Bessel Functions Of Fractional Order Research Articles

Retraction Note to: Approximation of Analytic Functions by Bessel Functions of Fractional Order

Retraction Note to: Approximation of Analytic Functions by Bessel Functions of Fractional Order

High accuracy approximation for the modified Bessel function of fractional order I1/3(x)

The modified Bessel functions of fractional order I1/3(x) has been approximated by an analytic function containing rational and hyperbolic fumctions. A technique using both power series and assymptotic expansion has been used. The accuracy of the approximation is very high using only three parameters., and the largest relative error is smaller than 0,004.

Fractured–Injection–Well Performance Under Non–Newtonian, Power–Law Fluids

Summary A new source function developed in this work is used to predict well responses to evaluate long-term injectivity and late-time responses of fractured wells after the injection of a non-Newtonian, pseudoplastic fluid. The source function yields integrals of modified Bessel functions of fractional order that are evaluated numerically by techniques developed by the authors. Consideration of well responses after the injection of pseudoplastic fluids into reservoirs through fractured wells requires the solution of a nonlinear differential equation and examination of flow fields in two dimensions. Consequently, numerical methods have been the only recourse. This work explores the applicability of new tools developed in recent times by analytical means. Calculating well injectivity accurately and rapidly for this complex situation by the methods outlined here is particularly convenient. The expressions we derive may be readily applied to concepts such as the Peaceman (1978) probe radius and transient well injectivity. Predictions by use of the new solution are compared with numerical solutions in the literature.

Full characterization of Airy beams under physical principles

The propagation characteristics of Airy beams is investigated and fully described under the traveling waves approach analogous to that used for non-diffracting Bessel beams. This is possible when noticing that Airy functions are in fact Bessel functions of fractional order 1 . We show how physical 3 principles impose restrictions such that the non-diffracting Airy beams cannot be of infinite extent as has been argued, and introduce for the first time quantitative expressions for the maximum transverse and longitudinal extent of Airy beams. We show that under the appropriate physical conditions it is possible to obtain higher-order Airy beams.

RETRACTED ARTICLE: Approximation of analytic functions by bessel functions of fractional order

We solve the inhomogeneous Bessel differential equation $$ {x^2}y''(x) + xy'(x) + \left( {{x^2} - {\nu^2}} \right)y(x) = \sum\limits_{m = 0}^\infty {{a_m}{x^m},} $$ where ν is a positive nonintegral number, and use this result for the approximation of analytic functions of a special type by the Bessel functions of fractional order.

Approximation of Analytic Functions by Bessel's Functions of Fractional Order

We will solve the inhomogeneous Bessel's differential equationx2y″(x)+xy′(x)+(x2-ν2)y(x)=∑m=0∞amxm, whereνis a positive nonintegral number and apply this result for approximating analytic functions of a special type by the Bessel functions of fractional order.

Evaluation of Fresnel integrals based on the continued fractions method

We present a simple algorithm for evaluating Fresnel integrals based on the continued fractions method: we use the relation between these integrals and first-kind Bessel functions of fractional order, and we apply a fast code to calculate them based on the continued fractions method. This latter code is especially useful for evaluating high order Bessel functions because it does not require recalculations using normalization relations. Comments on the same procedure but using Miller’s algorithm to evaluate the required Bessel functions are presented and a comparison with a standard code for evaluating Fresnel integrals (Numerical Recipes program FRENEL) is provided.

Easy and rapid computation of the transfer of heat from annular fins of nearly optimal profile with the finite‐difference technique and the shooting method

It is undeniable that the annular fin of hyperbolic profile with constant thermal conductivity and uniform convective coefficient is important in many applications of heat transfer engineering. The importance of this fin configuration stems from its close resemblance to the annular fin of optimal cross section capable of delivering maximum heat transfer for a given volume of material. This paper addresses two simple numerical procedures for solving the generalized Bessel equation that governs the temperature variation in annular fins of hyperbolic profile, one is the finite‐difference technique with an uncharacteristic coarse mesh and the other is the shooting method. Certainly, the central objective here is to avoid the evaluation of the elegant, but intricate exact analytic temperature distributions and companion fin efficiencies containing modified Bessel functions of fractional order.

A New Theory and Algorithm for Reconstructing Helical Structures with a Seam

Conventional helical reconstruction is a general method to obtain three-dimensional structures of many filamentous biological macromolecules. The method assumes helical symmetry, and generates the three-dimensional structures from two-dimensional projection images. However, the theory is inadequate to describe filamentous structures discontinuities, which are called seams in the case of microtubules or perturbations in the case of tobacco mosaic virus or the bacterial flagellar filament. To study such structures, a new theory and algorithm are required. To this aim, we developed a new algorithm, namely, asymmetric helical reconstruction, which is based on our new theory that describes a "helical" object with a seam. In the theory, "helical" objects with a seam are indexed with a non-integral order of nu. Like the conventional helical reconstruction, the layer-line data are extracted from the Fourier transform of the images. We show that the Fourier-Bessel transform using the Bessel functions of fractional order can, to good approximation, reconstruct the three-dimensional structure of the object. To test the new algorithm, we reconstructed three-dimensional structures of a kinesin-microtubule complex with a seam from both model data and experimental data from cryo-electron microscopic images. The reconstructed structures are almost identical with those reconstructed from conventional helical reconstruction demonstrating the validity of the algorithm. The algorithm enables the analysis of various "helical" specimens with seams and also significantly improves the throughput and the resolution of kinesin-microtubule complexes.

The evaluation of transition matrix elements for atomic excitations by a screened Coulomb potential

We have developed a FORTRAN code for calculating screened potential matrix elements for transitions between atomic bound states. In the applied procedure, the target wave functions can be arbitrarily chosen, and a wide class of screening potentials can be applied. The program includes subroutines for the calculation of the modified Bessel functions of fractional order and their derivatives for a broad range of argument and order.

On the optical properties of magnetic lenses with fields of the form B(z) ∝ z − n, n = 2, 3, 4

Summary The trajectories of electrons in magnetic lenses with field distributions in the form of an inverse power law can be expressed in terms of Bessel functions of fractional order. It is shown that the focal lengths and spherical and chromatic aberration coefficients can be evaluated explicitly. The results confirm and elucidate numerical results published by A. V. Crewe [1].

Quantum and classical solutions for a free particle in wedge billiards

We have studied the quantum and classical solutions of a particle constrained to move inside a sector circular billiard with angle θ w and its pacman complement with angle 2 π− θ w . In these billiards rotational invariance is broken and angular momentum is no longer a conserved quantum number. The `fractional' angular momentum quantum solutions are given in terms of Bessel functions of fractional order, with indices λ p= pπ θ w , p=1,2,… for the sector and μ q= qπ 2π−θ w , q=1,2… for the pacman. We derive a `duality' relation between both fractional indices given by λ p= pμ q 2μ q−q and μ q= qλ p 2λ p−p . We find that the average of the angular momentum L ̂ z is zero but the average of L ̂ 2 z has as eigenvalues λ p 2 and μ q 2. We also make a connection of some classical solutions to their quantum wave eigenfunction counterparts.

Circular sectoral waveguides

Hollow cylindrical waveguides that have pie-shaped cross sections are described. The explicit equations for the field components of the TE (transverse electric) and TM (transverse magnetic) waves are given, and the modal patterns for a number of modes are plotted. The relation between the field distributions inside the circular and sectoral waveguides is discussed. It is shown that the electric and magnetic fields of the modes TE/sub 1n/ and TM/sub 1n/ display edge singularity when the sectoral angle is greater than pi . Formulas convenient for the evaluation of Bessel functions of fractional order on a personal computer are given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Modified Bessel Functions of Fractional Order

Modified Bessel Functions of Fractional Order

Bessel Functions of Fractional Order

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Uniform inequalities for ultraspherical polynomials and Bessel functions of fractional order

Extension aux polynomes ultraspheriques et aux fonctions de Bessel d'ordre fractionnaire de quelques inegalites uniformes etablies par A. Martin pour les polynomes de Legendre

Polarization calculations for synchrotron radiation using integral representations of modified bessel functions of fractional order

This paper presents calculations of synchrotron radiation flux corresponding to both horizontal and vertical polarization of the electric field for different values of the vertical angle.

Equivalent Circuits of Binomial Form Nonuniform Coupled Transmission Lines

Equivalent circuits of nonuniform coupled transmission lines whose self and mutual characteristic admittance distributions obey binomial form are presented. Telegrapher's equations of these nonuniform coupled transmission lines can he solved exactly rising Bessel functions of fractional order. By decomposing the chain matrix, it is shown that equivalent circuits of these nonuniform conpled transmission lines consist of cascade connections of lumped reactance elements, uncoupled uniform transmission lines and ideal transformers.

Simple numerical evaluation of modified Bessel functions Kν( x) of fractional order and the integral [formula omitted

Theoretical expressions for the angular and spectral distributions of synchrotron radiation involve modified Bessel functions of fractional order and the integral ʃ x ∞K ν(η) dη . A simple series expressions for these quantities which can be evaluated numerically with hand-held programmable calculators is presented.

Axisymmetric buckling of certain annular composite plates

Axisymmetric buckling of polar-orthotropic quasi-heterogeneous laminated annular plates are studied. A closed type solution is obtained for the case of two particular ratios of inner and outer pressures in terms of first and second kind Bessel functions of fractional order and Lommel functions. Two ratios of pressures are chosen in such a way as to represent a solution of the second order differentia] equation for the stress resultant function when only one of the fundamental solutions is left. The general procedure presented herein is valid for symmetrically laminated plates as well as for plates composed of materials with proportional properties. Numerous examples are shown indicating the effect of plates' heterogeneity on their elastic stability.