Bessel Functions Of First Kind Research Articles

Control of chaos in Darcy–Bénard axisymmetric convection in a cylindrical enclosure using a uniform vertical cross-flow

The linear and weakly nonlinear stability analyses of Darcy–Bénard convection of a Newtonian fluid experiencing a uniform vertical cross-flow is investigated in the paper for various aspect ratios. Making use of the Maclaurin series representation, an expression for axial eigenfunctions is obtained with the radial eigenfunction being a Bessel function of first kind. These eigenfunctions are influenced by the Peclet number, Pe, the non-dimensional number that signifies the rate of vertical cross-flow. The modified-Vadasz-Lorenz model obtained in this paper has newly defined non-dimensional parameters that capture the influence of vertical cross-flow. From the linear stability analysis, it is found that the effect of introducing vertical cross-flow is to stabilize the system. Using a weakly nonlinear stability analysis, the closed-form expression of the Hopf-Rayleigh number as a function of Pe is obtained. Furthermore, the behavior of the modified-Vadasz–Lorenz model is analyzed using the largest Lyapunov exponent and the bifurcation diagram. This gives information about the intensity of chaos and occurrence of the periodic motion. We observe that the influence of vertical cross-flow is to increase the value of the Hopf–Rayleigh number and thereby to delay the onset of chaos. Furthermore, the appearance of the first periodic point is preponed due to the vertical cross-flow. As the rate of vertical cross-flow increases, the intensity of chaos decreases, thereby indicating that the effect of introducing vertical cross-flow is to suppress chaos.

Approximate controllability of a 2D linear system related to the motion of two fluids with surface tension

AbstractWe consider a coupled system of partial differential equations describing the interactions between a closed free interface and two viscous incompressible fluids. The fluids are assumed to satisfy the incompressible Navier‐Stokes equations in time‐dependent domains that are determined by the free interface. The mean curvature of the interface induces a surface tension force that creates a jump of the Cauchy stress tensor on both sides. It influences the behavior of the surrounding fluids, and therefore the deformation of this interface via the equality of velocities. In dimension 2, the steady states correspond to immobile interfaces that are circles with all the same volume. Considering small displacements of steady states, we are lead to consider a linearized version of this system. We prove that the latter is approximately controllable to a given steady state for any time by the means of additional surface tension type forces, provided that the radius of the circle of reference does not coincide with a scaled zero of the Bessel function of first kind.

Bessel functions of first kind and their Anuj transforms

Various problems of Statistics, Mathematics, Radio Physics, Nuclear Physics, Atomic Physics, Fluid Mechanics, Engineering and Science can easily handle by applying integral transform techniques on their mathematical models. Problems of heat equation, Schrodinger equation, Laplace equation, Helmholtz equation and wave equation have solutions in terms of Bessel functions. To solve such equations by integral transform methods, we need to know the integral transform of Bessel functions. In this paper, authors discuss Bessel functions of first kind and determine their Anuj transforms.

Inequalities for the modified Bessel function of first kind of non-negative order

Let Iν(x) be the modified Bessel function of order ν with real argument x. We present explicit error bounds for the asymptotic expansion of Iν(x) with x≥1. Two cases, ν an integer and ν a half-integer are considered separately. In addition, we present a short discussion on the error analysis for Iν(x) where ν is any non-negative real number.

Performance analysis of a fluid flow system modulated by a single server queue prone to catastrophic failures and repairs

<p style='text-indent:20px;'>This article deals with a new class of fluid flow queue regulated by an <inline-formula><tex-math id="M1">\begin{document}$ M/M/1 $\end{document}</tex-math></inline-formula> queueing system in the presence of active catastrophic failures and subsequent repairs of the server. The distribution of the stationary buffer-content is determined in terms of modified Bessel function of first kind. Besides, an explicit closed-form analytical expression for the probability density function of the fluid level in the stationary regime is derived in an elegant manner. Some important and essential performance measures are obtained. Finally, numerical results are provided to illustrate the behaviour of the buffer content distribution and the performance measures for the proposed fluid flow queueing model.</p>

Study on the Coincident-Loop Transient Electromagnetic Method in Seafloor Exploration—Taking Jiaodong Polymetallic Mine as a Model

The transient electromagnetic (TEM) method becomes more urgent than ever for marine exploration due to abundant resource reserves and the increasing undersea engineering construction activities, especially in the offshore exploration of mineral deposits such as Sanshandao gold mine. However, the research and application of TEM method in marine environment are still challenged by many problems. Such contradiction motivates our study on the coincident-loop TEM in seafloor exploration. The TEM response of coincident loops is firstly derived in the integral form, based on the potential functions in Helmholtz equations for a magnetic source locating in the whole-space layered model. The frequency-domain vertical magnetic field is described as the Hankel integral with double first-order Bessel functions of first kind. Secondly, the time-domain induced voltage is obtained by transforming the frequency-domain response through the cosine transform and then taking the derivative of time. To simultaneously solve the Hankel transform and the cosine transform, a novel algorithm is introduced by adapting the fixed-point quadrature and extrapolation via the Shanks transformation. Finally, a typical conductivity model for marine polymetallic deposit is designed to investigate the characteristic of TEM response under various conditions. Numerical results demonstrate that existence of conductive seawater causes the TEM response to increase significantly and decay slower. The air-sea reflected electromagnetic waves lead to a significantly large fake negative response (NR) in shallower seawater with depth less than 300 m. Increase in the height of loops will weaken and delay the anomaly response and shorten the observation time-window. The height of configuration should be no more than 100 m for shallower targets and 50 m for deeper targets, respectively. The observation time-window should cover 10-1 000 ms. Increase in the radius of loops only enhances the TEM response proportionally but hardly improves the relative anomaly. The vertical resolution on the low-resistivity target approximates 20 m for the configuration considered in the study. Decreases in D.C. resistivity and chargeability cause the positive response (PR) to increase significantly and decay more rapidly. Meanwhile, the NR is advanced and enlarged significantly and decays slower compared with the PR. The influence of time constant is not monotony and there exists an optimal value for producing the maximum NR. As the frequency parameter increases, the PR is caused to decay more rapidly without magnitude change and the NR is advanced and decays more rapidly with significant increase in magnitude. The influence of frequency parameter is more pronounced than that of time constant.

Radii of Starlikeness and Convexity of Some Entire Functions

A normalized analytic function f is lemniscate starlike if the quantity $$zf'(z)/f(z)$$ lies in the region bounded by the right half of the lemniscate of Bernoulli $$|w^2-1|=1$$ . It is Janowski starlike if the quantity $$zf'(z)/f(z)$$ lies in the disk whose diametric end points are $$(1-A)/(1-B)$$ and $$(1+A)/(1+B)$$ for $$-1\le B<A\le 1$$ . The radii of lemniscate starlikeness and Janowski starlikeness have been determined for normalizations of q-Bessel functions, Bessel functions of first kind of order $$\nu $$ and Lommel functions of first kind. Corresponding convexity radii are also determined.

Averaging Gaussian functionals

This paper consists of two parts. In the first part, we focus on the average of a functional over shifted Gaussian homogeneous noise and as the averaging domain covers the whole space, we establish a Breuer-Major type Gaussian fluctuation based on various assumptions on the covariance kernel and/or the spectral measure. Our methodology for the first part begins with the application of Malliavin calculus around Nualart-Peccati's Fourth Moment Theorem, and in addition we apply the Fourier techniques as well as a soft approximation argument based on Bessel functions of first kind. The same methodology leads us to investigate a closely related problem in the second part. We study the spatial average of a linear stochastic heat equation driven by space-time Gaussian colored noise. The temporal covariance kernel $\gamma_0$ is assumed to be locally integrable in this paper. If the spatial covariance kernel is nonnegative and integrable on the whole space, then the spatial average admits Gaussian fluctuation; with some extra mild integrability condition on $\gamma_0$, we are able to provide a functional central limit theorem. These results complement recent studies on the spatial average for SPDEs. Our analysis also allows us to consider the case where the spatial covariance kernel is not integrable: For example, in the case of the Riesz kernel, the first chaotic component of the spatial average is dominant so that the Gaussian fluctuation also holds true.

Impact of customers’ impatience on an M/M/1 queueing system subject to differentiated vacations with a waiting server

ABSTRACTImpatience behavior of an M/M/1 queueing system, which has differentiated server vacations, is considered. Arrivals follow a Poisson process, service is exponentially distributed, as are both vacation types. When the system is empty after waiting for a random period of time, the server takes a vacation and returns after a random duration. If there are still no customers in the system, server can go for a vacation of shorter duration. If the server is in a vacation state, arriving customers become impatient with an individual timer which is exponentially distributed. Explicit expressions for time-dependent probabilities of the system size are obtained in terms of the modified Bessel function of first kind by making use of Laplace transforms and probability generating function techniques along with continued fractions and the confluent hypergeometric function. Explicit expressions for the time-dependent mean and variance of the system size are also derived. Moreover, steady-state probabilities for the system size are derived from the transient state probabilities. Finally, a numerical example is presented to study the behavior of the system.

Inverses of generators of integrated fractional resolvent operator functions

Abstract This paper is devoted to the inverse generator problem in the setting of generators of integrated resolvent operator functions. It is shown that if the operator A is the generator of a tempered β-times integrated α-resolvent operator function ((α, β)-ROF) and it is injective, then the inverse operator A −1 is the generator of a tempered (α, γ)-ROF for all γ &gt; β + 1/2, by means of an explicit representation of the integrated resolvent operator function based in Bessel functions of first kind. Analytic resolvent operator functions are also considered, showing that A −1 is in addition the generator of a tempered (δ, 0)-ROF for all δ &lt; α. Moreover, the optimal decay rate of (α, β)-ROFs as t → ∞ is given. These result are applied to fractional Cauchy problem unsolved in the fractional derivative.

Transient Analysis of an M/M/1 Queue with Reneging, Catastrophes, Server Failures and Repairs

An M / M / 1 queue with reneging, catastrophes, server failures and repairs is considered. The arrivals follow a Poisson process and the servers serve according to an exponential distribution. On arrival a customer decides to join the queue and after joining the queue if a customer has to wait for the service longer than his expectation, he may renege. Explicit expression for the time-dependent probabilities of the system size is obtained in terms of the modified Bessel function of first kind by making use of Laplace transform and probability generating function techniques. The system queue length and failure distribution for steady state are derived. Additionally, time-dependent mean and variance are obtained. A numerical example is presented to study the behavior of the system.

Secrecy Mutual Information of the Independent and Identically Distributed Nakagami-q Fading SIMO Channel

ABSTRACTIn this paper, we consider an independent and identically distributed Nakagami-fading single-input multiple-output (SIMO) channel, where the transmitter sends some confidential message to the legitimate receiver in the presence of an eavesdropper. Both the receiver and the eavesdropper have multiple antennas. We formulated the probability of non-zero secrecy mutual information, secure outage probability, and ergodic secrecy mutual information for the SIMO wireless communication system. The calculations have been carried out using small limit argument approximation on zeroth-order modified Bessel function of first kind. This approximation provides decent results in the low signal-to-noise-ratio (SNR) regime. In the low-SNR regime, either the transmission power is small or the bandwidth is large. In either case, it is difficult for an eavesdropper to detect the signals in this regime. We present that in the presence of fading, positive secrecy mutual information is achievable even when the eavesdropper has better SNR than the main channel. We also found in our work that at low-SNR regime, the effect of number of eavesdropper's channel on average secrecy mutual information is almost negligible.

The Evaluation of Derivatives of Double Barrier Options of the Bessel Processes by Methods of Spectral Analysis

The paper deals with the spectral methods to calculate the value of the double barrier option generated by the Bessel diffusion process. This technique enables us to calculate the option price in the form of a Fourier-Bessel series with the corresponding ratio. The authors propose a simple method to estimate options using the Green’s expansion function for boundary value problem for a singular parabolic equation. Thus, the accuracy of the estimation coincides with the accuracy of the convergence of the Fourier-Bessel series. In this paper, the authors use the spectral theory to calculate the price of derivatives of financial assets considering that the processes are described by Markov and can be considered in Hilbert spaces. In this work, the authors use the diffusion process to find derivatives prices by introducing them through the Bessel functions of first kind. They also examine the Sturm-Liouville problem where the boundary conditions utilize the Bessel functions and their derivatives. All assumptions lead to analytical formulae that are consistent with the empirical evidence and, when implemented in practice, reflect adequately the passage of processes on stock markets. The authors also focus on the financial flows generated by Bessel diffusion processes which are presented in the system of Bessel functions of the first order under the condition that the linear combination of the flow and its spatial derivative are taken into account. Such a presentation enables us to calculate the market value of a share portfolio, provides the measurement of internal volatility in the market at any given time, and allows us to investigate the dynamics of the stock market. The splitting of Green’s function in the system of Bessel functions is presented by an analytical formula which is convenient for calculating the price level of options.

Radii Problems for Normalized Bessel Functions of First Kind

For three different normalizations of Bessel functions of first kind, the radius of k-parabolic starlikeness and k-uniform convexity of order $$\alpha $$ are determined. The radius of strong starlikeness and other related radius are also obtained for these functions. We also find optimal parameters for which these functions are k-parabolic starlike and k-uniformly convex in the open unit disk.

The evaluation of derivatives of double barrier options of the Bessel processes by methods of spectral analysis

The paper deals with the spectral methods to calculate the value of the double barrier option generated by the Bessel diffusion process. This technique enables us to calculate the option price in the form of a Fourier-Bessel series with the corresponding ratio. The autors propose a simple method to estimate options using the Green’s expansion function for boundary value problem for a singular parabolic equation. Thus, the accuracy of the estimation coincides with the accuracy of the convergence of the Fourier-Bessel series. In this paper, the authors use the spectral theory to calculate the price of derivatives of financial assets considering that the processes described are by Markov and can be considered in Hilbert spaces. In this work, the authors use the diffusion process to find derivatives prices by introducing them through the Bessel functions of first kind. They also examine the Sturm-Liouville problem where the boundary conditions utilize the Bessel functions and their derivatives. All assumptions lead to analytical formulae that are consistent with the empirical evidence and, when implemented in practice, reflect adequately the passage of processes on stock markets. The authors also focus on the financial flows generated by Bessel diffusion processes which are presented in the system of Bessel functions of the first order under the condition that the linear combination of the flow and its spatial derivative are taken into account. Such a presentation enables us to calculate the market value of a share portfolio, provides the measurement of internal volatility in the market at any given time, and allows us to investigate the dynamics of the stock market. The splitting of Green’s function in the system of Bessel functions is presented by an analytical formula which is convenient for calculating the price level of options.

Klein–Gordon equation for a charged particle in space-varying electromagnetic fields–A systematic study via the Laplace transform

Exact solutions of the Klein–Gordon equation for a charged particle in the presence of three spatially varying electromagnetic fields, namely, (i) E→=αβ0e−αx2x^2,B→=αβ1e−αx2x^3 (ii) E→=β0′x22x^2,B→=β1′x22x^3, and (iii) E→=2β0′x23x^2,B→=2β1′x23x^3, are studied. All these fields are generated from a systematic study of a particular type of differential equation whose coefficients are linear in the independent variable. The Laplace transform approach is used to find the solutions, and the corresponding eigenfunctions are expressed in terms of the hypergeometric functions 1F1(a′, b′; x) for the first two cases of the above configurations, while the same are expressed in terms of the Bessel functions of first kind, Jn(x), for the last case.

Numerical solutions of hyperbolic telegraph equation by using the Bessel functions of first kind and residual correction

In this study, a collocation method is presented to solve one-dimensional hyperbolic telegraph equation. The problem is given by hyperbolic telegraph equation under initial and boundary conditions. The method is based on the Bessel functions of the first kind. Using the collocation points and the operational matrices of derivatives, we reduce the problem to a set of linear algebraic equations. The determined coefficients from this system give the coefficients of the approximate solution. Also, an error estimation method is presented for the considered problem and the method. By using the residual function and the original problem, an error problem is constructed and thus the error function is estimated. By aid of the estimated function, the approximated solution is improved. Numerical examples are given to demonstrate the validity and applicability of the proposed method and also, the comparisons are made with the known results.

An Application of Generalized Bessel Functions on Certain Subclasses of Analytic Functions

The purpose of the present paper is to investigate some characterization for generalized Bessel functions of first kind to be in the new subclasses of β uniformly starlike and β uniformly convex functions of order α. Further we point out consequences of our main results.

Role of Heat Generation/Absorption on Mixed Convection Flow in a Vertical Tube Filled with Porous Material Having Time-Periodic Boundary Condition: Steady-Periodic Regime

This paper discusses the hydrodynamic and thermal behaviours of a fully developed mixed convective flow of a viscous incompressible heat generating/absorbing fluid in a vertical tube filled with isotropic porous material having time-periodic boundary condition. The dimensionless governing equations of motion and energy subjected to the Boussinesq approximation and asymmetric heat conditions are solved analytically. Closed-form solutions are expressed in terms of modified Bessel function of first kind. The solutions obtained are graphically represented, and the effects of the dimensionless heat generation/absorption H, the dimensionless frequency \(\Omega \) and the Darcy number Da on the fluid flow are investigated. It is discovered that velocity is maximum at two different locations in the flow domain, one near the surface of the tube and another at the axis of the tube. The presence of the heat generation parameter is seen to enhance the temperature distribution and this is reflected as increase in the magnitude of the oscillation dimensionless velocity, whereas in the presence of heat absorption a reversed trend occurs.

Connections between certain classes of harmonic univalent mappings involving generalized Bessel functions

In the present paper, we obtain some sufficient conditions of certain convolution operator involving generalized Bessel functions of first kind belonging to various subclasses of harmonic univalent functions. To be more precise, we investigate such connections with harmonic g-uniformly convex and harmonic g-uniformly starlike mappings in the plane.