Bessel Equation in the Semiunbounded Interval x∈[x0,∞]: Solving in the Neighbourhood of an Irregular Singular Point
This study expresses the solution of the Bessel equation in the neighbourhood of x=∞ 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...
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| Format: | Article |
| Language: | English |
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Wiley
2016-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2016/6826482 |
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| author | Qing-Hua Zhang Jian Ma Yuanyuan Qu |
| author_facet | Qing-Hua Zhang Jian Ma Yuanyuan Qu |
| author_sort | Qing-Hua Zhang |
| collection | DOAJ |
| description | This study expresses the solution of the Bessel equation in the neighbourhood of x=∞ 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 [x0,∞]. 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 x→∞ (not to mention a so-called formal solution), but a solution in the interval [x0,∞] 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. |
| format | Article |
| id | doaj-art-d52237f2bcda4b178048500281096799 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-d52237f2bcda4b1780485002810967992025-02-03T01:31:28ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252016-01-01201610.1155/2016/68264826826482Bessel Equation in the Semiunbounded Interval x∈[x0,∞]: Solving in the Neighbourhood of an Irregular Singular PointQing-Hua Zhang0Jian Ma1Yuanyuan Qu2The First Institute of Oceanography, State Oceanic Administration, Qingdao 266061, ChinaCollege of Marine Sciences, Shanghai Ocean University, Shanghai 201306, ChinaThe First Institute of Oceanography, State Oceanic Administration, Qingdao 266061, ChinaThis study expresses the solution of the Bessel equation in the neighbourhood of x=∞ 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 [x0,∞]. 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 x→∞ (not to mention a so-called formal solution), but a solution in the interval [x0,∞] 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.http://dx.doi.org/10.1155/2016/6826482 |
| spellingShingle | Qing-Hua Zhang Jian Ma Yuanyuan Qu Bessel Equation in the Semiunbounded Interval x∈[x0,∞]: Solving in the Neighbourhood of an Irregular Singular Point International Journal of Mathematics and Mathematical Sciences |
| title | Bessel Equation in the Semiunbounded Interval x∈[x0,∞]: Solving in the Neighbourhood of an Irregular Singular Point |
| title_full | Bessel Equation in the Semiunbounded Interval x∈[x0,∞]: Solving in the Neighbourhood of an Irregular Singular Point |
| title_fullStr | Bessel Equation in the Semiunbounded Interval x∈[x0,∞]: Solving in the Neighbourhood of an Irregular Singular Point |
| title_full_unstemmed | Bessel Equation in the Semiunbounded Interval x∈[x0,∞]: Solving in the Neighbourhood of an Irregular Singular Point |
| title_short | Bessel Equation in the Semiunbounded Interval x∈[x0,∞]: Solving in the Neighbourhood of an Irregular Singular Point |
| title_sort | bessel equation in the semiunbounded interval x∈ x0 ∞ solving in the neighbourhood of an irregular singular point |
| url | http://dx.doi.org/10.1155/2016/6826482 |
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