Modified Bessel Functions Research Articles

A high accuracy numerical approach for electro-hydrodynamic flow of a fluid in an ion-drag configuration in a circular cylindrical conduit

This paper deals with the construction of a computational approach based on B-spline functions for solving a nonlinear boundary value problem describing the electro-hydrodynamic flow (EHF) of a fluid in a circular cylindrical conduit. The radial dependence of the velocity field emerging in the EHF is computed. We study the effects of two relevant parameters, namely the Hartmann electric number H and the strength of the nonlinearity β, on the velocity field. Computational results show that the method is of sixth-order accuracy. It is shown that the Hartmann electric number (HEN) and the strength of the nonlinearity both have a profound impact on the velocity profile of EHF and that these effects can be understood from analytical considerations. In particular, quantitative results include: The velocity, taking its maximum at the center of the conduit, does not exceed the value 1/(1+β) (this confirms a previous result). At large HEN, a boundary layer develops near the outer radial boundary of the conduit (r=1). Its thickness is proportional to 1/(H1+β), being determined by both the HEN and the nonlinearity. Moreover, when a boundary layer is present, the flow velocity has a plug-like profile approaching the plateau value 1/(1+β) (from below) for r values smaller than those of the boundary layer. If both the nonlinearity and the HEN are too small for a boundary layer to develop, then the flow profile is essentially parabolic and describable via a modified Bessel function. The CPU time for our method is provided.

Improved algorithm for the quasi-discrete Hankel transform.

An improved algorithm for numerical evaluation of the Hankel transform is developed. The algorithm originally proposed by Yu et al. [Opt. Lett.23, 409 (1998)OPLEDP0146-959210.1364/OL.23.000409] uses the quadrature in which the nodes are zeros of the Bessel function. In this work, it is shown that the accuracy of the algorithm can be significantly improved, with virtually no increase in computation time, via two steps. One is to halve the weight of the last node, and the other is to extrapolate a function tail using the modified Bessel function of the second kind, which gives the analytical estimation of the integral remainder.

Effects of strain gradient and electromagnetic field gradient on potential and field distributions of multiferroic fibrous composites

The objective of this work is to study the effect of strain gradient, electric field gradient, and magnetic field gradient on the potential and field distributions of multiferroic fibrous composites subjected to generalized anti-plane shear deformation. A detailed energy variational formulation with strain, strain gradient, electric field, electric field gradient, magnetic field, and magnetic field gradient as independent variables is provided, and the equilibrium equations with the complete boundary conditions are simultaneously determined. Three internal characteristic lengths of the underlying microstructure are introduced in the constitutive equations. The general solutions in cylindrical polar coordinates consisting of the modified Bessel functions of first and second kind are derived. One inclusion problem’s solution is also obtained. The derived solutions are applied to a $$\hbox {CoFe}_{2}\hbox {O}_{4}$$ -PZT-4 composite to demonstrate the effect of strain gradient, electric field gradient, and magnetic field gradient. Numerical results provide insights into the role of characteristic lengths on the potential and field distributions of a multiferroic fibrous composite.

Sticky Bessel diffusions

We consider a Bessel process X of dimension δ∈(0,2) having 0 as a slowly reflecting (sticky) boundary point with a stickiness parameter 1/μ∈(0,∞). We show that (i) the process X can be characterised through its square Y=X2 as a unique weak solution to the SDE system dYt=δI(Yt>0)dt+2YtdBtI(Yt=0)dt=12μdℓt0(Y) where B is a standard Brownian motion and ℓ0(Y) is a diffusion local time process of Y at 0, and (ii) the transition density function of X can be expressed in the closed form as a convolution integral involving a Mittag-Leffler function and a modified Bessel function of the second kind. Appearance of the Mittag-Leffler function is novel in this context. We determine a (sticky) boundary condition at zero that characterises the transition density function of X as a unique solution to the Kolmogorov backward/forward equation of X. We also show that the convolution integral can be characterised as a unique solution to the generalised Abel equation of the second kind. Letting μ↓0 (absorption) and μ↑∞ (instantaneous reflection) the closed-form expression for the transition density function of X reduces to the ones found by Feller (1951) and Molchanov (1967) respectively.

Geometric Properties of the Products of Modified Bessel Functions of the First Kind

In this paper, the products of modified Bessel functions of the first kind are considered. Our main aim is to provide various conditions so that these products have certain geometric properties such as starlikeness, convexity, and close to convexity (univalency) with respect to certain functions in the unit disk. Subordinating factor sequences for these products are also obtained. Interesting consequences are provided. The results obtained are new in the literature.

A method of composition orthogonality and new sequences of orthogonal polynomials and functions for non-classical weights

A new method of composition orthogonality is introduced. It is applied to generate new sequences of orthogonal polynomials and functions. In particular, classical orthogonal polynomials are interpreted in the sense of composition orthogonality. Finally, new sequences of orthogonal polynomials with respect to the weight function ρν2(x),ρν(x)=2xν/2Kν(2x),x>0,ν≥0, where Kν(z) is the modified Bessel function or Macdonald function, are investigated. Differential properties, recurrence relations, explicit representations, generating functions and Rodrigues-type formulae are obtained. The corresponding multiple orthogonal polynomials are exhibited.

On new approximations for the modified Bessel function of the second kind \(K_0(x)\)

A new series representation of the modified Bessel function of the second kind \(K_0(x)\) in terms of simple elementary functions (Kummer's function) is obtained. The accuracy of different orders in this expansion is analysed and has been shown not to be so good as those of different approximations found in the literature. In the sequel, new polynomial approximations for \(K_0(x)\), in the limits \(0< x\leq 2\) and \(2\leq x < \infty\), are obtained. They are shown to be much more accurate than the two best classical approximations given by the Abramowitz and Stegun's Handbook, for those intervals.

Discrete Lebedev–Skalskaya transforms

Discrete analogs of the Lebedev–Skalskaya transforms are introduced and investigated. It involves series and integrals with respect to the kernels is the imaginary unit and is the modified Bessel function. The corresponding inversion formulas for suitable functions and sequences in terms of these series and integrals are established when . The case reduces to the Kontorovich–Lebedev transform.

Discrete Lebedev’s index transforms

Discrete analogs of the Lebedev transforms with the product of the modified Bessel functions are introduced and investigated. Several expansions of suitable functions and sequences in terms of the series and integrals, involving the modified Bessel and Struve functions are established.

Efficient spherical surface integration of Gauss functions in three-dimensional spherical coordinates and the solution for the modified Bessel function of the first kind

An efficient solution of calculating the spherical surface integral of a Gauss function defined as $$h\left( {s,{\mathbf{Q}}} \right) = \int_{0}^{2\pi } {\int_{0}^{\pi } {\left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{x}^{i} \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{y}^{j} \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{z}^{k} e^{{ - \gamma \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)^{2} }} } } \sin \theta d\theta d\varphi$$ is provided, where $$\gamma \ge 0$$ , and i, j, k are nonnegative integers. A computationally concise algorithm is proposed for obtaining the expansion coefficients of polynomial terms when the coordinate system is transformed from cartesian to spherical. The resulting expression for $$h\left( {s,{\mathbf{Q}}} \right)$$ includes a number of cases of elementary integrals, the most difficult of which is $$II\left( {n,\mu } \right) = \int_{0}^{\pi } {\cos^{n} \theta e^{ - \mu \cos \theta } } d\theta$$ , with a nonnegative integer n and positive μ. This integral can be formed by linearly combining modified Bessel functions of the first kind $$B(n,\mu ) = \frac{1}{\pi }\int\limits_{0}^{\pi } {e^{\mu \cos \theta } \cos \left( {n\theta } \right)d\theta }$$ , with a nonnegative integer n and negative μ. Direct applications of the standard approach using Mathematica and GSL are found to be inefficient and limited in the range of the parameters for the Bessel function. We propose an asymptotic function for this expression for n = 0,1,2. The relative error of asymptotic function is in the order of 10−16 with the first five terms of the asymptotic expansion. At last, we give a new asymptotic function of $$B\left( {n,\mu } \right)$$ based on the expression for $$e^{ - \mu } II\left( {n,\mu } \right)$$ when n is an integer and μ is real and large in absolute value.

The modified Bessel functions I 3 / 4(x) and I − 3 / 4(x) in certain fractional differential equations

Linear fracctional equations has been considered with modified Bessel functions I 3 / 4(x) and I − 3 / 4(x) as no-homogeneus part. The result have been found using Laplace transform. Dun results lead to generalized hypergeometric functions. Several Coincidences have found between the results for the modified Bessel functions of the some order by different sign. No limitations have been found for the fracctional derivative, with could be any real numbers. Some specific values of Alpha have been also consifered.

Calculation of thin-layer rubber-metal elements of machines and buildings considering the compressibility of the material

In some constructions with rubber-metal elements, the rubber is loaded in such a way that the hydrostatic stress is tens of times greater than the shear stress. In this case, ignoring the compressibility of rubber can lead to large errors in calculating the elastic and strength parameters of rubber parts. The elastic elements, for which consideration of the material compressibility is essential, include thin-layer rubber-metal elements, whose dimensions in plan are much larger than the thickness. Such rubber-metal elements are used for vibration and seismic protection of heavy machines, buildings and structures. In this paper, the compression of a thin-layer cylindrical rubber-metal element is considered with taking into account the compressibility of the material. In the calculation, it was assumed that the hydrostatic stress is constant over the thickness of the rubber layer, and also that the rate of change of the displacement components in the middle surface of the layer can be neglected in comparison with the corresponding speed in the direction normal to the layer. These assumptions were substantiated as a result of the analysis of the geometry of deformation and equilibrium conditions of a thin layer of rubber bonded to metal reinforcement. As a result of applying these assumptions to the solution of the problem set, a boundary value problem is obtained for an ordinary inhomogeneous linear differential equation of the second order. The corresponding homogeneous equation is solved using a power series, the coefficients of which are expressed in terms of the previous ones using a recurrent formula. Finally, a formula with a modified Bessel function of the first kind is obtained for the hydrostatic stress. The analysis of the obtained formula leads to some simplifications for different values of the dimensionless parameter, which connects the bulk modulus and shear modulus, and the radius of the cylindrical element and its thickness. The resulting axial load on the rubber element is represented by the force that would be perceived by the element in case of uniaxial deformation, i.e. if displacements in cross-sections were prohibited and compression would be performed only due to the change in volume, multiplied by a certain coefficient less than one, which takes into account the decrease in the stiffness of the element due to shear deformation. The paper also presents the results of experimental studies carried out by the authors for thin-layer cylindrical rubber-metal elements made of medium-filled rubber 2959. An example of calculating the axial load on a thin-layer element considering the compressibility of rubber is given. The obtained axial load value coincides with the experimental results with a sufficient accuracy value. Keywords: thin-layer rubber-metal element, rubber compressibility, hydrostatic stress, axial load.

Determinantal Expressions and Recursive Relations for the Bessel Zeta Function and for a Sequence Originating from a Series Expansion of the Power of Modified Bessel Function of the First Kind

In the paper, by virtue of a general formula for any derivative of the ratio of two differentiable functions, with the aid of a recursive property of the Hessenberg determinants, the authors establish determinantal expressions and recursive relations for the Bessel zeta function and for a sequence originating from a series expansion of the power of modified Bessel function of the first kind.

New Accurate Approximation for Average Error Probability

This paper proposes new accurate approximations for average error probability (AEP) of a communication system employing either $M$ -phase-shift keying (PSK) or differential quaternary PSK with Gray coding (GC-DQPSK) modulation schemes over generalized fading channel. Firstly, new accurate approximations of error probability (EP) of both modulation schemes are derived over additive white Gaussian noise (AWGN) channel. Leveraging the trapezoidal integral method, a tight approximate expression of symbol error probability for $M$ -PSK modulation is presented, while new upper and lower bounds for Marcum $Q$ -function (MQF) of the first order, and subsequently those for bit error probability (BEP) under DQPSK scheme, are proposed. Next, these bounds are linearly combined to propose a highly refined and accurate BE P’s approximation. The key idea manifested in the decrease property of modified Bessel function $I_{v}$ , strongly related to MQF, with its argument $v$ . As an application, these approximations are used to tackle AEP’s approximation under $\kappa -\mu $ shadowed fading. Numerical results show the accuracy of the presented approximations compared to the exact ones.

Geometric Brownian motion with affine drift and its time-integral

The joint distribution of a geometric Brownian motion and its time-integral was derived in a seminal paper by Yor (1992) using Lamperti’s transformation, leading to explicit solutions in terms of modified Bessel functions. In this paper, we revisit this classic result using the simple Laplace transform approach in connection to the Heun differential equation. We extend the methodology to the geometric Brownian motion with affine drift and show that the joint distribution of this process and its time-integral can be determined by a doubly-confluent Heun equation. Furthermore, the joint Laplace transform of the process and its time-integral is derived from the asymptotics of the solutions. In addition, we provide an application by using the results for the asymptotics of the double-confluent Heun equation in pricing Asian options. Numerical results show the accuracy and efficiency of this new method.

Classification Of Uniform Flag Triangulations Of The Boundary Of The Full Root Polytope Of Type A

The full root polytope of type A is the convex hull of all pairwise differences of the standard basis vectors which we represent by forward and backward arrows. We completely classify all flag triangulations of this polytope that are uniform in the sense that the edges may be described as a function of the relative order of the indices of the four basis vectors involved. These fifteen triangulations fall naturally into three classes: three in the lex class, three in the revlex class and nine in the Simion class. We also consider a refined face count where we distinguish between forward and backward arrows. We prove the refined face counts only depend on the class of the triangulations. The refined face generating functions are expressed in terms of the Catalan and Delannoy generating functions and the modified Bessel function of the first kind.

Nonlinear Hyperbolic Waves in Relativistic Gases of Massive Particles with Synge Energy

In this article, we study some fundamental properties of nonlinear waves and the Riemann problem of Euler’s relativistic system when the constitutive equation for energy is that of Synge for a monatomic rarefied gas or its generalization for diatomic gas. These constitutive equations are the only ones compatible with the relativistic kinetic theory for massive particles in the whole range from the classical to the ultra-relativistic regime. They involve modified Bessel functions of the second kind and this makes Euler’s relativistic system rather complex. Based on delicate estimates of the Bessel functions, we prove: (i) a limit on the speed of sound of $$1{/}\sqrt{3}$$ times the speed of light (which a fortiori implies subluminality, that is causality), (ii) the genuine non-linearity of the acoustic waves, (iii) the compatibility of Rankine–Hugoniot relations with the second law of thermodynamics (entropy growth through all Lax shocks), and (iv) the unique resolvability of the initial value problem of Riemann (if we include the possibility of vacuum as in the non-relativistic context).

Analytical Approximations of Dispersion Relations for Internal Gravity Waves Equation with Shear Flows

The problem of internal gravity waves fields in a stratified medium of finite depth is considered for model distributions of background shear currents. For the analytical solution of the problem, a constant distribution of the buoyancy frequency and various linear dependences of the background shear current on depth were used. The dispersion dependences are obtained, which are expressed in terms of the modified Bessel function of the imaginary index. Under the Miles–Howard stability condition and large Richardson numbers, the Debye asymptotics of the modified Bessel function of the imaginary index were used to construct analytical solutions. The dispersion equation is solved using the proposed analytical approximation. The properties of the dispersion equation are studied and the main analytical characteristics of the dispersion curves are investigated depending on the parameters of background shear flows.

Hall effect on MHD transient free convection flow of chemically reactive Casson fluid with heat source/sink past an infinite vertical cylinder

This article presents an analytical investigation of the effect of Hall current, magnetic field, and chemical reaction on an unsteady MHD transient free convection heat and mass transfer flow of viscous incompressible, electrically conducting Casson fluid past an infinite vertical cylinder subject to heat source and sink. Closed-form solutions are obtained by employing Laplace transform and inverted using Mellin-Fourier integral and presented them in terms of Bessel functions and modified Bessel functions. To get an insight into the model, we have depicted the graphs of velocity, temperature and concentration profiles for various values of the pertinent flow parameters. The values of Sherwood number, Nusselt number, and skin friction are calculated and presented in the tabular form. Further, we have also considered a special case and investigate the velocity profiles of the Newtonian fluid for both heat source and sink as the Casson parameter α → ∞. From the investigation, it is perceived that with an increase in the values of chemical reaction parameters the species concentration and the velocity of the Casson fluid get reduced while its reverse trend has been observed on Sherwood number and skin friction. With an increase in the Prandtl number the fluid temperature decreases whereas, the Nusselt number increases. A crossing over point is observed in velocity distribution for the Hall current parameter. This fluid flow model will be applicable in geophysics, mining industry, chemical and biomedical engineering such as underground energy system, electromagnetic material production, and automotive fuel level indication, etc.

Transient thermoelastic analysis of a semi-infinite circular plate with the power-law nonhomogeneity subjected to ramp-type sectional heating

The main objective is to investigate the transient thermoelastic response on a nonhomogeneous semi-infinite elastic circular plate that is subjected to ramp-type heating over the concentric circular region of the upper face. The nonhomogeneous plate has thermo-mechanical properties expressed in the form of power-law through the plate thickness with variable z and with constant Poisson's ratio. The transient nonhomogeneous heat conduction differential equation is solved using a modified integral transformation technique in terms of modified Bessel functions. The associated thermal stress components are evaluated theoretically by incorporating the thermoelastic displacement potential feature based on the Boussinesq-Papkovich method. Numerical calculation is carried out the functionally graded ceramic – metal based material plate, in which partially stabilized zirconia (PSZ) and austenitic stainless steel (SUS304) are subjected to ramp-type heating on one of the plane boundaries; whereas the other plane boundary is held at zero temperature. The analyzed findings indicate strong consent with those available in the literature.