The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges
The Dirichlet problem for the 2D Helmholtz equation in an exterior domain with cracks is studied. The compatibility conditions at the tips of the cracks are assumed. The existence of a unique classical solution is proved by potential theory. The integral representation for a solution in the form of...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2012-01-01
|
| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2012/340310 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832563551522258944 |
|---|---|
| author | P. A. Krutitskii |
| author_facet | P. A. Krutitskii |
| author_sort | P. A. Krutitskii |
| collection | DOAJ |
| description | The Dirichlet problem for the 2D Helmholtz equation in an exterior domain with cracks is studied. The compatibility conditions at the tips of the cracks are assumed. The existence of a unique classical solution is proved by potential theory. The integral representation for a solution in the form of potentials is obtained. The problem is reduced to the Fredholm equation of the second kind and of index zero, which is uniquely solvable. The asymptotic formulae describing singularities of a solution gradient at the edges (endpoints) of the cracks are presented. The weak solution to the problem may not exist, since the problem is studied under such conditions that do not ensure existence of a weak solution. |
| format | Article |
| id | doaj-art-3fe5bf251e244ee88e4fb4d3a9462ec5 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-3fe5bf251e244ee88e4fb4d3a9462ec52025-02-03T01:13:10ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252012-01-01201210.1155/2012/340310340310The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the EdgesP. A. Krutitskii0KIAM, Miusskaya Square 4, Moscow 125047, RussiaThe Dirichlet problem for the 2D Helmholtz equation in an exterior domain with cracks is studied. The compatibility conditions at the tips of the cracks are assumed. The existence of a unique classical solution is proved by potential theory. The integral representation for a solution in the form of potentials is obtained. The problem is reduced to the Fredholm equation of the second kind and of index zero, which is uniquely solvable. The asymptotic formulae describing singularities of a solution gradient at the edges (endpoints) of the cracks are presented. The weak solution to the problem may not exist, since the problem is studied under such conditions that do not ensure existence of a weak solution.http://dx.doi.org/10.1155/2012/340310 |
| spellingShingle | P. A. Krutitskii The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges International Journal of Mathematics and Mathematical Sciences |
| title | The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges |
| title_full | The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges |
| title_fullStr | The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges |
| title_full_unstemmed | The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges |
| title_short | The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges |
| title_sort | 2d dirichlet problem for the propagative helmholtz equation in an exterior domain with cracks and singularities at the edges |
| url | http://dx.doi.org/10.1155/2012/340310 |
| work_keys_str_mv | AT pakrutitskii the2ddirichletproblemforthepropagativehelmholtzequationinanexteriordomainwithcracksandsingularitiesattheedges AT pakrutitskii 2ddirichletproblemforthepropagativehelmholtzequationinanexteriordomainwithcracksandsingularitiesattheedges |