International Journal of Mathematics and Mathematical Sciences

Volume 2016 (2016), Article ID 6826482, 7 pages

http://dx.doi.org/10.1155/2016/6826482

## Bessel Equation in the Semiunbounded Interval : Solving in the Neighbourhood of an Irregular Singular Point

^{1}The First Institute of Oceanography, State Oceanic Administration, Qingdao 266061, China^{2}College of Marine Sciences, Shanghai Ocean University, Shanghai 201306, China

Received 28 April 2016; Accepted 28 June 2016

Academic Editor: Theodore E. Simos

Copyright © 2016 Qing-Hua Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This study expresses the solution of the Bessel equation in the neighbourhood of as the product of a known-form singular divisor and a specific nonsingular function, which satisfies the corresponding derived equation. Considering the failure of the traditional irregular solution constructed with the power series, we adopt the corrected Fourier series with only limited smooth degree to approximate the nonsingular function in the interval . In order to guarantee the series’ uniform convergence and uniform approximation to the derived equation, we introduce constraint and compatibility conditions and hence completely determine all undetermined coefficients of the corrected Fourier series. Thus, what we found is not an asymptotic solution at (not to mention a so-called formal solution), but a solution in the interval with certain regularities of distribution. During the solution procedure, there is no limitation on the coefficient property of the equation; that is, the coefficients of the equation can be any complex constant, so that the solution method presented here is universal.

#### 1. Introduction

The general form of the Bessel equation is where are arbitrary complex constants, with and , and .

The Bessel equation only has two singular points on the real axis: and . For convenience, set division point at , to separate the entire solution interval into two parts, and , the latter of which is the semiunbounded interval that this study will solve the equation in. It is easy to verify that is a regular singular point, but is irregular. Previous studies showed that the solution in the neighbourhood of the isolated singular point could be written as a combination of a known-form singular divisor and a specific nonsingular function. To find the solution in the neighbourhood of the regular singular point, one often uses the power series extension to reconstruct the nonsingular function. In the neighbourhood of the irregular singular point, attempts have been made with power series traditionally, but already existing efforts cannot find a uniformly convergent solution, which can only give an asymptotic solution at . This so-called regular formal solution can only indicate the variation tendency of the solution [1–4].

However, the failure to obtain a solution with certain regularities of distribution using power series method does not necessarily mean inexistence of this kind of solution. As is known, the power series with uniform convergence in the complex number space converges to a complex analytic function, so that it is full differentiable. Yet, the Bessel equation is only a second-order differential equation, which demands limited smoothness (e.g., second-order derivative) only, but not full smoothness. Overdemanded smoothness could be the reason for the inability to find the solution with the uniformly convergent power series.

This study adopts the corrected Fourier series [5–10] with limited smoothness to construct the nonsingular function and insert it into the derived equations from the Bessel equation. By introducing the constraint and compatibility conditions at the same time, all coefficients of the corrected Fourier series can be determined completely. Thus, one can obtain the solution with certain regularities of distribution in the interval , but not an asymptotic or formal solution. There is no limitation on the equation parameter in the whole solution procedure, which could be any arbitrary complex constant, so that the solution method given in this study is universal.

#### 2. The Solution of the Bessel Equation in the Semiunbounded Interval

##### 2.1. Obtaining the Derived Equation

For convenience, we introduce coordinate transformationwhich transforms the semiunbounded interval to a bounded one . Equation (1) turns into the following form with the new coordinate (Appendix A):It is easy to find that is an irregular singular point of the equation, and then the irregular solution can be written aswhere , in are constants to be determined.

is a nonsingular function with , which satisfies the following equation:whereUsing the differential relationship,one can obtainHence, (5) can be written as the following form:whereLet , and, fromone can obtainthat is,which determines thatwhereSo, the derived equation is obtained:where

##### 2.2. Nonsingular Classic Solution of the Derived Solution

The form of the derived equation (see (16)) still seems to be a singular equation, but it can be guaranteed to have nonsingular classic solution after introducing certain constraint conditions (Appendix B). That is,that is,

Because , can be expanded to the following corrected Fourier series:which is uniformly convergent and where . At the same time, we introduce the Fourier projection:Inserting expansion equation (19) to the derived equation (see (16)), one can obtain (Appendix C)After the Fourier projection,whereIt can be derived thatwhere is the solution of the following equation:

Inserting solution (24) into extension (19), one can obtain whereThe compatibility condition is required [5, 6] in Fourier projection of (16) (Appendix B):that is,Insert solution (26) into constraint (18a) and (18b) and consistency (28a) and (28b) conditions to obtain the equation determining (Appendix D):It can be solved that hence, Note thatFinally, the solution of (3) is obtained:

#### 3. Examples and Discussion

As mentioned before, the solution method given here satisfies the equation with each parameter being any arbitrary complex constant, so that the first set of examples we will show are to provide the results of the functions and according to several sets of different complex constants. Their accuracy of approximation to satisfy the derived equation will also be estimated. The second set of examples aim to solve , and, in turn, the solution of the Bessel equation, and , with several sets of certain common real constants. Noting their linear combination as and , we will compare them with already known Bessel functions and modified Bessel functions.

##### 3.1. First Set of Examples

*Example 1. *Let , , , , and , and calculate and . To examine the accuracy of the solution to approximate the derived equation, notewith complex module And then, we can define the relative accuracy of the solution to satisfy the equationValues of the solution and relative error are shown in Table 1.