Classical and Mild Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential
We study the first mixed problem for the telegraph equation with a nonlinear potential in the first quadrant. We pose the Cauchy conditions on the lower base of the domain and the Dirichlet condition on the lateral boundary. By the method of characteristics, we obtain an expression for the solution...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Irkutsk State University
2023-03-01
|
| Series: | Известия Иркутского государственного университета: Серия "Математика" |
| Subjects: | |
| Online Access: | http://mathizv.isu.ru/en/article/file?id=1440 |
| _version_ | 1797868041955966976 |
|---|---|
| author | V. I. Korzyuk J. V. Rudzko |
| author_facet | V. I. Korzyuk J. V. Rudzko |
| author_sort | V. I. Korzyuk |
| collection | DOAJ |
| description | We study the first mixed problem for the telegraph equation with a nonlinear potential in the first quadrant. We pose the Cauchy conditions on the lower base of the domain and the Dirichlet condition on the lateral boundary. By the method of characteristics, we obtain an expression for the solution of the problem in an implicit analytical form as a solution of some integral equations. To solve these equations, we use the method of sequential approximations. The existence and uniqueness of the classical solution under specific smoothness and matching conditions for given functions are proved. Under inhomogeneous matching conditions, we consider a problem with conjugation conditions. When the given data is not smooth enough, we construct a mild solution. |
| first_indexed | 2024-04-09T23:50:55Z |
| format | Article |
| id | doaj.art-f46ac22dfc27495d8d988572a30f014b |
| institution | Directory Open Access Journal |
| issn | 1997-7670 2541-8785 |
| language | English |
| last_indexed | 2024-04-09T23:50:55Z |
| publishDate | 2023-03-01 |
| publisher | Irkutsk State University |
| record_format | Article |
| series | Известия Иркутского государственного университета: Серия "Математика" |
| spelling | doaj.art-f46ac22dfc27495d8d988572a30f014b2023-03-17T12:18:39ZengIrkutsk State UniversityИзвестия Иркутского государственного университета: Серия "Математика"1997-76702541-87852023-03-014314863https://doi.org/10.26516/1997-7670.2023.43.48Classical and Mild Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear PotentialV. I. KorzyukJ. V. RudzkoWe study the first mixed problem for the telegraph equation with a nonlinear potential in the first quadrant. We pose the Cauchy conditions on the lower base of the domain and the Dirichlet condition on the lateral boundary. By the method of characteristics, we obtain an expression for the solution of the problem in an implicit analytical form as a solution of some integral equations. To solve these equations, we use the method of sequential approximations. The existence and uniqueness of the classical solution under specific smoothness and matching conditions for given functions are proved. Under inhomogeneous matching conditions, we consider a problem with conjugation conditions. When the given data is not smooth enough, we construct a mild solution.http://mathizv.isu.ru/en/article/file?id=1440nonlinear wave equationclassical solutionmixed problemmatching conditionsgeneralized solution |
| spellingShingle | V. I. Korzyuk J. V. Rudzko Classical and Mild Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential Известия Иркутского государственного университета: Серия "Математика" nonlinear wave equation classical solution mixed problem matching conditions generalized solution |
| title | Classical and Mild Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential |
| title_full | Classical and Mild Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential |
| title_fullStr | Classical and Mild Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential |
| title_full_unstemmed | Classical and Mild Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential |
| title_short | Classical and Mild Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential |
| title_sort | classical and mild solution of the first mixed problem for the telegraph equation with a nonlinear potential |
| topic | nonlinear wave equation classical solution mixed problem matching conditions generalized solution |
| url | http://mathizv.isu.ru/en/article/file?id=1440 |
| work_keys_str_mv | AT vikorzyuk classicalandmildsolutionofthefirstmixedproblemforthetelegraphequationwithanonlinearpotential AT jvrudzko classicalandmildsolutionofthefirstmixedproblemforthetelegraphequationwithanonlinearpotential |