Kummer Confluent Hypergeometric Functions Research Articles

On the properties of electrostatic drift and sound modes in radially and axially inhomogeneous bounded plasmas

The behavior of electrostatic drift and ion sound waves is discussed in plasmas with an equilibrium density gradient both perpendicular and parallel to the ambient magnetic field. In spatially unlimited plasmas, and in Cartesian geometry, the mode amplitude is calculated for several profiles of the equilibrium density. Exact analytical solutions are found. Cylindric plasmas bounded in the radial direction and bounded or unbounded in the axial direction are also investigated in the presence of equilibrium density gradients in both the radial and axial directions. Exact analytical solutions are found in terms of the Kummer confluent hypergeometric functions, describing standing oscillatory patterns in the radial and axial directions, associated with the drift mode propagating in the poloidal (θ) direction.

Optimal Dividend Strategy in the Compound Poisson Model with Constant Interest

In this paper, we study optimal dividend problems in a compound Poisson risk process with constant interest on investments. Throughout the paper, we assume that the rate of dividend pay out is bounded by some positive constant. It is shown that the optimal value function of the dividends can be characterized by the Hamilton–Jacobi–Bellman equation. For the case of an exponential claim amount distribution, it is shown that the optimal dividend strategy is a threshold strategy. Then the Laplace transform of the time of ruin, under a threshold strategy, can be calculated. Kummer's confluent hypergeometric equation and confluent hypergeometric functions play a key role in this paper.

Some peculiarities of the asymmetric Graetz problem

A numerical simulation and an analysis of the steady state forced convection heat transfer with plane laminar flow confined by two parallel plates that are kept at constant but different temperatures are presented. We name this heat transfer configuration shortly the asymmetric Graetz problem. The essential features of the asymmetric in comparison to the symmetric Graetz problem are the reversal of the heat flux and the jump of the Nusselt number from positive to negative region at the plate having the temperature closer to the fluid inlet temperature. These phenomena occur at different axial positions, which depend on the thermal asymmetry and the fluid inlet conditions. The numerical results agree excellently with an analytical solution obtained in terms of Kummer confluent hypergeometric function and Hermite polynomials.

Joule-heating effects in mixed electroosmotic and pressure-driven microflows under constant wall heat flux

Heat-transfer characteristics of mixed electro-osmotic and pressure-driven flows are obtained in a two-dimensional straight microchannel by a solving steady-state energy equation. Both thermally developing and fully developed regions are considered for hydraulically fully developed mixed flows under isoflux channel wall conditions. The steady temperature distribution is obtained from the superposition of homogeneous solution and particular function. The particular solution is derived based on the constant heat-flux condition in the fully developed region, while the homogeneous solution is presented in terms of infinite series containing Kummer confluent hypergeometric functions due to the existence of non-self-adjoint eigenvalues. The coefficients of the homogeneous solution are found by utilizing the Gram-Schmidt orthogonalization procedure, and the Secant method is utilized to obtain the corresponding eigenvalues. Our analytical techniques are verified by obtaining an excellent agreement with existing literature for slug, pressure-driven, and pure electroosmotic flow cases. In the fully developed region, the Nusselt number of mixed flow is independent of the thermal Peclet number for a particular Joule heating and imposed surface heat flux. The entry length of mixed flow significantly depends on the applied/induced pressure gradient to the electro-osmotic flow

The Kummer confluent hypergeometric function and some of its applications in the theory of azimuthally magnetized circular ferrite waveguides

The Kummer confluent hypergeometric function and some of its applications in the theory of azimuthally magnetized circular ferrite waveguides

Exact solutions for perturbed confined hydrogen atoms: Polarizabilities and nuclear shielding factors

We extend our recent treatment of a well-known model of a nonrelativistic hydrogen-like atom confined within an impenetrable spherical ``box'' of radius . Analytical closed-form solutions are presented for the first-order perturbation corrections appropriate to the dipole polarization of states. solutions are expressed conveniently in terms of Kummer (confluent hypergeometric) functions, exactly as for the field-free atom, by exploiting several well-known recurrence and differential relations for these functions. The functions and all necessary integrals are evaluated using standard routines, which exploit analytical properties of these functions. The accuracy of our procedures has been checked by a similar calculation of the dipole shielding factors. Our procedures are easily extended to states of nonzero angular momentum, as well as to higher-multipole perturbations. Our exact values may be of interest in assessing the utility of this simple model of atomic confinement.

Electrostatic waves in bounded dusty magnetoplasma

Electrostatic perturbations propagating obliquely to the magnetic field lines are studied in a cylindric magnetized dusty plasma configuration. A set of two coupled equations describing the perturbed electrostatic and gravity potentials in a radially nonuniform plasma is derived. The equations are discussed and solved in two limits, without and with the self-gravity effects included. Without the gravity the corresponding equation for the electrostatic potential is solved analytically for a Gaussian-type equilibrium density profile. The general solution, which can be applied to various plasma situations, is expressed in terms of the Kummer confluent hypergeometric functions. In the self-gravitating case the equations are solved analytically for a homogeneous plasma and the general solutions are written in terms of the Bessel functions, describing well-behaving radially localized wave amplitudes.

WAVE FUNCTIONS WITH DISCRETE AND WITH CONTINUOUS SPECTRUM FOR QUANTUM DAMPED HARMONIC OSCILLATOR PERTURBED BY A SINGULARITY

The quantum states with discrete and continuous spectrum for the damped harmonic oscillator perturbed by a singularity have been investigated using invariant operator and unitary operator together. The eigenvalue of the invariant operator for ω0≤β/2 is continuous while for ω0>β/2 is discrete. The wave functions for ω0=β/2 expressed in terms of the Bessel function and for ω0<β/2 in terms of the Kummer confluent hypergeometric function. The convergence of the probability density is more rapid for over-damped harmonic oscillator than that of the other two cases due to the large damping constant.

Approximate analytic solutions for diffraction by non-uniform reflection geometry fiber Bragg gratings

Approximate analytic expressions for the transmitted and diffracted fields from a non-uniform in-fiber Bragg grating are presented, in terms of Hermite and Kummer confluent hypergeometric functions. These novel approximate analytic solutions are derived for the case of a grating having both a weak linear variation of grating period with length (weak linear chirp) and a quadratic approximation to a Gaussian refractive index modulation variation. The solution is derived using the Beta-value first-order two-wave coupled differential equations and describes both on- and off-Bragg replay. The equations are shown to be useful, when compared to more exact numerical results, for a wide range of practical grating parameter values.

Generalized mittag-leffler function and generalized fractional calculus operators

The paper is devoted to the study of the function E γ ρ,μ(z) defined for complex ρ, μ, γ (Re(ρ) > 0) by which is a generalization of the classical Mittag-Leffler function E ρ,μ(z) and the Kummer confluent hypergeometric function Φ(γ, μ; z). The properties of E γ ρ,μ(z) including usual differentiation and integration, and fractional ones are proved. Further the integral operator with such a function kernel is studied in the space L(a, b). Compositions of the Riemann–Liouville fractional integration and differentiation operators with E γ ρ,μ,ω;a+ are established. An analogy of the semigroup property for the composition of two such operators with different indices is proved, and the results obtained are applied to construct the left inversion operator to the operator E γ ρ,μ,ω;a+. Since, for γ = 0, E 0 ρ,μ,ω;a+ coincides with the Riemann–Liouville fractional integral of order μ, the above operator and its inversion can be considered as generalized fractional calculus operators involving the generalized Mittag-Leffler function E γ ρ,μ(z) in the kernels. Similar assertions are presented for the integral operators containing the Mittag-Leffler and Kummer functions, E ρ,μ(z) and Φ(γ, μ; z), in the kernels, and applications are given to obtain solutions in closed form of the integral equations of the first kind.

Analytical solutions for unsteady free convection in porous media

Analytic solutions for two of the similarity cases identified by Johnson and Cheng (1978) for the unsteady free-convection boundary-layer flow over an impermeable vertical flat plate adjacent to a fluid saturated porous medium are given in the present paper. These are the solutions corresponding to an exponential (e sup a 2 t sup) and a power-law (t m) variation of the surface temperature, respectively. They represent exact solutions for doubly infinite plates and approximate solutions for semi-infinite plates. In the latter cases their validity is restricted to the so-called `conduction regime' of the flow. It is shown that in the power law case, physical solutions only exist in the range m>−1 of the temperature exponent and they can be expressed in terms of Kummer's confluent hypergeometric functions. For m ≥ 0 exponentially decaying unique solutions were found, while in the range −1<m<0 both exponentially and algebraically decaying multiple solutions occur. The origin of the multiple solutions as well as the feasibility conditions of all the above mentioned solutions is discussed in detail.

Exact Quantum State for the Time-Dependent Repulsive Oscillator with a Driving Force and with a Singular Perturbation

We derived the quantum solutions of a repulsive oscillator driven by a timedependent force and a repulsive oscillator perturbed by a singularity using unitary transformation method. The wave functions of the system perturbed by a singularity are expressed in terms of the Kummer confluent hypergeometric function while that of the driven system are expressed in terms of the Weber function. Since the repulsive oscillator is unbound, its spectrum of eigenvalues is continuous. We applied our general development to the repulsive Caldirola–Kanai oscillator.

Compositions of Hadamard-type fractional integration operators and the semigroup property

This paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of Hadamard in the space Xcp of functions f on R+=(0,∞) such that ∫0∞ucf(u)pduu<∞(1⩽p<∞),esssupu>0uc|f(u)|<∞(p=∞), for c∈R=(−∞,∞), in particular in the space Lp(0,∞) (1⩽p⩽∞). The semigroup property and its generalizations are established for the generalized Hadamard-type fractional integration operators under consideration. Conditions are also given for the boundedness in Xcp of these operators; they involve Kummer confluent hypergeometric functions as kernels.

Estimates on the Green's function of second-order elliptic operators in ℝN

Sharp upper and lower pointwise bounds are obtained for the Green's function of the equationfor λ&gt; 0. Initially, in a Cartesian frame, it is assumed that . Pointwise estimates for the heat kernel of ut + Lu = 0, recently obtained under this assumption, are Laplace-transformed to yield corresponding elliptic results. In a second approach, the coordinate-free constraint is imposed. Within this class of operators, the equations defining the maximal and minimal Green's functions are found to be simple ODEs when written in polar coordinates, and these are soluble in terms of the singular Kummer confluent hypergeometric function. For both approaches, bounds on are derived as a consequence.

Some properties of the circular waveguide with azimuthally magnetized ferrite

A comprehensive analysis of normal rotationally symmetric TE modes in a circular waveguide, filled with ferrite, magnetized azimuthally to remanence by a coaxial switching conductor of finite radius, is presented. The characteristic equation of the structure, derived in terms of Kummer and Tricomi confluent hypergeometric functions of complex parameter and variable, is solved numerically, using specially compiled tables of wave functions. Families of theoretically calculated nonreciprocal phase characteristics of the gyrotropic waveguide are shown in normalized form for the two latched states of remanent magnetization, a variety of ferrite parameters, and different values of switching conductor to waveguide radius ratio. The influence of structure geometry and parameters of anisotropic ferrite on normalized differential phase shift and cutoff frequency spectrum of the TE01 mode is discussed.

Technical memorandum. Kummer and Tricomi functions for analysis of circular gyrotropic waveguides with azimuthal magnetisation

The wave functions for propagation of rotationally symmetric modes in circular gyrotropic waveguides with azimuthal magnetisation (the complex Kummer and Tricomi confluent hypergeometric functions) are studied analytically and numerically. Based on specially compiled tables, families of curves are presented, illustrating the behaviour of both higher transcendental functions. The analysis allows a comprehensive treatment of closed and opened circular guiding structures with radially stratified gyrotropic layers, magnetised azimuthally.

On a Class of Electromagnetic Wave Functions for Propagation Along the Circular Gyrotropic Waveguide

The properties of confluent hypergeometric functions as exact electromagnetic wave fnnctions for propagation in a circular waveguide containing azimuthally magnetized remanent ferrite are investigated. Two different forms of solutions of the propagation problem for angular symmetric transverse electric modes are constructed-one in terms of Kummer and Tricomi confluent hypergeometric functions of complex parameter and variable and a second in terms of Whittaker functions. An evaluation of this class of wave functions is performed to sufficient extent, followed by tabulation of their imaginary zeros, providing computation of eigenvalue spectrum and phase characteristics of the gyrotropic guide.

Asymptotic study of rotationally symmetric waves in a circular anisotropic waveguide

The nonreciprocal propagation of rotationally symmetric TE-modes along a circular guide, uniformly filled with latching ferrite magnetised azimuthally to remanence, is studied theoretically. The boundary-value problem is solved in terms of Kummer confluent hypergeometric functions of complex parameter and variable. An asymptotic method for solution of the characteristic equation is developed, providing a basis for numerical computation of the eigenvalue spectrum and phase characteristics of the structure with a fair degree of engineering accuracy. The normalised nonreciprocal phase characteristics and the differential phase shift of the waveguide under rotationally symmetric TE-mode excitation are presented for both latched states of ferrite remanent magnetisation. The potentialities of the structure as digital nonreciprocal phase shifter and/or bistable ferrite switch using intrinsic advantages of ferrite remanent magnetisation are discussed.

Evaluation of the single-electron density of states in a non Debye plasma

The single-electron density of states in a non Debye plasma is calculated in terms of parabolic cylinder and Kummer's confluent hypergeometric functions. An asymptotic expansion is given.