Solving differential equations in terms of bessel functions (original) (raw)
On the solutions of a class of nonlinear ordinary differential equations by the Bessel polynomials
Journal of Numerical Mathematics, 2012
In this study, we suggest a collocation method to solve a class of the nonlinear differential equations under the mixed conditions in terms of the Bessel polynomials. The method is based on the matrix forms of the Bessel polynomials and their derivatives by means of collocation points. To illustrate the usefulness of this technique we apply it to some test problems and the comparisons are made with existing results. The results show the efficiently and accuracy of the present work. All of the numerical computations have been performed on computer using a program written in Maple.
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.
arXiv: Classical Analysis and ODEs, 2020
We obtain a class of exact solutions of a Bessel-type differential equation, which is a six-parameter linear ordinary differential equation of the second order with irregular (essential) singularity at the origin. The solutions are obtained using the Tridiagonal Representation Approach (TRA) as bounded series of square integrable functions written in terms of the Bessel polynomial on the real line. The expansion coefficients of the series are orthogonal polynomials in the equation parameters space. We use our findings to obtain solutions of the Schrodinger equation for some novel potential functions.
Use of bessel polynomials for solving differential difference equations
Arab Journal of Basic and Applied Sciences
In this paper, the linear differential difference equation subject to the mixed conditions has been solved numerically using Bessel polynomials. The solution is obtained in terms of Bessel polynomials. In addition, the accuracy and error analysis of the method are considered using residual function. Some examples have also been considered to examine the applicability and reliability of the method.
A note on the theory ofn-variable generalized bessel functions
Il Nuovo Cimento B, 1991
In this note we introduce a further generalization of Bessel-type functions, discussing the case of a multivariables and one-index function. This kind of function can be usefully exploited in problems in which the dipole approximation does not hold and many higher harmonics are simultaneously operating. We analyse the relevant recurrence properties, the modified forms and the generating functions.
Fractional powers of Bessel operators
Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics, 2020
In this paper we study fractional powers of the Bessel differential operator. The fractional powers are defined explicitly in the integral form without use of integral transforms in its definitions. Some general properties of the fractional powers of the Bessel differential operator are proved and some are listed. Among them are different variations of definitions, relations with the Mellin and Hankel transforms, group property, generalized Taylor formula with Bessel operators, evaluation of resolvent integral operator in terms of the Wright or generalized Mittag-Leffler functions. At the end, some topics are indicated for further study and possible generalizations. Also the aim of the paper is to attract attention and give references to not widely known results on fractional powers of the Bessel differential operator.
A Neumann series of Bessel functions representation for solutions of perturbed Bessel equations
Applicable Analysis, 2017
A new representation for a regular solution of the perturbed Bessel equation of the form Lu = -u ′′ + l(l+1) x 2 + q(x) u = ω 2 u is obtained. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to ω. For the coefficients of the series explicit direct formulas are obtained in terms of the systems of recursive integrals arising in the spectral parameter power series (SPPS) method, as well as convenient for numerical computation recurrent integration formulas. The result is based on application of several ideas from the classical transmutation (transformation) operator theory, recently discovered mapping properties of the transmutation operators involved and a Fourier-Legendre series expansion of the transmutation kernel. For convergence rate estimates, asymptotic formulas, a Paley-Wiener theorem and some results from constructive approximation theory were used. We show that the analytical representation obtained among other possible applications offers a simple and efficient numerical method able to compute large sets of eigendata with a nondeteriorating accuracy.
The Conversion a Bessel’s Equation to a Self-Adjoint Equation and Applications
2011
In many applications of various Self-adjoint differential equations, whose solutions are complex, are produced . In this paper, a method for the conversion Bessel equation to Self-adjoint equation to provide a method and then, as the inverse of the transformation Self-adjoint equations of the Bessel equation. By which one can obtain analytical solutions to Self-adjoint equations. Because this solution, an exact analytical solution can provide to us, we benefited from the solution of numerical Self-adjoint equations .