| Title:
|
On the boundary conditions associated with second-order linear homogeneous differential equations (English) |
| Author:
|
Das, J. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
40 |
| Issue:
|
3 |
| Year:
|
2004 |
| Pages:
|
301-313 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The ideas of the present paper have originated from the observation that all solutions of the linear homogeneous differential equation (DE) $y^{\prime \prime }(t) + y(t)=0$ satisfy the non-trivial linear homogeneous boundary conditions (BCs) $y(0) + y(\pi )=0$, $y^{\prime }(0) + y^{\prime }(\pi )=0$. Such a BC is referred to as a natural BC (NBC) with respect to the given DE, considered on the interval $[0, \pi ]$. This observation suggests the following queries : (i) Will each second-order linear homogeneous DE possess a natural BC ? (ii) How many linearly independent natural BCs can a DE possess ? The present paper answers these queries. It also establishes that any non-trivial homogeneous mixed BC, which is not a NBC with respect to the given linear homogeneous DE, determines uniquely (up to a constant multiplier), the solution of the DE. Two BCs are said to be compatible with respect to a given DE if both of them determine the same solution of the DE. Conditions for the compatibility of sets of two and three BCs with respect to a given DE have also been determined. (English) |
| Keyword:
|
natural BC |
| Keyword:
|
compatible BCs with respect to a given DE |
| MSC:
|
34B05 |
| MSC:
|
34B24 |
| idZBL:
|
Zbl 1117.34008 |
| idMR:
|
MR2107026 |
| . |
| Date available:
|
2008-06-06T22:44:04Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107913 |
| . |
| Reference:
|
[1] Das J. (neé Chaudhuri): On the solution spaces of linear second-order homogeneous ordinary differential equations and associated boundary conditions.J. Math. Anal. Appl. 200, (1996), 42–52. Zbl 0851.34008, MR 1387967 |
| Reference:
|
[2] Ince E. L.: Ordinary Differential Equations.Dover, New York, 1956. MR 0010757 |
| Reference:
|
[3] Eastham M. S. P.: Theory of Ordinary Differential Equations.Van Nostrand Reinhold, London, 1970. Zbl 0195.37001 |
| . |
