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.
Example 2. Let , , , and , and the results are presented in Table 2.
Example 3. Let , , and , and the results are in Table 3.
From the above three examples, it is easy to see that functions and vary very slowly with ; thus, it is hard to imagine to approximate them with uniform convergence by using the power series. On the other hand, when letting and , they satisfy the derived equation at relative errors smaller than 4 × 10−4, meaning accuracy is significant enough.
3.2. Second Set of Examples
Example 1. To compare with Bessel function of the first kind, let , , and . For these parameters, the Bessel equation has solutions of the Bessel function of the first kind, and , in the neighbourhood of , which has infinite points of zero on the real axis. And when , they have the following Asymptotic Formula to determine the distribution of the zero points:The solutions obtained in this study and are conjugate complex (; Appendix E); however, real solution compatible with and can be obtained by linear combination of certain form (Appendix F):with the zero points noted as and .
Table 4 compares the zero points of and of and the zero points of and of . It is easy to find that the solution obtained here (after rearranging) has consistent distribution of zero points with Bessel function of the first kind, indicating the credibility of the solution method provided by the present study [1, 2].
Example 2. Let , , , , and , which represent the Bessel equation with modified Bessel functions and in the neighbourhood of . With this set of parameters, the solutions obtained by this paper are and in the interval or , with their linear combination also being solution of the equationFrom the table of Bessel functions, one can look up the values of functions and in the interval . At the two arbitrarily selected points in the interval , we linearly fit and and and , to determine constant . The obtained distributions of functions and are compared with those of functions and in Table 5, which shows apparent consistency between the solution obtained with known modified Bessel functions and that calculated in this study.
Example 3. Let , , , , and , and this set of parameters corresponds to Bessel equation with modified Bessel functions and as the solution in the neighbourhood of . The corresponding solution obtained here is and , whose linear combination can construct new solution functions, after linear fit with functions and :Comparisons between their distributions and those of and are illustrated in Table 6. It can be easily seen that the solution obtained here is completely consistent with the known solution with modified Bessel functions [1, 2].
4. Conclusions
The present study proves that the solution of the Bessel equation in the interval with irregular singular point can be expressed as the product of a singular divisor and a nonsingular function, the latter of which can be approximated with uniform convergence by corrected Fourier series with limited smoothness in the interval. Thus, we are able to obtain the solution with certain regularities of distribution of the Bessel equation in the semiunbounded interval with irregular singular point, but no more an asymptotic solution at , or a regular formal solution. This indicates that it is incorrect to conclude existence of only asymptotic or regular formal solution in the neighbourhood of an irregular singular point only because of the inadequacy of certain solution method.
Appendix
A. Coordinate Transformation
There is a relationship between the old and new coordinates as follows: Inserting these relations into (1), one can derive (3) using the new coordinate after rearrangements.
B. Constraint and Compatibility
Based on Appendix of [4], although the differential equation still has a form of singular equation, its nonsingular solution can exist with certain constraint condition; that is,On the other hand, the classic solution of the equation not only demands the existence of its Fourier projected equationbut also requires the following compatibility condition:
C. Parameters for (21) and (22)
Considerwhere .
D. Determining Coefficients of (26)
In (29),Let , which can deriveIt can be solved that , whereat the same time, let .
E. Proof of Conjugate Relation
Because parameters and are all real numbers,and the parameters in (16) satisfy the following relation:It is known that and satisfy (16), noted asSolving the conjugate complex for (E.4), Comparing (E.5) and (E.3), a conjugate relation is obtained:It is known thatso the conjugate relation is obtained:
F. Linear Combination for Real Solution
Taking large enough, , for example, the corresponding . Demandingor and can be determined.
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
The authors acknowledge the support from the National Natural Science Foundation of China (no. 41076013) and Shanghai Chair Professor-Eastern Scholar Program, China.