Modified Bessel Functions Research Articles

Analysis of (P,m)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(P,\\mathrm{m})$\\end{document}-superquadratic function and related fractional integral inequalities with applications

In the present work we establish for the first time a class of (P,m)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(P,\\mathrm{m})$\\end{document}-superquadratic functions and look into its features. Using them, we come up with the Jensen and Hermite–Hadamard inequalities, as well as the fractional versions of Hermite–Hadamard inequalities with respect to Riemann–Liouville fractional integral operators. The findings are confirmed by certain numerical calculations and graphical depictions that take a few appropriate examples into account. The study is enhanced by the addition of applications of special means, moments of random variables, and modified Bessel functions of the first kind. This is achieved by considering new functions pertaining to the uniform probability density function and taking the modified Bessel functions of the first kind into consideration. The new results clearly provide extensions and improvements of the work given in the literature.

ANALYSIS ON MULTIPLICATIVELY (P,m)-SUPERQUADRATIC FUNCTIONS AND RELATED FRACTIONAL INEQUALITIES WITH APPLICATIONS

In this work, we, for the first time, establish a class of multiplicatively [Formula: see text]-superquadratic function and look into its various features. In the light of these features, we come up with the several integer order integral inequalities in the frame of multiplicative calculus. Moreover, we develop the fractional version of Hermite–Hadamard’s type inequalities involving midpoints and end points for multiplicatively [Formula: see text]-superquadratic function with respect to multiplicatively [Formula: see text]-Riemann–Liouville fractional integrals. By choosing different values for the parameters of such integral operators, we acquire a simple version of integral inequalities of Hermite–Hadamard’s type as well as its fractional form via multiplicatively Riemann–Liouville fractional integrals for multiplicatively [Formula: see text]-superquadratic function. The findings are confirmed by graphical illustration by taking appropriate examples into account. The study is further enhanced by the addition of applications of special means and first-type modified Bessel functions. The new results clearly provide extensions and improvements of the work available in the literature.

Transient and Steady-State Analysis of an M/PH2/1 Queue with Catastrophes

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.

Analytical solutions for the magneto-fluid–structure interaction dynamics with two degrees of freedom

The study focuses on the interaction between fluid and structure under an external magnetic field. The system consists of two rigid cylinders, with the inner cylinder having either translating vibrating or rotating vibrating freedoms. The solutions describing the coupling characteristics among the multi-physics fields are derived, using the zero- and first-order complex modified Bessel functions. The study analyzes the effects of the magnetic field on flow fields, electrical potential distributions, and cylinder vibration characteristics. It also reveals the physical mechanisms behind the coupled effects among fluid flow, cylinder vibration, and the electrical potential field. The study finds that for initial tangential rotating vibration, the magnetic field can stabilize both the flow field and cylinder vibration. Additionally, an increasing magnetic field can transform the cylinder vibration from underdamping to overdamping. On the other hand, for initial translating vibration, the magnetic field can induce instability in the system, with larger magnetic fields leading to divergent, negatively damped vibration. The critical magnetic fields for both rotating and translating vibrations are determined using phase maps. The study also shows that the combined effects of rotating and translating vibrations lead to asymmetry in the flow field. While translating vibration plays a dominant role in the combined vibration, it nonlinearly and non-monotonously influences the system dynamics.

On the Containment of the Unit Disc Image by Analytical Functions in the Lemniscate and Nephroid Domains

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.

Time-dependent behaviour of an M/M/3 heterogeneous server queueing system with multiple vacations subject to system disaster

This research paper investigates the dynamic behaviour of a heterogeneous three-server queuing model with multiple vacation. The model also accounts for the possibility that a disaster could occur during busy periods or on vacation, in which case the restoration process would begin immediately. The time-dependent probabilities of system size are presented in the work with explicit equations computed in terms of modified Bessel functions, employing generating functions. To support the theoretical conclusions, numerical examples are given.

Absolute moments of the variance-gamma distribution

We obtain exact formulas for the absolute raw and central moments of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. When the skewness parameter is equal to zero (the symmetric variance-gamma distribution), the infinite series reduces to a single term. Moreover, for the case that the shape parameter is a half-integer (in our parameterisation of the variance-gamma distribution), we obtain a closed-form expression for the absolute moments in terms of confluent hypergeometric functions. As a consequence, we deduce new exact formulas for the absolute raw and central moments of the asymmetric Laplace distribution and the product of two correlated zero mean normal random variables, and more generally the sum of independent copies of such random variables.

Some formulas for Bessel functions related to diag(1, −1, −1)-matrices and an intertwining operator

In this paper, we have developed novel proofs of two known formulas for the Meijer integral transform and obtained non-trivial extensions of these formulas (in one of them, the Bessel function is replaced by the Coulomb wave function). Also we have derived a new formula for a series involving product of Whittaker functions and modified Bessel functions of the second kind. Our proofs are based on a group theoretical approach and involves the use of an intertwining operator for two representations of one matrix group.

Asymptotics of the partition function of the perturbed Gross–Witten–Wadia unitary matrix model

AbstractWe consider the asymptotics of the partition function of the extended Gross–Witten–Wadia unitary matrix model by introducing an extra logarithmic term in the potential. The partition function can be written as a Toeplitz determinant with entries expressed in terms of the modified Bessel functions of the first kind and furnishes a ‐function sequence of the Painlevé equation. We derive the asymptotic expansions of the Toeplitz determinant up to and including the constant terms as the size of the determinant tends to infinity. The constant terms therein are expressed in terms of the Riemann zeta‐function and the Barnes ‐function. A third‐order phase transition in the leading terms of the asymptotic expansions is also observed.

A theoretical framework for robust implementation of in situ measurements of ocean currents and waves in dynamics of mooring systems

We develop an approximated method to solve analytically the equations of motion that describe mooring line dynamics in a one-dimensional model. For the first time, we derive integral closed-form expressions to compute dynamic properties of mooring lines subject to ocean currents and waves of arbitrary time and spatial dependence, in terms of modified Bessel functions. This is done by decomposing the mooring line in three regions where different approximations and mathematical techniques of solution are carried out. Our analytical results provide a robust framework to simulate and analyze extreme realistic oceanic events when data from in situ ocean observation systems are available, regardless of the resolution or coarseness of subsurface measurements and even for long acquisition times. In order to prove the advantages of this approach, we have processed data from two stations in the National Data Buoy Center of the National Oceanic and Atmospheric Administration. From simulations with ocean currents data, we have gained insights into the coupling of the spatial modulation of ocean currents with the characteristic wavelengths of elastic lines. From simulations with ocean waves data, we have defined a scheme to analyze wave data and identify the contribution of each subset of frequency peaks to the net fluctuation of mooring line tension. This could be useful for classification of irregular waves based on their impact on mooring line tension. The development of better tools that integrate theoretical and experimental findings is necessary for the assessment of marine structures under the environmental conditions associated with climate change.

Extreme eigenvalues of random matrices from Jacobi ensembles

Two-term asymptotic formulæ for the probability distribution functions for the smallest eigenvalue of the Jacobi β-Ensembles are derived for matrices of large size in the régime where β > 0 is arbitrary and one of the model parameters α1 is an integer. By a straightforward transformation this leads to corresponding results for the distribution of the largest eigenvalue. The explicit expressions are given in terms of multi-variable hypergeometric functions, and it is found that the first-order corrections are proportional to the derivative of the leading order limiting distribution function. In some special cases β = 2 and/or small values of α1, explicit formulæ involving more familiar functions, such as the modified Bessel function of the first kind, are presented.

Integral inequalities of h‐superquadratic functions and their fractional perspective with applications

The purpose of this article is to provide a number of Hermite–Hadamard and Fejér type integral inequalities for a class of ‐superquadratic functions. We then develop the fractional perspective of inequalities of Hermite–Hadamard and Fejér types by use of the Riemann–Liouville fractional integral operators and bring up with few particular cases. Numerical estimations based on specific relevant cases and graphical representations validate the results. Another motivating component of the study is that it is enriched with applications of modified Bessel function of first type, special means, and moment of random variables by defining some new functions in terms of modified Bessel function and considering uniform probability density function. The results in this paper have not been initiated before in the frame of ‐superquadraticity. We are optimistic that this effort will greatly stimulate and encourage additional research.

Influence of chemical reaction on transient natural convective flow in an infinite vertical cylinder packed with porous material

A study of mass and heat transport on fully developed laminar and transient free convective flow in a vertical cylinder containing a porous substance inside it. The Laplace Transform Method (LTM) is used to solve the governing equations with suitable boundary conditions. A compact form solution represented by using Bessel and modified Bessel functions. The numerical results were thoroughly scrutinized using graphs and tables to illustrate the effects of newly discovered fluid characteristics, including time, Grashof, mass Grashof, Schmidt, Prandtl, Darcy numbers, viscosity ratio on the concentration and the velocity profiles. The velocity and concentration profiles progressively increase with varying time values until reaching a steady state. Furthermore, it is found that the velocity profiles are decreased by the Prandtl, Schmidt number, and viscosity ratio. In contrast, the Darcy, Grashof, and mass Grashof numbers exhibit an opposite effect. The rate of mass transport, skin friction, and mass flux are presented in tables. The major finding of the current research is that the viscosity ratio parameter decelerates the skin friction and mass flux. This study has potential applications in innovative fields such as solar energy systems, chemical engineering processes, etc.

Asymptotic matching the self-consistent expansion to approximate the modified Bessel functions of the second kind

The self-consistent expansion (SCE) is a powerful technique for obtaining perturbative solutions to problems in statistical physics but it suffers from a subtle problem—too much freedom! The SCE can be used to generate an enormous number of approximations but distinguishing the superb approximations from the deficient ones can only be achieved after the fact by comparison to experimental or numerical results. Here, we propose a method of using the SCE to a priori obtain uniform approximations, namely asymptotic matching. If the asymptotic behaviour of a problem can be identified, then the approximations generated by the SCE can be tuned to asymptotically match the desired behaviour and this can be used to obtain uniform approximations over the entire domain of consideration, without needing to resort to empirical comparisons. We demonstrate this method by applying it to the task of obtaining uniform approximations of the modified Bessel functions of the second kind, Kα(x) .

Monotonicity rules for the ratio of two Laplace transforms with second-order oscillation

In this paper, we establish several monotonicity rules for the ratio of Laplace transforms when the ratio of two functions exhibits second-order oscillations. Before that, we first explore its discrete form, with the Z -transform being a specific case. Additionally, applications in the context of modified Bessel functions of the second kind are presented.

Effects of multiple relaxation times in the annular flow of pulsatile electro-osmotic flow of a complex biological fluid: blood with low and high cholesterol

This study investigates the electro-osmotic flow of a biological fluid (blood with varying cholesterol levels) in annular flow to simulate a first approximation to arterial occlusion. The fluid´s rheology is characterized by a multi-modal convected Maxwell model equation. The charge density follows the Boltzmann distribution, governing the electrical field. Mathematically, this scenario can be modeled by the Poisson–Boltzmann partial differential equation. Assuming a small zeta potential (less than 25 mV) using the Debye–Huckel approximation and considering a pulsatile electrical field, analytical solutions are derived using the Fourier transform formalism. These solutions, expressed in terms of the modified Bessel function, provide transfer functions for axial velocity and volumetric flow as functions of material parameters represented by characteristic dimensionless numbers. This study further analyzes thermal, electric, inertial, viscoelastic, and various interactions within the plasma, hematocrit, hematocrit–cholesterol, and cholesterol–cholesterol as well as weight concentration through numerical simulations. Finally, the flow and rheology predictions are validated using experimental data on human blood with varying cholesterol levels. The obtained transfer functions reveal that the electric–thermal–viscoelastic effects and the multiple geometric relationships contribute to the dynamic response of the interactions between the input electrical field and output volumetric flow and shear stress functions, leading to and evolution of resonance curves. It is noteworthy that electro-osmotic flow in blood with pathologies associated with low and high cholesterol has been scarcely reported in the literature on rheology. Thus, this work represents a significant contribution to the field.

Modelling and analysis of production management system using vacation queueing theoretic approach

This paper explores a queueing model in a production management context, featuring periods of working vacations and Bernoulli vacation. When there are no pending orders, the manufacturing unit transits into a maintenance phase, also termed as working vacation, during which production continues, albeit at a slower rate. This diminished productivity could result in longer lead times and potential customer dissatisfaction. Depending on the influx of demand, the maintenance phase could be interrupted, propelling the unit back to full operational capacity. Conversely, if no orders are received during the maintenance phase, the unit has the choice of switching to standby mode in anticipation of incoming orders, or initiating an extended break. We utilize mathematical techniques such as continued fractions, Modified Bessel functions, and Laplace transforms to precisely compute the transient state probabilities of this model. To further illustrate the impact of these operational dynamics on production management, we present corroborative numerical examples.

Feynman-Kac formula for general diffusion equations driven by TFBM with Hurst index H ∈ (0,1)

We consider the general diffusion equation driven by tempered fractional Brownian motion (TFBM)∂u(t,x)∂t=Lu(t,x)+u(t,x)dBH,λ(t)dt,(t,x)∈R+×Rd, where L is the infinitesimal generator of some time-homogeneous Markov process {Xt}t≥0 and {BH,λ(t)}t∈R is a tempered fractional Brownian motion with Hurst index H∈(0,12)∪(12,1) and tempering parameter λ>0 which is independent of {Xt}t≥0. Based on approximating TFBM with a family of Gaussian processes possessing absolutely continuous sample paths, a unified framework of the Feynman-Kac formulau(t,x)=Ex[f(Xt)]eBH,λ(t) is established for the general stochastic diffusion equation driven by TFBM with the initial value f, Hurst index H∈(0,12)∪(12,1) and tempering parameter λ>0. The difficulty is that the Hurst parameter H can be allowed to be less than 1/2. The idea is to explore the above simplest form, to utilize the techniques of Malliavin calculus. By using the properties of TFBM and especially that of the modified Bessel function of the second kind, we prove that the process defined by the Feynman-Kac formula is the mild and weak solutions of the general diffusion equation driven by TFBM. From the Feynman-Kac formula, we exhibit the smoothness of the density of the solution and the sharp Hölder regularity. We further show that limt→∞⁡ln⁡‖u(t,⋅)‖pg(t)=0 in the almost surely sense where p∈R and the function g:R+→R+ grows at least as fast as a linear function. The Feynman-Kac formula for the stochastic general diffusion equation in the Skorohod sense is also achieved by exploring the relationship between the Stratonovich and Skorohod integrals.

Multi-Dimensional Integral Transform with Fox Function in Kernel in Lebesgue-Type Spaces

This paper is devoted to the study of the multi-dimensional integral transform with the Fox H-function in the kernel in weighted spaces with integrable functions in the domain R+n with positive coordinates. Due to the generality of the Fox H-function, many special integral transforms have the form studied in this paper, including operators with such kernels as generalized hypergeometric functions, classical hypergeometric functions, Bessel and modified Bessel functions and so on. Moreover, most important fractional integral operators, such as the Riemann–Liouville type, are covered by the class under consideration. The mapping properties in Lebesgue-weighted spaces, such as the boundedness, the range and the representations of the considered transformation, are established. In special cases, it is applied to the specific integral transforms mentioned above. We use a modern technique based on the extensive use of the Mellin transform and its properties. Moreover, we generalize our own previous results from the one-dimensional case to the multi-dimensional one. The multi-dimensional case is more complex and needs more delicate techniques.

Love-like wave fields at the interface of sliding contact with non-local elastic heterogeneous fluid-saturated fractured poro-viscoelastic layer

The investigation delves into the propagation of Love-like waves within a stratified medium incorporating the impact of non-local elasticity and a sliding interface. The medium is characterized by an orthotropic viscoelastic layer with fractures and matrix porosity saturated with fluid positioned over an orthotropic viscoelastic half-space under initial stresses. The media is vertically heterogeneous with a binomial function of depth featuring a real positive exponent. Two distinct non-local parameters are considered for the layer and half-space, along with a sliding parameter accounting for interface sliding. The particle displacement components are expressed using modified Bessel functions of first and second kinds. A finite number of terms of the asymptotic representation of the modified Bessel functions and their derivatives have been used and compared with the exact solution. The dispersion equation is derived by neglecting fourth and subsequent powers of non-local parameters under suitable boundary conditions. Various cases are explored based on welded, partially, and fully sliding interfaces and compared with classical Love wave propagation. An interesting phenomenon has been found that under a fully sliding interface, the existence of Love-like wave vanishes, and the medium behaves as a separate layer and half-space with shear wave propagation.