The Goursat problem for hyperbolic linear third order equations
The third order hyperbolic linear differential equation is considered in the non‐cylindrical domain of multidimensional Euclidean space. The equation operator is a composition of a differentiation operator of the first order and second order operator, which is hyperbolic with respect to the prescrib...
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| Format: | Article |
| Language: | English |
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Vilnius Gediminas Technical University
2001-12-01
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| Series: | Mathematical Modelling and Analysis |
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| Online Access: | https://journals.vgtu.lt/index.php/MMA/article/view/9913 |
| _version_ | 1818871560686010368 |
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| author | V. I. Korzyuk |
| author_facet | V. I. Korzyuk |
| author_sort | V. I. Korzyuk |
| collection | DOAJ |
| description | The third order hyperbolic linear differential equation is considered in the non‐cylindrical domain of multidimensional Euclidean space. The equation operator is a composition of a differentiation operator of the first order and second order operator, which is hyperbolic with respect to the prescribed vector field. Apart from the equation, Goursat and Cauchy conditions are defined for an unknown function. Thus the boundary of the domain, where this hyperbolic equation is defined, consists of characteristic hypersurfaces, the hypersur‐faces, where Cauchy conditions are prescribed, and hypersurfaces with no conditions. For the mentioned problem the existence and uniqueness of the strong solution are proved using mollifying operators with a variable step and functional analysis methods on the base of the previously proved energy inequality.
Trečios eilės tiesinių hiperbolinių lygčių Goursat uždavinys
Santrauka
Daugiamate Euklido erdves necilindrineje srityje nagrinejama trečios eiles tiesine hiper‐boline lygtis. Lygties operatorius yra pirmos eiles diferencialinio operatoriaus ir antros eiles operatoriaus, kuris yra hiperbolinis apibrežto vektorinio lauko atžvilgiu, kompozicija. Srities kontūra sudaro charakteristinis hiperpaviršius (jame formuojama Goursat salyga), hiperpaviršiaus, kuriame formuluojama Caushy salyga, ir laisvas nuo bet kokiu salygu hiperpaviršius. Naudojantis kintamojo žingsnio suvidurkinto operatoriaus bei funkcines analizes metodais, paremtais energetine nelygybe, irodytas šio uždavinio stipriojo sprendinio egzistavimas ir vienatis.
First Published online: 14 Oct 2010 |
| first_indexed | 2024-12-19T12:24:52Z |
| format | Article |
| id | doaj.art-ca1cb357e8bf480380fc9b3ed77c2258 |
| institution | Directory Open Access Journal |
| issn | 1392-6292 1648-3510 |
| language | English |
| last_indexed | 2024-12-19T12:24:52Z |
| publishDate | 2001-12-01 |
| publisher | Vilnius Gediminas Technical University |
| record_format | Article |
| series | Mathematical Modelling and Analysis |
| spelling | doaj.art-ca1cb357e8bf480380fc9b3ed77c22582022-12-21T20:21:36ZengVilnius Gediminas Technical UniversityMathematical Modelling and Analysis1392-62921648-35102001-12-016210.3846/13926292.2001.9637166The Goursat problem for hyperbolic linear third order equationsV. I. Korzyuk0Belarusian State University , Skoryna Ave. 4, Minsk, 220050, Belarus; Institute of Mathematics , NAS of Belarus , Surganov St., 11, Minsk, 220072, BelarusThe third order hyperbolic linear differential equation is considered in the non‐cylindrical domain of multidimensional Euclidean space. The equation operator is a composition of a differentiation operator of the first order and second order operator, which is hyperbolic with respect to the prescribed vector field. Apart from the equation, Goursat and Cauchy conditions are defined for an unknown function. Thus the boundary of the domain, where this hyperbolic equation is defined, consists of characteristic hypersurfaces, the hypersur‐faces, where Cauchy conditions are prescribed, and hypersurfaces with no conditions. For the mentioned problem the existence and uniqueness of the strong solution are proved using mollifying operators with a variable step and functional analysis methods on the base of the previously proved energy inequality. Trečios eilės tiesinių hiperbolinių lygčių Goursat uždavinys Santrauka Daugiamate Euklido erdves necilindrineje srityje nagrinejama trečios eiles tiesine hiper‐boline lygtis. Lygties operatorius yra pirmos eiles diferencialinio operatoriaus ir antros eiles operatoriaus, kuris yra hiperbolinis apibrežto vektorinio lauko atžvilgiu, kompozicija. Srities kontūra sudaro charakteristinis hiperpaviršius (jame formuojama Goursat salyga), hiperpaviršiaus, kuriame formuluojama Caushy salyga, ir laisvas nuo bet kokiu salygu hiperpaviršius. Naudojantis kintamojo žingsnio suvidurkinto operatoriaus bei funkcines analizes metodais, paremtais energetine nelygybe, irodytas šio uždavinio stipriojo sprendinio egzistavimas ir vienatis. First Published online: 14 Oct 2010https://journals.vgtu.lt/index.php/MMA/article/view/9913- |
| spellingShingle | V. I. Korzyuk The Goursat problem for hyperbolic linear third order equations Mathematical Modelling and Analysis - |
| title | The Goursat problem for hyperbolic linear third order equations |
| title_full | The Goursat problem for hyperbolic linear third order equations |
| title_fullStr | The Goursat problem for hyperbolic linear third order equations |
| title_full_unstemmed | The Goursat problem for hyperbolic linear third order equations |
| title_short | The Goursat problem for hyperbolic linear third order equations |
| title_sort | goursat problem for hyperbolic linear third order equations |
| topic | - |
| url | https://journals.vgtu.lt/index.php/MMA/article/view/9913 |
| work_keys_str_mv | AT vikorzyuk thegoursatproblemforhyperboliclinearthirdorderequations AT vikorzyuk goursatproblemforhyperboliclinearthirdorderequations |