Kummer Confluent Hypergeometric Functions Research Articles

Effect of MHD and radiation on biviscous Bingham fluid flow on Marangoni boundary for heat source/sink with chemical reaction

A significance of Marangoni-driven convective magnetohydrodynamics (MHD) was investigated in a planned study. The present study examines the impact of radiation and chemical reactions on the two dimensional boundary layer flow of bi-viscous Bingham fluid on a thermosolutal Marangoni boundary with magnetic field and heat source/sink. The physical flow problem is mathematically modelled into Navier–Stokes equations. These nonlinear partial differential equations are then mapped into a set of nonlinear ordinary differential equations using similarity transformation. The elegant analytical method is used to obtain analytic approximations for the resulting system of nonlinear differential equations. The analytical solution for the temperature and concentration profiles is expressed in terms of Kummer's confluent hypergeometric function. The features of flow characteristics such as velocity, temperature, and concentration profiles in response to the variations of the emerging parameters are simulated and examined with a physical explanation through graphs. Surface tension is considered to be corresponding to heat and mass. The results reveal that the velocity profile decreases with increasing the magnetic field, while the Marangoni number value increases with increasing the velocity profile. Moreover, the temperature profile rises with rising the radiation parameter and whereas the value chemical reaction parameter upsurges with decline concentration profile. The present problem has great interest due to its applications in industrial applications such as drying of silicon wafers, thin layers of paint, glues, in heat exchangers, crystal growth in space, biological fluids, blood, synovial liquids, plasma, lubricants, many pharmaceutical products, proteins, sewage sludge, china clay, and many more.

IMPACT OF POROUS AND MAGNETIC DISSIPATION ON DISSIPATIVE FLUID FLOW AND HEAT TRANSFER IN THE PRESENCE OF DARCY-BRINKMAN POROUS MEDIUM

In this article, the boundary layer flow of an electrically-conducting fluid through a porous medium attached with a radiative permeable stretching sheet is analyzed. Following the Brinkman theory, an extended Darcy model (Darcy-Brinkman model) is utilized for the model momentum equation. Heat transfer analysis is also performed in the presence of viscous and Joule dissipation. Moreover, in the modeling of the energy equation, the effects of internal heating resulting from the mechanical effort required to squeeze out the fluid through the porous medium are also included in porous dissipation. Suitable dimensionless variables are introduced to convert the governing boundary layer equations into a dimensionless form, which are then converted into self-similar, nonlinear ordinary differential equations by utilizing similarity transformations. The exact solution of the nonlinear self-similar momentum equation is obtained in the form of the exponential function. In contrast, the solution of the energy equation is computed through the Laplace transform technique in the form of Kummer confluent hypergeometric functions. Effects of involved physical parameters on the momentum boundary layer (MBL), thermal boundary layer (TBL), wall shear stress, and local Nusselt number are explored through graphs and tables. Moreover, the slope linear regression (SLR) technique is used to calculate the rate of decrease/increase in shear stress and the rate of heat transfer at the boundary. The velocity and momentum boundary layer decreases for large values of porosity parameter and increases by increasing the viscosity ratio. The shear stress increases by increasing the porosity parameter, Hartman number, and suction parameter, while the opposite effect is examined with increasing values of viscosity ratio parameter. Heat transfer rate also enhances by increasing the Brinkman viscosity ratio parameter and wall suction velocity.

Hitting times for sticky skew CIR process

In this paper, we consider an extended skew CIR processes with sticky points, which is referred to as the sticky skew CIR process. We first calculate the infinitesimal generator and its domain. To explore its trajectory properties, we compute the Laplace transforms and the expectations of first hitting times over a constant boundary. The solutions of Laplace transforms are expressed in terms of Tricomi and Kummer confluent hypergeometric functions.

Exact solution of the 1D Dirac equation for a pseudoscalar interaction potential with the inverse-square-root variation law

We present the exact solution of the one-dimensional stationary Dirac equation for the pseudoscalar interaction potential, which consists of a constant and a term that varies in accordance with the inverse-square-root law. The general solution of the problem is written in terms of irreducible linear combinations of two Kummer confluent hypergeometric functions and two Hermite functions with non-integer indices. Depending on the value of the indicated constant, the effective potential for the Schrödinger-type equation to which the problem is reduced can form a barrier or well. This well can support an infinite number of bound states. We derive the exact equation for the energy spectrum and construct a rather accurate approximation for the energies of bound states. The Maslov index involved turns out to be non-trivial; it depends on the parameters of the potential.

Summation identities for the Kummer confluent hypergeometric function 1F1(a; b;z)

The role which hypergeometric functions have in the numerical and symbolic calculation, especially in the fields of applied mathematics and mathematical physics motivated research in this paper. In this note, a general formula for the sum, with the Kummer confluent hypergeometric function 1F1(a; b; z) is derived and given in terms of the function 2F2(a; b; z). The idea for this investigation comes from the theory of generalized Gauss-Rys quadrature formulas developed recently by (Milovanović, 2018) and (Milovanović and Vasović, 2022). Several results are obtained as special cases of the main result.

Asymptotic behavior for wave equations with space-dependent damping in a weighted energy class

In this paper, we discuss diffusion phenomena for the wave equation with space-dependent damping. It is known that this phenomenon occurs in the constant damping case with initial data belonging to a suitable energy class. This paper clarifies that diffusion phenomenon also occurs when the damping is effective and space-dependent, and initial data belong to a certain energy class. The proof relies on an energy method involving Kummer's confluent hypergeometric functions with a modified version of the technique of decomposition of solutions introduced in [20].

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.

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.

A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions

The confluent hypergeometric equation, also known as Kummer's equation, is one of the most important differential equations in physics, chemistry, and engineering. Its two power series solutions are the Kummer function, M(a,b,z), often referred to as the confluent hypergeometric function of the first kind, and M ≡ z1-bM(1+a-b, 2-b,z), where a and b are parameters that appear in the differential equation. A third function, the Tricomi function, U(a,b,z), sometimes referred to as the confluent hypergeometric function of the second kind, is also a solution of the confluent hypergeometric equation that is routinely used. Contrary to common procedure, all three of these functions (and more) must be considered in a search for the two linearly independent solutions of the confluent hypergeometric equation. There are situations, when a, b, and a - b are integers, where one of these functions is not defined, or two of the functions are not linearly independent, or one of the linearly independent solutions of the differential equation is different from these three functions. Many of these special cases correspond precisely to cases needed to solve problems in physics. This leads to significant confusion about how to work with confluent hypergeometric equations, in spite of authoritative references such as the NIST Digital Library of Mathematical Functions. Here, we carefully describe all of the different cases one has to consider and what the explicit formulas are for the two linearly independent solutions of the confluent hypergeometric equation. The procedure to properly solve the confluent hypergeometric equation is summarized in a convenient table. As an example, we use these solutions to study the bound states of the hydrogenic atom, correcting the standard treatment in textbooks. We also briefly consider the cutoff Coulomb potential. We hope that this guide will aid physicists to properly solve problems that involve the confluent hypergeometric differential equation.

Some remarks on the location of non-asymptotic zeros of Whittaker and Kummer hypergeometric functions

This paper focuses on the location of the non-asymptotic zeros of Whittaker and Kummer confluent hypergeometric functions. Based on a technique proposed by E. Hille for the analysis of solutions of some second-order ordinary differential equations, we characterize the sign of the real part of zeros of Whittaker and Kummer functions and provide estimates on the regions of the complex plane where those zeros can be located. Our main result is a correction of a previous statement by G. E. Tsvetkov whose propagation has induced mistakes in the literature. In particular, we review some results of E. B. Saff and R. S. Varga on the error of Padé's rational approximation of the exponential function, which are based on the latter.

Quasinormal modes of scalar field coupled to Einstein's tensor in the non-commutative geometry inspired black hole

We investigate the quasinormal modes (QNMs) of the scalar field coupled to the Einstein's tensor in the non-commutative geometry inspired black hole spacetime. It is found that the lapse function of the non-commutative black hole metric can be represented by a Kummer's confluent hypergeometric function, which can effectively solve the problem that the numerical results of the QNMs are sensitive to the model parameters and make the QNMs values more reliable. We make a careful analysis of the scalar QNM frequencies by using several numerical methods, and find that the numerical results obtained by the new WKB method (the Padé approximants) and the Mashhoon method (Pöschl-Teller potential method) are quite different from those obtained by the asymptotic iterative method (AIM) and time-domain integration method when the non-commutative parameter θ and coupling parameter η are large. The most obvious difference is that the numerical results obtained by the AIM and the time-domain integration method appear a critical value ηc with an increase of η, which leads to the dynamical instability. After carefully analyzing the numeral results, we conclude that the numerical results obtained by the AIM and the time-domain integration method are closer to the theoretical values than those obtained by the WKB method and the Mashhoon method, when the θ and η are large. Moreover, through a numerical fitting, we obtain that the functional relationship between the threshold ηc and the non-commutative parameter θ satisfies ηc=aθb+c for a fixed l approximately. We find that the stability of dynamics can be ensured in the η<ηc(θ,l) region.

A Sturm‐Liouville equation on the crossroads of continuous and discrete hypercomplex analysis

The paper studies discrete structural properties of polynomials that play an important role in the theory of spherical harmonics in any dimensions. These polynomials have their origin in the research on problems of harmonic analysis by means of generalized holomorphic (monogenic) functions of hypercomplex analysis. The Sturm‐Liouville equation that occurs in this context supplements the knowledge about generalized Vietoris number sequences , first encountered as a special sequence (corresponding to ) by Vietoris in connection with positivity of trigonometric sums. Using methods of the calculus of holonomic differential equations, we obtain a general recurrence relation for , and we derive an exponential generating function of expressed by Kummer's confluent hypergeometric function.

Multiplicity-induced-dominancy for delay-differential equations of retarded type

An important question of ongoing interest for linear time-delay systems is to provide conditions on its parameters guaranteeing exponential stability of solutions. Recent works have explored spectral techniques to show that, for some low-order delay-differential equations of retarded type, spectral values of maximal multiplicity are dominant, and hence determine the asymptotic behavior of the system, a property known as multiplicity-induced-dominancy. This work further explores such a property and shows its validity for general linear delay-differential equations of retarded type of arbitrary order including a single delay in the system's representation. More precisely, an interesting link between characteristic functions with a real root of maximal multiplicity and Kummer's confluent hypergeometric functions is exploited. We also provide examples illustrating our main result.

Off-Axis Vortex Beam Propagation through Classical Optical System in Terms of Kummer Confluent Hypergeometric Function

The analytical solution for the propagation of the laser beam with optical vortex through the system of lenses is presented. The optical vortex is introduced into the laser beam (described as Gaussian beam) by spiral phase plate. The solution is general as it holds for the optical vortex of any integer topological charge, the off-axis position of the spiral phase plate and any number of lenses. Some intriguing conclusions are discussed. The higher order vortices are unstable and split under small phase or amplitude disturbance. Nevertheless, we have shown that off-axis higher order vortices are stable during the propagation through the set of lenses described in paraxial approximation, which is untypical behavior. The vortex trajectory registered at image plane due to spiral phase plate shift behaves like a rigid body. We have introduced a new factor which in our beam plays the same role as Gouy phase in pure Gaussian beam.

Indefinite integrals of special functions from integrating factors

ABSTRACTSome general integrals are presented which were obtained from two integrating factors and for the first two and last two terms, respectively, of the second-order linear ordinary differential equation obeyed by many special functions. The integrating factor is identical to the Lagrangian factor introduced in Conway [A Lagrangian method for deriving new indefinite integrals of special functions. Integral Transforms Spec Funct. 2015;26(10):812–824; Indefinite integrals of some special functions from a new method. Integral Transforms Spec Funct. 2015;26(11):845–858] and sometimes provides textbook integrals. The integrals provided by the second factor seem to have been little explored. Results are given for Gauss hypergeometric functions and Kummer confluent hypergeometric functions. All results presented have been checked using Mathematica.

Density and distribution evaluation for convolution of independent gamma variables

Convolutions of independent gamma variables are encountered in many applications such as insurance, reliability, and network engineering. Accurate and fast evaluations of their density and distribution functions are critical for such applications, but no open source, user-friendly software implementation has been available. We review several numerical evaluations of the density and distribution of convolution of independent gamma variables and compare them with respect to their accuracy and speed. The methods that are based on the Kummer confluent hypergeometric function are computationally most efficient. The benefit of employing the confluent hypergeometric function is further demonstrated by a renewal process application, where the gamma variables in the convolution are Erlang variables. We derive a new computationally efficient formula for the probability mass function of the number of renewals by a given time. An R package coga provides efficient C++ based implementations of the discussed methods and are available in CRAN.

Exact solutions of initial boundary value problems for Sobolev type equations of elastic oscillations

The article is devoted to the problem of the unique solvability of initial boundary value problems for partial differential equations. These initial boundary value problems are dissipative dynamic mathematical models from the field of linear elasticity. The considered equations are not resolved with respect to the highest time derivative. Therefore, these equations are referred to the so-called Sobolev type partial differential equations. The research of initial boundary value problems is carried out from general position in the form of the complete second order differential equation in abstract spaces. The theory of the distributions with values in Banach spaces and the concept of a fundamental solution of differential operator are used. Theorems of existence and uniqueness of the solutions of considered initial boundary value problems in the class of functions of finite smoothness with respect to time are proved. These solutions are obtained as an functional series of the Kummer confluent hypergeometric function.

Moments of Student's t-distribution: A Unified Approach

In this paper, we derive the closed form formulae for moments of Student's t-distribution in the one dimensional case as well as in higher dimensions through a unified probability framework. Interestingly, the closed form expressions for the moments of Student's t-distribution can be written in terms of the familiar Gamma function, Kummer's confluent hypergeometric function, and the hypergeometric function.

A Lambert-W Exactly Solvable Level-Crossing Confluent Hypergeometric Two-State Model

We introduce a new exactly integrable level-crossing model of quantum semiclassical two-state problem for which the analytic solution is written in terms of the Kummer confluent hypergeometric functions. This is a constant-amplitude field-configuration describing an asymmetricin- time level-crossing process for which the laser field frequency detuning is given in terms of the Lambert-W function. The general solution of the problem for this model is written as a linear combination, with arbitrary constant coefficients, of two fundamental solutions each of which presents an irreducible linear combination of two confluent hypergeometric functions. We present the fundamental solutions and analyze the behavior of the system in the external field defined by the specific field configuration.

On the sum of the Lah numbers and zeros of the Kummer confluent hypergeometric function

Abstract In the paper, the authors find the sum of the Lah numbers and make sure that the Kummer confluent hypergeometric function 1F1(n+ 1; 2; z) has only n− 1real and negative zeros.