Solving differential equations in terms of bessel functions (original) (raw)
Related papers
A unified point of view on the theory of generalized Bessel functions
Computers & Mathematics with Applications, 1995
Bessel functions have been generalized in a number of ways and many of these generalizations have been proved to be important tools in applications. In this paper we present a unified treatment, thus proving that many of the seemingly different generalizations may be viewed as particular cases of a two-variable function of the type introduced by Miller during the sixties.
The Fourth-order Bessel–type Differential Equation
Applicable Analysis, 2004
The Bessel-type functions, structured as extensions of the classical Bessel functions, were de…ned by Everitt and Markett in 1994. These special functions are derived by linear combinations and limit processes from the classical orthogonal polynomials, classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials. These Bessel-type functions are solutions of higher-order linear di¤erential equations, with a regular singularity at the origin and an irregular singularity at the point of in…nity of the complex plane.
Exponential generating functions for the associated Bessel functions
Journal of Physics A: Mathematical and Theoretical, 2008
Similar to the associated Legendre functions, the differential equation for the associated Bessel functions B l,m (x) is introduced so that its form remains invariant under the transformation l → −l − 1. A Rodrigues formula for the associated Bessel functions as squared integrable solutions in both regions l < 0 and l ≥ 0 is presented. The functions with the same m but with different positive and negative values of l are not independent of each other, while the functions with the same l + m (l − m) but with different values of l and m are independent of each other. So, all the functions B l,m (x) may be taken into account as the union of the increasing (decreasing) infinite sequences with respect to l. It is shown that two new different types of exponential generating functions are attributed to the associated Bessel functions corresponding to these rearranged sequences.
On the coefficients of integrated expansions of Bessel polynomials
Journal of Computational and Applied Mathematics, 2006
A new formula expressing explicitly the integrals of Bessel polynomials of any degree and for any order in terms of the Bessel polynomials themselves is proved. Another new explicit formula relating the Bessel coefficients of an expansion for infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is also established. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed.
On the Regular Integral Solutions of a Generalized Bessel Differential Equation
Advances in Mathematical Physics, 2018
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.
Applied Mathematics and Computation, 2008
A formula expressing explicitly the derivatives of ordinary Bessel polynomials of any degree and for any order in terms of the ordinary Bessel polynomials themselves is proved. Another explicit formula, which expresses the ordinary Bessel expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of its original ordinary Bessel coefficients, is also given. A formula for the ordinary Bessel coefficients of the moments of one single ordinary Bessel polynomial of certain degree is proved. Formula for the ordinary Bessel coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its ordinary Bessel coefficients is also obtained. Application of these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between generalized Bessel and ordinary Bessel polynomials is described. Explicit formula for these coefficients between Jacobi and ordinary Bessel polynomials is given, of which the ultraspherical polynomials and its consequences are important special cases. Two analytical formulae for the connection coefficients between Laguerre-ordinary Bessel and Hermite-ordinary Bessel are also developed.
The simultaneous computation of Bessel functions of first and second kind
Computers & Mathematics with Applications, 1996
Based on the qualitative properties of Bessel's differential equation and its solutions, a method is proposed for the simultaneous evaluation of Bessel functions of first and second kind. Special attention is paid to the numerical properties of the method and to the errors of approximation. geywords-Bessel function evaluation, Singular ODE, ODE on infinite interval, Priifer transformation.
Studies on the Generalized and Reverse Generalized Bessel Polynomials
2004
, 73 pages The special functions and, particularly, the classical orthogonal polynomials encountered in many branches of applied mathematics and mathematical physics satisfy a second order differential equation, which is known as the equation of the hypergeometric type. The variable coefficients in this equation of the hypergeometric type are of special structures. Depending on the coefficients the classical orthogonal polynomials associated with the names Jacobi, Laguerre and Hermite can be derived as solutions of this equation. In this thesis, these well known classical polynomials as well as another class of polynomials, which receive less attention in the literature called Bessel polynomials have been studied.
This paper is a deep exploration of the project Bessel Functions by Martin Kreh of Pennsylvania State University. We begin with a derivation of the Bessel functions Ja(x) and Ya(x), which are two solutions to Bessel's differential equation. Next we find the generating function and use it to prove some useful standard results and recurrence relations. We use these recurrence relations to examine the behavior of the Bessel functions at some special values. Then we use contour integration to derive their integral representations, from which we can produce their asymptotic formulae. We also show an alternate method for deriving the first Bessel function using the generating function. Finally, a graph created using Python illustrates the Bessel functions of order 0, 1, 2, 3, and 4.