared to the previous one, it can provide the concentration profile, the deposition rate and the determination of the PM profile in the isokinetic sampling region. Our solution proved to be more stable close to the tube wall, which may improve the techniques for measuring the flow of particles in tubes. Furthermore, using more eigenvalues improved the estimation of PM2.5 and PM10 deposition. The results presented here show that the proposed new analytic solution has the potential to contribute to numerical and experimental environmental analyses.

Holomorphic Family of Dirac–Coulomb Hamiltonians in Arbitrary Dimension

AbstractWe study massless one-dimensional Dirac–Coulomb Hamiltonians, that is, operators on the half-line of the form $$D_{\omega ,\lambda }:=\begin{bmatrix} -\frac{\lambda +\omega }{x} &{} - \partial _x \\ \partial _x &{} -\frac{\lambda -\omega }{x} \end{bmatrix}$$ D ω , λ : = - λ + ω x - ∂ x ∂ x - λ - ω x . We describe their closed realizations in the sense of the Hilbert space $$L^2(\mathbb {R}_+,\mathbb {C}^2)$$ L 2 ( R + , C 2 ) , allowing for complex values of the parameters $$\lambda ,\omega $$ λ , ω . In physical situations, $$\lambda $$ λ is proportional to the electric charge and $$\omega $$ ω is related to the angular momentum. We focus on realizations of $$D_{\omega ,\lambda }$$ D ω , λ homogeneous of degree $$-1$$ - 1 . They can be organized in a single holomorphic family of closed operators parametrized by a certain two-dimensional complex manifold. We describe the spectrum and the numerical range of these realizations. We give an explicit formula for the integral kernel of their resolvent in terms of Whittaker functions. We also describe their stationary scattering theory, providing formulas for a natural pair of diagonalizing operators and for the scattering operator. We describe the point spectrum of their nonhomogeneous realizations. It is well-known that $$D_{\omega ,\lambda }$$ D ω , λ arise after separation of variables of the Dirac–Coulomb operator in dimension 3. We give a simple argument why this is still true in any dimension. Furthermore, we explain the relationship of spherically symmetric Dirac operators with the Dirac operator on the sphere and its eigenproblem. Our work is mainly motivated by a large literature devoted to distinguished self-adjoint realizations of Dirac–Coulomb Hamiltonians. We show that these realizations arise naturally if the holomorphy is taken as the guiding principle. Furthermore, they are infrared attractive fixed points of the scaling action. Beside applications in relativistic quantum mechanics, Dirac–Coulomb Hamiltonians are argued to provide a natural setting for the study of Whittaker (or, equivalently, confluent hypergeometric) functions.

A study of generalized hypergeometric Matrix functions via two-parameter Mittag–Leffler matrix function

Abstract The main aim of this article is to study a new generalizations of the Gauss hypergeometric matrix and confluent hypergeometric matrix functions by using two-parameter Mittag–Leffler matrix function. In particular, we investigate certain important properties of these extended matrix functions such as integral representations, differentiation formulas, beta matrix transform, and Laplace transform. Furthermore, we introduce an extension of the Jacobi matrix orthogonal polynomial by using our generalized Gauss hypergeometric matrix function, which is very important in scattering theory and inverse scattering theory.

Some Double Integrals Stemming from the Boltzmann Equation in the Kinetic Theory of Gasses

The main object of this article is to revisit a certain double integral involving Kummer’s confluent hypergeometric function 1F1 , which arose in the study of the collision terms of the celebrated Boltzmann equation in the kinetic theory of gases. Here, in this article, we propose to investigate some novel extensions and generalizations of this family of double integrals. We also point out some relevant connections of the results, which are presented here, with other related recent developments in the theory and applications of hypergeometric functions.

Reciprocal Data Transformations and Their Back-Transforms

Variable transformations have a long and celebrated history in statistics, one that was rather academically glamorous at least until generalized linear models theory eclipsed their nurturing normal curve theory role. Still, today it continues to be a covered topic in introductory mathematical statistics courses, offering worthwhile pedagogic insights to students about certain aspects of traditional and contemporary statistical theory and methodology. Since its inception in the 1930s, it has been plagued by a paucity of adequate back-transformation formulae for inverse/reciprocal functions. A literature search exposes that, to date, the inequality E(1/X) ≤ 1/(E(X), which often has a sizeable gap captured by the inequality part of its relationship, is the solitary contender for solving this problem. After documenting that inverse data transformations are anything but a rare occurrence, this paper proposes an innovative, elegant back-transformation solution based upon the Kummer confluent hypergeometric function of the first kind. This paper also derives formal back-transformation formulae for the Manly transformation, something apparently never done before. Much related future research remains to be undertaken; this paper furnishes numerous clues about what some of these endeavors need to be.

An investigation of space distributed-order models for simulating anomalous transport in a binary medium

Recent studies highlight that diffusion processes in highly heterogeneous, fractal-like media can exhibit anomalous transport phenomena, which motivates us to consider the use of generalised transport models based on fractional operators. In this work, we harness the properties of the distributed-order space fractional potential to provide a new perspective on dealing with boundary conditions for nonlocal operators on finite domains. Firstly, we consider a homogeneous space distributed-order model with Beta distribution weight. An a priori estimate based on the L2 norm is presented. Secondly, utilising finite Fourier and Laplace transform techniques, the analytical solution to the model is derived in terms of Kummer’s confluent hypergeometric function. Moreover, the finite volume method combined with Jacobi-Gauss quadrature is applied to derive the numerical solution, which demonstrates high accuracy even when the weight function shows near singularity. Finally, a one-dimensional two-layered problem involving the use of the fractional Laplacian operator and space distributed-order operator is developed and the correct form of boundary conditions to impose is analysed. Utilising the idea of ‘geometric reconstruction’, we introduce a ‘transition layer’ whereby the fractional operator index varies from fractional order to integer order across a fine layer at the boundary of the domain when transitioning from the complex internal structure to the external conditions exposed to the medium. An important observation is that for a fractional dominated case, the diffusion behaviour in the main layer is similar to fractional diffusion, while near the boundary the behaviour transitions to the case of classical diffusion.

THE CARBON NANOTUBE-EMBEDDED BOUNDARY LAYER THEORY FOR ENERGY HARVESTING

Single-walled carbon nanotubes (SWNTs) and multi-walled nanotubes (MWNTs) are gaining appeal in mechanical engineering and industrial applications due to their direct influence on enhancing the thermal conductivity of base fluids. With such intriguing properties of carbon nanotubes in mind, our goal in this work is to investigate radiation effects on the flow of carbon nanotube suspended nanofluids in the presence of a magnetic field past a stretched sheet impacted by slip state. CNTs flow and heat transmission are frequently modelled in practice using nonlinear differential equation systems. This system has been precisely solved, and an accurate analytical expression for the fluid velocity in terms of an exponential function has been derived, while the temperature distribution is stated in terms of a confluent hypergeometric function. The impact of the radiation parameter, slip parameter, sloid volume fraction, magnetic parameter, Eckart and Prandtl numbers on the velocity, temperature, and heat transfer rate profiles are demonstrated using a parametric analysis. When compared to the two types of nanoparticles (Cooper and Silver) in earlier published articles, temperature profiles for single-walled nanotubes (SWNTs) and multi-walled nanotubes (MWNTs) are revealed to be particularly sensitive to radiation, solid volume fraction, and slip parameters. Nanomechanical gears, nanosensors, nanocomposite materials, resonators, and thermal materials are only a few of the present problem's technical applications.

Fractional Integral of the Confluent Hypergeometric Function Related to Fuzzy Differential Subordination Theory

The fuzzy differential subordination concept was introduced in 2011, generalizing the concept of differential subordination following a recent trend of adapting fuzzy sets theory to other already-established theories. A prolific tool in obtaining new results related to operators is the fractional integral applied to different functions. The fractional integral of the confluent hypergeometric function was previously investigated using means of the classical theory of subordination. In this paper, we give new applications of this function using the theory of fuzzy differential subordination. Fuzzy differential subordinations are established and their best dominants are also provided. Corollaries are written using particular functions, in which the conditions for the univalence of the fractional integral of the confluent hypergeometric function are given. An example is constructed as a specific application of the results obtained in this paper.

FRW model with two-fluid source in fractal cosmology

Present paper deals with flat Friedmann - Robertson - Walker (FRW) model with two - fluid source in fractal cosmology. In this model one fluid represents matter content of the universe and another fluid is radiation field modeling the cosmic microwave background. To get the deterministic model, we have used the relation between pressure and density for matter through the gamma law equation of state $p_{m} = (\gamma-1)\, \rho_{m}, \,\, 1\leq \gamma\leq2$ . The solutions of fractal field equations are obtained in terms of Kummer's confluent hypergeometric function of the first kind. Some physical parameters of the models are obtained and discussed their behavior with the help of graphs in detail.

Generalized Helicity Operator for a Spin 2 Particle in the Presence of an External Uniform Magnetic Field

The eigenvalue problem for generalized helicity operator for a spin 2 particle in the presence of a uniform magnetic field is solved. After separating variables in the basis of cylindrical tetrad the system of 10 first order differential equations is derived, it is split into two independent subsystems of four and six equations. First, the free particle is studied. The system of four equations is solved straightforwardly in terms of the confluent hypergeometric functions, there are found corresponding eigenvalues and eigenfunctions. Subsystem of six equations leads to one ordinary 4-th order differential equation. Corresponding operator is factorized into the product of two commuting 2-nd order operators, so the problem reduces to solving two differential equations of the 2-nd order. Their solutions are constructed in terms of the Bessel functions. The problem is extended to a presence of an external uniform magnetic field, the method is much the same, the explicit solutions are constructed in terms of the confluent hypergeometric functions. The helicity eigenvalues are described in an implicit form, as solutions of the polynomial equations of 3-rd and 5-th orders.

Self-similarity in turbulence and its applications.

First, we discuss the non-Gaussian type of self-similar solutions to the Navier-Stokes equations. We revisit a class of self-similar solutions which was studied in Canonne et al. (1996 Commun. Partial. Differ. Equ. 21, 179-193). In order to shed some light on it, we study self-similar solutions to the one-dimensional Burgers equation in detail, completing the most general form of similarity profiles that it can possibly possess. In particular, on top of the well-known source-type solution, we identify a kink-type solution. It is represented by one of the confluent hypergeometric functions, viz. Kummer's function [Formula: see text]. For the two-dimensional Navier-Stokes equations, on top of the celebrated Burgers vortex, we derive yet another solution to the associated Fokker-Planck equation. This can be regarded as a 'conjugate' to the Burgers vortex, just like the kink-type solution above. Some asymptotic properties of this kind of solution have been worked out. Implications for the three-dimensional (3D) Navier-Stokes equations are suggested. Second, we address an application of self-similar solutions to explore more general kind of solutions. In particular, based on the source-type self-similar solution to the 3D Navier-Stokes equations, we consider what we could tell about more general solutions. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

A Generalization of the Bivariate Gamma Distribution Based on Generalized Hypergeometric Functions

In this paper, we provide a new bivariate distribution obtained from a Kibble-type bivariate gamma distribution. The stochastic representation was obtained by the sum of a Kibble-type bivariate random vector and a bivariate random vector builded by two independent gamma random variables. In addition, the resulting bivariate density considers an infinite series of products of two confluent hypergeometric functions. In particular, we derive the probability and cumulative distribution functions, the moment generation and characteristic functions, the Hazard, Bonferroni and Lorenz functions, and an approximation for the differential entropy and mutual information index. Numerical examples showed the behavior of exact and approximated expressions.

Applications of Confluent Hypergeometric Function in Strong Superordination Theory

In the research presented in this paper, confluent hypergeometric function is embedded in the theory of strong differential superordinations. In order to proceed with the study, the form of the confluent hypergeometric function is adapted taking into consideration certain classes of analytic functions depending on an extra parameter previously introduced related to the theory of strong differential subordination and superordination. Operators previously defined using confluent hypergeometric function, namely Kummer–Bernardi and Kummer–Libera integral operators, are also adapted to those classes and strong differential superordinations are obtained for which they are the best subordinants. Similar results are obtained regarding the derivatives of the operators. The examples presented at the end of the study are proof of the applicability of the original results.

Another Method for Proving Certain Reduction Formulas for the Humbert Function ψ2 Due to Brychkov et al. with an Application

Recently, Brychkov et al. established several new and interesting reduction formulas for the Humbert functions (the confluent hypergeometric functions of two variables). The primary objective of this study was to provide an alternative and simple approach for proving four reduction formulas for the Humbert function ψ2. We construct intriguing series comprising the product of two confluent hypergeometric functions as an application. Numerous intriguing new and previously known outcomes are also achieved as specific instances of our primary discoveries. It is well-known that the hypergeometric functions in one and two variables and their confluent forms occur naturally in a wide variety of problems in applied mathematics, statistics, operations research, physics (theoretical and mathematical) and engineering mathematics, so the results established in this paper may be potentially useful in the above fields. Symmetry arises spontaneously in the abovementioned functions.

Solutions of Fractional Kinetic Equations using the $(p,q;l)$-Extended τ -Gauss Hypergeometric Function

The main objective of this paper is to use the newly proposed $(p,q;l)$-extended beta function to introduce the $(p,q;l)$-extended $τ$-Gauss hypergeometric and the $(p,q;l)$-extended $τ$-confluent hypergeometric functions with some of their properties, such as the Laplace-type and the Euler-type integral formulas. Another is to apply them to fractional kinetic equations that appear in astrophysics and physics using the Laplace transform method.

Non-commutativity and non-inertial effects on a scalar field in a cosmic string space-time: II. Spin-zero Duffin–Kemmer–Petiau-like oscillator

We study the non-inertial effects of a rotating frame on a spin-zero, Duffin–Kemmer–Petiau-like oscillator in a cosmic string space-time with non-commutative geometry in the momentum space. The spin-zero DKP-like oscillator is obtained from the Klein–Gordon Lagrangian with a non-standard prescription for the oscillator coupling. We find that the solutions of the time-independent radial equation with the non-zero non-commutativity parameter parallel to the string are related to the confluent hypergeometric function. We find the quantized energy eigenvalues of the non-commutative oscillator.

Fractional Calculus and Confluent Hypergeometric Function Applied in the Study of Subclasses of Analytic Functions

The study done for obtaining the original results of this paper involves the fractional integral of the confluent hypergeometric function and presents its new applications for introducing a certain subclass of analytic functions. Conditions for functions to belong to this class are determined and the class is investigated considering aspects regarding coefficient bounds as well as partial sums of these functions. Distortion properties of the functions belonging to the class are proved and radii estimates are established for starlikeness and convexity properties of the class.

On distributions of covariance structures

The derivation of the sample covariance is difficult compared to that of the distribution of the sample correlation coefficient. This paper deals with the distributions of covariance structures appearing in real scalar/vector/matrix variables. Covariance structure is a bilinear structure. Consider a bilinear form where X and Y are and real vectors and A is a constant p × q matrix. The basic aim in this paper is to derive the distribution of such a structure when the components are scalar/vector/matrix Gaussian variables. The procedure used is to examine the Laplace transform or the moment generating function (mgf) coming from such a bilinear form in real scalar/vector/matrix variables. Covariance structures in several situations are shown to produce a mgf of the type where t is the mgf parameter, means the real part of and λ and α are real scalar parameters. Explicit evaluation of the density of u is considered when α is a positive integer as well as for a general α. It is shown that the exact densities can be written as linear functions of double gamma densities and double exponential or Laplace densities when α is a positive integer. For the general value of α, it is shown that the exact density can be written in terms of double Mittag-Leffler or a double confluent hypergeometric function.

Fractional Integral of a Confluent Hypergeometric Function Applied to Defining a New Class of Analytic Functions

The study on fractional integrals of confluent hypergeometric functions provides interesting subordination and superordination results and inspired the idea of using this operator to introduce a new class of analytic functions. Given the class of functions An=f∈HU:fz=z+an+1zn+1+…,z∈U written simply A when n=1, the newly introduced class involves functions f∈A considered in the study due to their special properties. The aim of this paper is to present the outcomes of the study performed on the new class, which include a coefficient inequality, a distortion theorem and extreme points of the class. The starlikeness and convexity properties of this class are also discussed, and partial sums of functions from the class are evaluated in order to obtain class boundary properties.

Certain implementations in fractional calculus operators involving Mittag-Leffler-confluent hypergeometric functions

The Mittag-Leffler function and confluent hypergeometric functions were created to approximate interpolation in exponential functions. The researchers noted that Prabhakar’s integral transformation, which involves extended multi-parameter Mittag-Leffler functions, may be used to create and explore different fractional calculus models. This four-parameter function is further illustrated in graphs using MATLAB. With classical (Riemann–Liouville) fractional integrals, the research shows a set of formulations for these fractional differintegral operators. Moreover, using AB Model results of Prabhakar and the generalized Prabhakar models, the authors use a series of formulae to come up with new results. This paper demonstrates how this series formula may be used to provide simple alternative evidence for numerous well-known effects of Prabhakar differintegrals.