Bessel Functions Research Articles

Focussing of concentric free-surface waves

Gravito–capillary waves at free surfaces are ubiquitous in several natural and industrial processes involving quiescent liquid pools bounded by cylindrical walls. These waves emanate from the relaxation of initial interface distortions, which often take the form of a cavity (depression) centred on the symmetry axis of the container. The surface waves reflect from the container walls leading to a radially inward propagating wavetrain converging (focussing) onto the symmetry axis. Under the inviscid approximation and for sufficiently shallow cavities, the relaxation is well-described by the linearised potential-flow equations. Naturally, adding viscosity to such a system introduces viscous dissipation that enervates energy and dampens the oscillations at the symmetry axis. However, for viscous liquids and deeper cavities, these equations are qualitatively inaccurate. In this study, we decompose the initial localised interface distortion into several Bessel functions and study their time evolution governing the propagation of concentric gravito–capillary waves on a free surface. This is carried out for inviscid as well as viscous liquids. For a sufficiently deep cavity, the inward focussing of waves results in large interfacial oscillations at the axis, necessitating a second-order nonlinear theory. We demonstrate that this theory effectively models the interfacial behaviour and highlights the crucial role of nonlinearity near the symmetry axis. This is rationalised via demonstration of the contribution of bound wave components to the interface displacement at the symmetry axis Contrary to expectations, the addition of slight viscosity further intensifies the oscillations at the symmetry axis although the mechanism of wavetrain generation here is quite different compared with bubble bursting where such behaviour is well known (Duchemin et al., Phys. Fluids, vol. 14, issue 9, 2002, pp. 3000–3008). This finding underscores the limitations of the potential flow model and suggests avenues for more accurate modelling of such complex free-surface flows.

Photon vortex generation from nonlinear Compton scattering in Feynman approach

In the present study, we show the calculation of nonlinear Compton scattering with circularly polarized photons in a cylindrical coordinate using Feynman diagram to calculate photon vortex generation in intermediate states considering conservation of angular momentum. We take two different vortex wave functions based on Bessel function for the emitted photon and the electron after emission of the photon, which are the eigenstate of the z component of the total angular momentum (zTAM) when a particle propagates along the z axis. The result shows that when an electron absorbs N photons a photon vortex with a zTAM=N is predominantly radiated, but there are still very small contributions of photons with a zTAM=(N−1) and (N+1) due to the spin flip of the initial electron. This means even when an electron absorbs only a single photon, the electron may emit a photon vortex of a zTAM of 2ℏ. However, the numerical calculations show that the contribution of the spin flip is four orders of magnitude smaller than that of the dominant radiation. We also discuss the circular polarization for the generated photon vortices. Published by the American Physical Society 2025

A WKB method based on parabolic cylinder function for very-low-frequency sound propagation in deep ocean

A WKB method based on parabolic cylinder function for very-low-frequency sound propagation in deep ocean

Impact of Variable Viscosity on Unsteady Couette Flow

ABSTRACTThis research presents the effect of space‐dependent and viscosity variation on the time‐dependent flow of viscous, incompressible fluid in a finite channel (Couette flow). The viscosity of the fluids is assumed to grow exponentially. The governing equations and the boundary conditions are nondimensionalized with the aid of dimensionless parameters and solved semi‐analytically using the Laplace transformation method and its numerical inversion approach called Riemann‐sum approximation (RSA). Time‐dependent velocity profiles, time‐dependent skin frictions, and time‐dependent mass flow rates of the fluids with variables viscosity and of the fluids with constant viscosity are obtained exactly in the Laplace domain in terms of modified Bessel functions of the first and second kinds. Due to the complexity of the solutions to the problem, the analytical procedure could not be obtained in the time domain, so the RSA is sought. For the validation of the method used, steady‐state solutions of the velocity profiles, skin frictions, and the mass flow rates of the fluids with variable viscosity and constant viscosity are obtained analytically and compared with the numerical results. Graphs are plotted and tables are tabulated for the analysis of the effect of space‐dependent and viscosity variation of the fluid flows. In the course of analysis, it is observed that the velocity profile is higher in the case where the fluid viscosity is constant compared with the variable viscosity.

Heat transfer in the entrance region of annular laminar flow with constant and different heat flux on two walls

AbstractHeat transfer differs in the regions where the flow is developed and developing thermally. These regions can be differentiated by using the thermal entry length. Many researchers have presented correlations to determine the thermal entry length for natural and forced convection. In this study, heat transfer in the entrance region of a concentric annuli is investigated. It is accepted that beginning from the inlet of annuli the flow is developed hydrodynamically and it is developing thermally. Heat transfer is investigated where the internal or external surfaces of the annuli are at constant but different heat fluxes. The fluid velocity is assumed to be constant or radially variable. Due to thermal boundary conditions, one thermal boundary layer appears on the outer cylinder surface, another on the inner cylinder surface. The edge of two boundary layers will be adiabatic and naturally, the temperature of fluid between the two edges will be equal to free stream temperature. Transformation, Separation of Variables method, eigenvalue problem, Sturm‐Liouville system, Bessel differential equation and properties of orthogonal functions are used in solution of the problem. Exact and analytical solutions of the momentum and energy equations are presented. Velocity and temperature distributions, local Nusselt numbers and convection heat transfer coefficients are calculated for the internal and external surfaces of annuli.

Some properties of Bessel functions in multiplicative calculus

Some properties of Bessel functions in multiplicative calculus

Modeling of shear wave propagation in piezo-viscoelastic microbeam over quadratic heterogeneous viscoelastic plate with sliding contact

Abstract This study analyzes the phase and attenuation dynamic behavior of piezo-viscoelastic microbeam overlying quadratic heterogeneous viscoelastic plate under sliding contact. Using the Kelvin-Voigt model, the material properties of the system are assumed to be viscoelastic. Maxwell’s relations are used to incorporate the electric potential function. The solutions for both media are derived separately by solving the second-order hyperbolic differential equation using the method of separation of variables and asymptotic expansions of Bessel functions. The system of linear homogeneous equations is obtained by applying admissible boundary conditions to determine fundamental physical quantities. The key contribution of the current work is demonstrating the influence of dissipation factors, sliding contact, micro-length, heterogeneity, and thickness ratio parameters on shear wave propagation. The micro-length effect is found to suppress the attenuation of shear waves through the analysis and discussion of the dispersion and attenuation curves.

Summed Series Involving 1F2 Hypergeometric Functions

Summation of infinite series has played a significant role in a broad range of problems in the physical sciences and is of interest in a purely mathematical context. In a prior paper, we found that the Fourier–Legendre series of a Bessel function of the first kind JNkx and modified Bessel functions of the first kind INkx lead to an infinite set of series involving 1F2 hypergeometric functions (extracted therefrom) that could be summed, having values that are inverse powers of the eight primes 1/(2i3j5k7l11m13n17o19p) multiplying powers of the coefficient k, for the first 22 terms in each series. The present paper shows how to generate additional, doubly infinite summed series involving 1F2 hypergeometric functions from Chebyshev polynomial expansions of Bessel functions, and trebly infinite sets of summed series involving 1F2 hypergeometric functions from Gegenbauer polynomial expansions of Bessel functions. That the parameters in these new cases can be varied at will significantly expands the landscape of applications for which they could provide a solution.

Properties and integral inequalities of P-superquadratic functions via multiplicative calculus with applications

This manuscript explores the idea of a multiplicatively P-superquadratic function and its properties. Utilizing these properties, we derive the inequalities of Hermite–Hadamard type for such functions in the framework of multiplicative calculus. In addition to this, we derive integral inequalities of Hermite–Hadamard type for the product and quotient of multiplicatively P-superquadratic and multiplicatively P-subquadratic functions. Moreover, we develop the fractional version of Hermite–Hadamard type inequalities involving midpoints and end points for multiplicatively P-superquadratic functions with respect to multiplicatively Riemann–Liouville (R.L) fractional integrals. Graphical illustrations based on specific relevant examples validate the credibility of the findings. The study is further stimulating by being pushed with potential applications in terms of moment of random variables, special means, and modified Bessel functions of the first kind. Regarding superquadraticity, the results presented in this work are new, therefore they clearly provide extensions and improvements of the work available in the literature.

Geometric Nature of the Turánian of Modified Bessel Function of the First Kind

This work explores the geometric properties of the Turanian of the modified Bessel function of the first kind (TMBF). Using the properties of the digamma function, we establish conditions under which the normalized TMBF satisfies starlikeness, convexity, k-starlikeness, k-uniform convexity, pre-starlikeness, lemniscate starlikeness, and convexity, and under which exponential starlikeness and convexity are obtained. By combining methods from complex analysis, inequalities, and functional analysis, the article advances the theory of Bessel functions and hypergeometric functions. The established results could be useful in approximation theory and bounding the behavior of functions.

Efficient numerical methods of integrals with products of two Bessel functions and their error analysis

Efficient numerical methods of integrals with products of two Bessel functions and their error analysis

Self-Convolution and Its Invariant Properties for the Kernel Function of the Aortic Fractal Operator

In this paper, we explore the self-convolution of the kernel function of the aortic fractal operator. Previous research has established a model named “physical fractal”, and confirmed that the hemodynamics of the aorta can be inscribed by a fractal operator and that the dominant component of the kernel function of the fractal operator is a weighted first-order Bessel function. These studies primarily focus on solving the fractal operator kernel function and examining the overall properties of the physical fractal. As we began to investigate the internal structure of physical fractals, we discovered that studying the powers of fractal operators is a necessary step. In this paper, we introduce the concept of kernel function self-convolution, establish its connection with the power of the fractal operator, and derive a series of invariant properties for the self-convolution of the aortic operator kernel function. These invariant properties, in turn, are deeply and intrinsically related to the invariant properties of the Bessel functions. The research findings of this paper enrich hemodynamics and biomechanics in physical fractal space and extend the scope of using fractal operators to characterize the dynamics of living organisms.

BLAST: Beyond Limber Angular power Spectra Toolkit. A fast and efficient algorithm for 3x2 pt analysis

The advent of next-generation photometric and spectroscopic surveys is approaching, bringing more data with tighter error bars. As a result, theoretical models will become more complex, incorporating additional parameters, which will increase the dimensionality of the parameter space and make posteriors more challenging to explore. Consequently, the need to improve and speed up our current analysis pipelines will grow. In this work, we focus on the 3×2pt statistics, a summary statistic that has become increasingly popular in recent years due to its great constraining power. These statistics involve calculating angular two-point correlation functions for the auto- and cross-correlations between galaxy clustering and weak lensing. The corresponding model is determined by integrating the product of the power spectrum and two highly-oscillating Bessel functions over three dimensions, which makes the evaluation particularly challenging. Typically, this difficulty is circumvented by employing the so-called Limber approximation, which is an important source of error. We present 𝙱𝚕𝚊𝚜𝚝.𝚓𝚕, an innovative and efficient algorithm for calculating angular power spectra without employing the Limber approximation or assuming a scale-dependent growth rate, based on the use of Chebyshev polynomials. The algorithm is compared with the publicly available beyond-Limber codes, whose performances were recently tested by the Rubin Observatory Legacy Survey of Space and Time Dark Energy Science Collaboration. At similar accuracy, 𝙱𝚕𝚊𝚜𝚝.𝚓𝚕 is ≈10- 15× faster than the winning method of the challenge, also showing excellent scaling with respect to various hyper-parameters. 𝙱𝚕𝚊𝚜𝚝.𝚓𝚕 is publicly available on GitHub, and we release a repository where we explain how to use the code.

The role of spatial dimension in the emergence of localized radial patterns from a Turing instability

The emergence of localized radial patterns from a Turing instability has been well studied in two- and three-dimensional settings and predicted for higher spatial dimensions. We prove the existence of localized ( n + 1 ) -dimensional radial patterns in general two-component reaction–diffusion systems near a Turing instability, where n > 0 is taken to be a continuous parameter. We determine the explicit dependence of each pattern’s radial profile on the dimension n through the introduction of ( n + 1 ) -dimensional Bessel functions, revealing a deep connection between the formation of localized radial patterns in different spatial dimensions.

Addendum to “On new identities involving zeros of Bessel functions” [J. Math. Anal. Appl. 542 (2025) 128828

Addendum to “On new identities involving zeros of Bessel functions” [J. Math. Anal. Appl. 542 (2025) 128828

Performance Analysis of Cardioid and Omnidirectional Microphones in Spherical Sector Arrays for Coherent Source Localization.

Traditional spherical sector microphone arrays using omnidirectional microphones face limitations in modal strength and spatial resolution, especially within spherical sector configurations. This study aims to enhance array performance by developing a spherical sector array employing first-order cardioid microphones. A model based on spherical sector harmonic (SSH) functions is introduced to extend the benefits of spherical harmonics to sector arrays. Modal strength analysis demonstrates that cardioid microphones in open spherical sectors enhance nonzero-order strengths and eliminate the nulls associated with spherical Bessel functions. We find that the spatial resolution of spherical cap arrays depends on the array's maximum order and the limiting polar angle, but is independent of the microphone gain pattern. We assess direction-of-arrival (DOA) estimation performance for coherent wideband sources using the array manifold interpolation method, and compare cardioid and omnidirectional arrays through simulations in both open and rigid hemispherical configurations. The results indicate that cardioid arrays outperform omnidirectional ones on DOA estimation tasks, with performance improving alongside increased microphone directivity in the open hemispherical configuration. Specifically, hypercardioid microphones yielded the best results in the open configuration, while subcardioid microphones (without nulls) were optimal in rigid configurations. These findings demonstrate that spherical sector arrays of first-order cardioid microphones offer improved modal strength and DOA estimation capabilities over traditional omnidirectional arrays, providing significantly enhancing performance in spherical sector array processing.

Analytical treatment of the Yukawa screened Coulomb interaction in a plane-wave basis

The calculation of the many-electron (screened) Coulomb and exchange integrals is a very common task to perform in modern electronic structure theory. While an analytical treatment of the complete integrals is too complicated to be attempted directly, essential insight can be gained by focusing on the most significant contributions, as determined by a physical criterion. The monopole term of the screened Coulomb interaction is particularly important, since it defines the Hubbard interaction U among a set of localized electrons, which is an essential parameter for effective models of strongly correlated systems. Here, we derive an analytical solution for the matrix elements of the screened Coulomb interaction on a plane-wave basis, which is routinely used in the most common methods for electronic structure calculations. Screening is treated using a Yukawa potential, which is suitable to describe the valence electrons in the Fermi level region. For the solution of the integrals, the plane waves and the Yukawa potential are first expanded in Bessel functions of first and second kind. Then, by means of the lower and upper incomplete gamma functions, we are able to obtain closed-form integrals of a series that remains convergent for realistic parameters. Our exact solution for the radial integrals of the monopole term can find usage in plane-wave codes for electronic structure calculations, both as an output tool as well as within the computational cycle, as e.g., for many-body extensions of density-functional theory. Considering the importance of Bessel functions in solid state physics and electronic structure theory, it is also easy to foresee that our solution to the various integrals across this work may become useful to several other problems in the field.

A Novel Family of q-Mittag-Leffler-Based Bessel and Tricomi Functions via Umbral Approach

Many properties of special polynomials, such as recurrence relations, sum formulas, integral transforms and symmetric identities, have been studied in the literature with the help of generating functions and their functional equations. In this paper, we introduce hybrid forms of q-Mittag-Leffler functions. The q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are constructed using a q-symbolic operator. The generating functions, series definitions, q-derivative formulas and q-recurrence formulas for q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are obtained. The Nq-transforms and Nq-transforms of q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are obtained. These hybrid q-special functions are also studied by plotting their graphs for specific values of the indices and parameters.

Uniform Enclosures for the Phase and Zeros of Bessel Functions and their Derivatives

Uniform Enclosures for the Phase and Zeros of Bessel Functions and their Derivatives

Globality of the DPW construction for Smyth potentials in the case of SU1,1

We construct harmonic maps into SU1,1/U1 starting from Smyth potentials ξ, by the DPW method. In this method, harmonic maps are obtained from the Iwasawa factorization of a solution L of L−1dL=ξ. However, the Iwasawa factorization in the case of a noncompact group is not always global. We show that L can be expressed in terms of Bessel functions and from the asymptotic expansion of Bessel functions we solve a Riemann-Hilbert problem to give a global Iwasawa factorization. In this way we give a more direct proof of the globality of our solution than in the work of Dorfmeister-Guest-Rossman [5], while avoiding the general isomonodromy theory used by Guest-Its-Lin [11], [12].