Starlikeness of Normalized Bessel Functions with Symmetric Points

Bessel functions are related with the known Bessel differential equation. In this paper, we determine the radius of starlikeness for starlike functions with symmetric points involving Bessel functions of the first kind for some kinds of normalized conditions. Our prime tool in these investigations is the Mittag-Leffler representation of Bessel functions of the first kind.

On a solution of a fractional hyper-Bessel differential equation by means of a multi-index special function

In this paper we introduce a new multiple-parameters (multi-index) extension of the Wright function that arises from an eigenvalue problem for a case of hyper-Bessel operator involving Caputo fractional derivatives. We show that by giving particular values to the parameters involved in this special function, this leads to some known special functions (as the classical Wright function, the α-Mittag-Leffler function, the Tricomi function, etc.) that on their turn appear as cases of the so-called multi-index Mittag-Leffler functions. As an application, we mention that this new generalization Wright function nis an isochronous solution of a nonlinear fractional partial differential equation.

Solving Second-Order Linear Differential Equations with Random Analytic Coefficients about Regular-Singular Points

In this contribution, we construct approximations for the density associated with the solution of second-order linear differential equations whose coefficients are analytic stochastic processes about regular-singular points. Our analysis is based on the combination of a random Fröbenius technique together with the random variable transformation technique assuming mild probabilistic conditions on the initial conditions and coefficients. The new results complete the ones recently established by the authors for the same class of stochastic differential equations, but about regular points. In this way, this new contribution allows us to study, for example, the important randomized Bessel differential equation.

On the Regular Integral Solutions of a Generalized Bessel Differential Equation

The original Bessel differential equation that describes, among many others, cylindrical acoustic or vortical waves, is a particular case of zero degree of the generalized Bessel differential equation that describes coupled acoustic-vortical waves. The solutions of the generalized Bessel differential equation are obtained for all possible combinations of the two complex parameters, order and degree, and finite complex variable, as Frobenius-Fuchs series around the regular singularity at the origin; the series converge in the whole complex plane of the variable, except for the point-at-infinity, that is, the only other singularity of the differential equation. The regular integral solutions of the first and second kinds lead, respectively, to the generalized Bessel and Neumann functions; these reduce to the original Bessel and Neumann functions for zero degree and have alternative expressions for nonzero degree.

On the asymptotic analysis of the JWKB method via change of dependent variable in the first-order Bessel’s equation

The first-order Jeffreys–Wentzel–Kramers–Brillouin method (called (JWKB)1) is a conventional semi-classical approximation method used in quantum mechanical systems for accurate solutions. It is known to give accurate energy and wave-function in the classically accessible region of the related quantum mechanical system defined by Schroedinger’s equation whereas the solutions in the classically inaccessible region require special treatment, conventionally known as the asymptotic matching rules. In this work, (JWKB)1 solution of the Bessel differential equation of the first order (called (BDE)1), chosen as a mathematical model, is studied by being transformed into the normal form via the change of dependent variable. General JWKB solution of the initial value problem where appropriately chosen initial values are applied is studied in both normal and standard form representations to be analyzed by the generalized JWKB asymptotic matching rules regarding the Sij matrix elements defined in the literature. Consequently, regions requiring first-order and zeroth-order JWKB approximations are determined successfully.

Calculating Galois groups of third-order linear differential equations with parameters

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel’s equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In this paper, we give the first known algorithm that calculates the differential Galois group of a third-order parameterized linear differential equation.

On nabla discrete fractional calculus operator for a modified Bessel equation

In thermal sciences, it is possible to encounter topics such as Bessel beams, Bessel functions or Bessel equations. In this work, we also present new discrete fractional solutions of the modified Bessel differential equation by means of the nabla-discrete fractional calculus operator. We consider homogeneous and non-homogeneous modified Bessel differential equation. So, we acquire four new solutions of these equations in the discrete fractional forms via a newly developed method