Exact Asymptotic Expansion of Singular Solutions for the (2+1)-D Protter Problem
We study three-dimensional boundary value problems for the nonhomogeneous wave equation, which are analogues of the Darboux problems in ℝ2. In contrast to the planar Darboux problem the three-dimensional version is not well posed, since its homogeneous adjoint problem has an infinite number of class...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2012-01-01
|
| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/278542 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832547519489376256 |
|---|---|
| author | Lubomir Dechevski Nedyu Popivanov Todor Popov |
| author_facet | Lubomir Dechevski Nedyu Popivanov Todor Popov |
| author_sort | Lubomir Dechevski |
| collection | DOAJ |
| description | We study three-dimensional boundary value problems for the nonhomogeneous wave equation, which are analogues of the Darboux problems in ℝ2. In contrast to the planar Darboux problem the three-dimensional version is not well posed, since its homogeneous adjoint problem has an infinite number of classical solutions. On the other hand, it is known that for smooth right-hand side functions there is a uniquely determined generalized solution that may have a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic light cone and does not propagate along the cone. The present paper describes asymptotic expansion of the generalized solutions in negative powers of the distance to this singular point. We derive necessary and sufficient conditions for existence of solutions with a fixed order of singularity and give a priori estimates for the singular solutions. |
| format | Article |
| id | doaj-art-86ea23beadb444dc8e29af235801fafc |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-86ea23beadb444dc8e29af235801fafc2025-02-03T06:44:31ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/278542278542Exact Asymptotic Expansion of Singular Solutions for the (2+1)-D Protter ProblemLubomir Dechevski0Nedyu Popivanov1Todor Popov2Faculty of Technology, Narvik University College, Lodve Langes Gate 2, 8505 Narvik, NorwayFaculty of Mathematics and Informatics, University of Sofia, 1164 Sofia, BulgariaFaculty of Mathematics and Informatics, University of Sofia, 1164 Sofia, BulgariaWe study three-dimensional boundary value problems for the nonhomogeneous wave equation, which are analogues of the Darboux problems in ℝ2. In contrast to the planar Darboux problem the three-dimensional version is not well posed, since its homogeneous adjoint problem has an infinite number of classical solutions. On the other hand, it is known that for smooth right-hand side functions there is a uniquely determined generalized solution that may have a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic light cone and does not propagate along the cone. The present paper describes asymptotic expansion of the generalized solutions in negative powers of the distance to this singular point. We derive necessary and sufficient conditions for existence of solutions with a fixed order of singularity and give a priori estimates for the singular solutions.http://dx.doi.org/10.1155/2012/278542 |
| spellingShingle | Lubomir Dechevski Nedyu Popivanov Todor Popov Exact Asymptotic Expansion of Singular Solutions for the (2+1)-D Protter Problem Abstract and Applied Analysis |
| title | Exact Asymptotic Expansion of Singular Solutions for the (2+1)-D Protter Problem |
| title_full | Exact Asymptotic Expansion of Singular Solutions for the (2+1)-D Protter Problem |
| title_fullStr | Exact Asymptotic Expansion of Singular Solutions for the (2+1)-D Protter Problem |
| title_full_unstemmed | Exact Asymptotic Expansion of Singular Solutions for the (2+1)-D Protter Problem |
| title_short | Exact Asymptotic Expansion of Singular Solutions for the (2+1)-D Protter Problem |
| title_sort | exact asymptotic expansion of singular solutions for the 2 1 d protter problem |
| url | http://dx.doi.org/10.1155/2012/278542 |
| work_keys_str_mv | AT lubomirdechevski exactasymptoticexpansionofsingularsolutionsforthe21dprotterproblem AT nedyupopivanov exactasymptoticexpansionofsingularsolutionsforthe21dprotterproblem AT todorpopov exactasymptoticexpansionofsingularsolutionsforthe21dprotterproblem |