Modified Bessel Functions Research Articles

On new formulae for cumulative distribution function for McKay Bessel distribution

Motivated by recent results by Nadarajah, Srivastava and Gupta (2007) in which the authors introduced a new formula for CDF FK of the McKay Bessel function distribution containing the modified Bessel function we aim to present new formulae for FK, especially for certain values of parameters for which their formula fails to be calculated by in-built numerical routines. Also, several computational series formulae are obtained for the CDF of McKay Bessel distribution which contains modified Bessel function one is given in terms of the Horn confluent hypergeometric function and the others by Kampé de Fériet function Marcum Q-function and modified Bessel functions Horn confluent hypergeometric function and finally in terms of incomplete Gamma function γ and Exton’s double hypergeometric X–function.

On squares of odd zeta values and analogues of Eisenstein series

A Ramanujan-type formula involving the squares of odd zeta values is obtained. The crucial part in obtaining such a result is to conceive the correct analogue of the Eisenstein series involved in Ramanujan's formula for ζ(2m+1). The formula for ζ2(2m+1) is then generalized in two different directions, one, by considering the generalized divisor function σz(n), and the other, by studying a more general analogue of the aforementioned Eisenstein series, consisting of one more parameter N. A number of important special cases are derived from the first generalization. For example, we obtain a series representation for ζ(1+ω)ζ(−1−ω), where ω is a non-trivial zero of ζ(z). We also evaluate a series involving the modified Bessel function of the second kind in the form of a rational linear combination of ζ(4k−1) and ζ(4k+1) for k∈N.

A micropolar-Newtonian blood flow model through a porous layered artery in the presence of a magnetic field

In this work, we present a two-phase model of blood flow through a porous layered artery in the presence of a uniform magnetic field. The characteristic of suspensions in blood allows us to assume blood as a micropolar fluid in the core region and plasma as a Newtonian fluid in the peripheral region of a blood vessel. The wall of a blood vessel is porous and composed of a thin Brinkman transition layer followed by a Darcy porous layer of different permeabilities. A magnetic field of uniform strength is transversally applied to the direction of blood flow. The authors obtained an analytical solution of the problem of blood flow through the composite porous walled artery. Analytical expressions for the flow velocity, microrotational velocity, flow rate, and stresses at the wall have been obtained in the closed form using the modified Bessel function. The effects of various flow parameters on the two-fluid model of blood flow are analyzed graphically. An important conclusion which is drawn from the solution of the present problem is that the different permeabilities of Darcy and Brinkman regions of the porous layered artery have a significant effect on the flow. The present work is validated from the previously published literature studies.

On the Rayleigh-Taylor unstable dynamics of three-dimensional interfacial coherent structures with time-dependent acceleration

Rayleigh-Taylor instability (RTI) occurs in a range of natural and industrial processes. Whereas the majority of existing studies have considered constant acceleration, RTI is in many instances driven by variable acceleration. Here we focus on RTI driven by acceleration with a power-law time-dependence, and, by applying a group theoretic method, find solutions to this classical nonlinear boundary value problem. We deduce that the dynamics is dominated by the acceleration and that the solutions depend critically on the acceleration parameters for values of the acceleration exponent greater than (-2). We find that in the early-time dynamics, the RTI growth-rate is defined by modified Bessel functions. For the late-time dynamics, we link the interface dynamics with an interfacial shear function, find a continuous family of regular asymptotic solutions and identify invariant properties of nonlinear RTI. The essentially interfacial and multi-scale character of the dynamics is also demonstrated. The velocity field is potential in the bulk, and vortical structures may appear at the interface due to interfacial shear. The multi-scale character becomes clear from the invariance properties of the dynamics. We achieve good agreement with existing observations and elaborate new benchmarks for future research.

Analytical Solutions of the Internal Gravity Wave Equation in a Stratified Medium with Shear Flows

The problem of constructing internal gravity wave fields generated by an oscillating localized point source of disturbances in a stratified medium with an average shear flow is considered. A model distribution of shear flow over depth is considered, and an analytical solution of the problem is obtained in the form of a characteristic Green function expressed in terms of modified Bessel functions of imaginary index. Analytical expressions for the dispersion relations are obtained using Debye asymptotics of the modified Bessel functions. Integral representations of solutions are constructed. The wave characteristics of the excited fields are investigated depending on the basic parameters of the used stratification models, shear flows, and generation modes.

Axially Symmetric Vibrations of a Liquid-Filled Poroelastic Thin Cylinder Saturated with Two Immiscible Liquids Surrounded by a Liquid

This paper studies axially symmetric vibrations of a liquid-filled poroelastic thin cylinder saturated with two immiscible liquids of infinite extent that is surrounded by an inviscid elastic liquid. By considering the stress free boundaries, the frequency equation is obtained. Particular case, namely, liquid-filled poroelastic cylinder saturated with single liquid is discussed. When the wavenumber is large, the frequency equation is reduced to that of Rayleigh-type surface wave at the plane boundary of a poroelastic half-space. In this case, the asymptotic expressions of Bessel functions and modified Bessel functions are used. In both general and particular cases, the case of the propagation of Rayleigh waves in a poroelastic half-space is obtained. The parameter values of Columbia fine sandy loam saturated with air-water mixture are used for the numerical evaluation. In all the cases, phase velocity as a function of wavenumber is computed and presented graphically. From the numerical results, some inferences are drawn.

PEP Analysis of AF Relay NOMA Systems Employing Order Statistics of Cascaded Channels

The precise error performance analysis is challenging for non-orthogonal multiple access (NOMA) systems due to nonlinear successive interference cancellation (SIC) processing among NOMA users. In this paper, the pairwise error probability (PEP) performance of different users is investigated for relay NOMA simultaneous wireless information and power transfer (SWIPT) systems. By employing the order statistics theory, we obtain the ordered probability density function of the cascaded channel through Source-to-Relay-to-User links. Then we derive the analytical closed-form PEP expressions for NOMA users. To obtain the approximate closed-form PEP, we explore the finite series representation of the power of the modified Bessel function to replace the integrand terms. Monte Carlo simulation results show that the approximate analytical PEP of each user is basically in agreement with the simulated PEP. Furthermore, on the basis of the closed-form PEP, the influence of relevant system parameters on the error performance is examined via numerical simulations, which manifests that the choice of power allocation coefficients should be balanced between the users’ channel conditions and the demanded quality of service.

Approximating the Partially Coherent Additive White Gaussian Noise Channel in Polar Coordinates

We consider the partially coherent additive white Gaussian noise channel (PCAWGN) in optical communications and review the derivation of the exact channel conditional probability model in a closed-form solution in polar coordinates. In addition, we derive a reduced-complexity approximation by replacing the Rician and Tikhonov distributions describing amplitude and phase components, respectively, with their Gaussian approximation under certain assumptions of high SNR and low phase noise or jitter . Our proposal significantly reduces the hardware complexity by removing the modified Bessel functions involved in the exact solution. Furthermore, we compare the proposed approximation with a different metric previously found in the literature and observe that for maximum-likelihood hard symbol decision, both models are in perfect agreement with the optimal detector. However, our model not only reduces the required number of multiplications from 12 to 8 and additions from 9 to 3 (per computed symbol) but also reduces the information loss by at least one and up to several orders of magnitude with respect to the previously published metric when used to compute the channel achievable information rate (AIR). In all the simulation cases, we use QAM constellations of orders 4, 8, 16, and 32 as test input symbol sets.

The Dirichlet problem for an elliptic singular equation

ABSTRACTSecond-order singular elliptic equations of the form are considered. For a b>0 we find an analytical solution to the Dirichlet problem for this equation. A procedure for obtaining its solution is proposed based on the using of a modified Bessel function of the second kind.

An Asymptotic Expansion for the Error Term in the Brent-McMillan Algorithm for Euler’s Constant

The Brent-McMillan algorithm is the fastest known procedure for the high-precision computation of Euler’s constant γ and is based on the modified Bessel functions I_0(2x) and K_0(2x). An error estimate for this algorithm relies on the optimally truncated asymptotic expansion for the product I_0(2x)K_0(2x) when x assumes large positive integer values. An asymptotic expansion for this optimal error term is derived by exploiting the techniques developed in hyperasymptotics, thereby enabling more precise information on the error term than recently obtained bounds and estimates.

Continuum Limit of Random Matrix Products in Statistical Mechanics of Disordered Systems

We consider a particular weak disorder limit ("continuum limit") of matrix products that arise in the analysis of disordered statistical mechanics systems, with a particular focus on random transfer matrices. The limit system is a diffusion model for which the leading Lyapunov exponent can be expressed explicitly in terms of modified Bessel functions, a formula that appears in the physical literature on these disordered systems. We provide an analysis of the diffusion system as well as of the link with the matrix products. We then apply the results to the framework considered by Derrida and Hilhorst [J. Phys. A (1983)], which deals in particular with the strong interaction limit for disordered Ising model in one dimension and that identifies a singular behavior of the Lyapunov exponent (of the transfer matrix), and to the two dimensional Ising model with columnar disorder (McCoy-Wu model). We show that the continuum limit sharply captures the Derrida and Hilhorst singularity. Moreover we revisit the analysis by McCoy and Wu [Phys. Rev. 1968] and remark that it can be interpreted in terms of the continuum limit approximation. We provide a mathematical analysis of the continuum approximation of the free energy of the McCoy-Wu model, clarifying the prediction (by McCoy and Wu) that, in this approximation, the free energy of the two dimensional Ising model with columnar disorder is $C^\infty$ but not analytic at the critical temperature.

Study on MHD Cylindrical Couette Flow and Rheological Properties of Some Magnetic Suspensions

The study of magnetic suspensions (MS) and magnetic field effects on their rheological properties is of evident practical importance due to its ability to orient and change their physical properties, especially their viscosity, by magnetic fields. This research presents the effect of a uniform magnetic field on the flow of MS in the annular region between two concentric cylinders. The motion of the fluid is due to the rotation of the inner cylinder with a constant angular velocity. An exact solution of the governing equations is obtained in the form of modified Bessel functions of the first and second kinds. The torque, which must be applied to the inner cylinder in order to maintain the rotation, is also calculated. The results show that as the magnetic parameter increases, the velocity profile decreases, while the torque increases due to the effect of magnetic force against the flow direction.
 In order to model the magnetoviscous effects, experiments were performed for different shear rates and different magnetic field strengths by using specially designed rheometers. The studied samples are iron oxide-water-glycerol system, ferrofluid nanoparticles, MAG DX biocompatible ferrofluid. The theoretical analysis is based on Giesekus model for MS. This model gives more accurate results and takes into account the effects of viscoelastic shear thinning characteristics. It is found that a magnetic field increases the viscosity of all suspensions under consideration. Finally, new proposed correlation for the viscosity of MS as a function of both shear rate and magnetic field has been suggested.

Using Modified Bessel Functions for Analysis of Nonlinear Effects in a MOS Transistor Operating in Moderate Inversion

This paper describes analysis of nonlinear effects in a MOS transistor operating in moderate inversion and saturation. The dependence of the drain current on the gate-source and drain-source voltages is described using a modified version of the “reconciliation” model developed by Y. Tsividis. In the new model, the current components, which correspond to the terms depending exponentially on normalized gate-source or drain-source modulating sinusoidal voltages, are presented using modified Bessel functions. This approach allows one to find the first, second, and third harmonics of the drain current caused by the gate–source or drain–source voltage sinusoidal modulation and find also the intermodulation terms produced by these two modulating voltages. The results are applied to set the requirements to the gate–source and drain–source bias voltages in design of low-distortion and/or low-voltage amplifiers. It is shown that the realization of the stage with the zero value of third-order harmonic requires extremely tight tolerances for the threshold voltage. The suppression of intermodulation terms requires increased drain–source voltage. These recommendations are confirmed by simulations.

Life-span of semilinear wave equations with scale-invariant damping: Critical Strauss exponent case

The blow up problem of the semilinear scale-invariant damping wave equation with critical Strauss type exponent is investigated. The life span is shown to be: $T(\varepsilon)\leqslant \exp(C\varepsilon^{-p(p-1)})$ when $p=p_S(n+\mu)$ for $0 < \mu < \frac{n^2+n+2}{n+2}$. This result completes our previous study [9] on the sub-Strauss type exponent $p < p_S(n+\mu)$. Different from the work of M. Ikeda and M. Sobajima [5], we construct the suitable test function by introducing the modified Bessel function of second type. We note this method can be easily extended to some other scale-invariant wave models even with the Laplacian of variable coefficients.

Fast Evaluation of the Lightning Horizontal Electric Field Over a Lossy Ground Based on the Time-Domain Cooray–Rubinstein Formula

Cooray–Rubinstein formula has been extensively used for the evaluation of the horizontal component of the electric field generated by a lightning return stroke over a lossy ground. Considering the good characteristics of the formulation in terms of “step response” kernel $K_{\varepsilon }(t)$ corresponding to the ground surface impedance, proposed by Andreotti et al . recently, a new method for evaluating the Cooray–Rubinstein formula in the time domain is proposed in this paper. In this method, $K_{\varepsilon }(t)$ is first expressed by an integral of exponential function based on the integral formula of the modified Bessel function of the first kind, and then approximated by a piecewise exponential function using the Gauss numerical integration. Finally, taking advantage of the piecewise recursive convolution, the time-domain Cooray–Rubinstein formula for evaluating the lightning horizontal electric field can be calculated accurately and efficiently. The validity and efficiency of the proposed method are demonstrated by the comparisons with the methods in other literature.

Free Convective Oldroyd Fluid Flow through an Annulus under Transverse Magnetic Field Using Modified Bessel Functions

Free Convective Oldroyd Fluid Flow through an Annulus under Transverse Magnetic Field Using Modified Bessel Functions

An application of L1 estimates for oscillating integrals to parabolic like semi-linear structurally damped σ-evolution models

We study the following Cauchy problems for semi-linear structurally damped σ-evolution models:utt+(−Δ)σu+μ(−Δ)δut=f(u,ut),u(0,x)=u0(x),ut(0,x)=u1(x) with σ≥1, μ>0 and δ∈(0,σ2). Here the function f(u,ut) stands for the power nonlinearities |u|p and |ut|p with a given number p>1. We are interested in investigating L1 estimates for oscillating integrals in the presentation of the solutions to the corresponding linear models with vanishing right-hand sides by applying the theory of modified Bessel functions and Faà di Bruno's formula. By assuming additional Lm regularity on the initial data, we use (Lm∩Lq)−Lq and Lq−Lq estimates with q∈(1,∞) and m∈[1,q), to prove the global (in time) existence of small data Sobolev solutions to the above semi-linear models from suitable function spaces basing on Lq spaces.

Eigenvalue approach to a magneto-thermoelastic problem in transversely isotropic hollow cylinder: comparison of three theories

ABSTRACT The present article deals with the magneto-thermoelastic interactions in a transversely isotropic hollow cylinder. In the presence of the magnetic field, the governing equations are formulated for three-phase-lag model of generalized thermoelasticity. A vector-matrix differential equation is formed by using Laplace transform which is solved by the eigenvalue approach. In order to obtain the stress, displacement and temperature field, the field functions are expressed in terms of the modified Bessel functions in the Laplace transformed domain. When the outer radius of the hollow cylinder tends to infinity, the corresponding results are discussed. Finally, an appropriate Laplace transform inversion technique is adopted and comparison of three generalized thermoelastic theories for the considered parameters is presented graphically.

Inequalities for some integrals involving modified Bessel functions

Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases these inequalities are tight in certain limits. As a consequence, we deduce a tight double inequality, involving the modified Bessel function of the first kind, for a generalized hypergeometric function. We also present some open problems that arise from this research.

Turbulent wind field representation and conditional mean-field simulation.

The covariance structure of a homogeneous isotropic turbulent wind velocity field is derived in terms of modified Bessel functions for an extended form of the Kàrmàn velocity spectrum, including explicit expressions for the transverse coherence functions. A concept of transformed isotropic turbulence is introduced to account for differences in the axial, transverse and vertical fluctuating wind velocities and length scales in natural wind. A special form of the auto-regressive simulation format is developed for convected turbulence with exponentially increasing intervals to the regression planes. In each step, the wind velocity field in a transverse plane is represented by a conditional mean field and a stochastic contribution determined explicitly by the time-space covariances. Simulation results are presented for a square area of dimension less than the integral length scale, representative of buildings and wind turbines, and a horizontal line of length six times the length scale, representative of a long-span bridge. The simulations demonstrate high accuracy of simulated spectral densities, covariance functions and transverse coherence functions. The simulated results do not show visible dependence on the specific points used for the simulated records. The efficiency and the free simulation point configuration suggest high competitiveness compared to fast Fourier transform-based spectral methods.