On the class of pointwise and integrally loaded differential equations
We investigate a system of linear ordinary differential equations containing point and integral loadings with nonlocal boundary conditions. Boundary conditions include integral and point values of the unknown function. An essential feature of the problem is that the kernels of the integral terms in...
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| Format: | Article |
| Language: | English |
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Academician Ye.A. Buketov Karaganda University
2024-03-01
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| Series: | Қарағанды университетінің хабаршысы. Математика сериясы |
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| Online Access: | https://mathematics-vestnik.ksu.kz/index.php/mathematics-vestnik/article/view/680 |
| _version_ | 1797234198463905792 |
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| author | K.R. Aida-zade V.M. Abdullayev |
| author_facet | K.R. Aida-zade V.M. Abdullayev |
| author_sort | K.R. Aida-zade |
| collection | DOAJ |
| description |
We investigate a system of linear ordinary differential equations containing point and integral loadings with nonlocal boundary conditions. Boundary conditions include integral and point values of the unknown function. An essential feature of the problem is that the kernels of the integral terms in the differential equations depend only on the integration variable. It is shown that similar problems arise during feedback control of objects with both lumped and distributed parameters during point and integral measurements of the current state for the controllable object. The problem statement considered in the paper generalizes a lot of previously studied problems regarding loaded differential equations with nonlocal boundary conditions. By introducing auxiliary parameters, we obtain necessary conditions for the existence and uniqueness of a solution to the problem under consideration. To solve the problem numerically, we propose to use a representation of the solution to the original problem, which includes four matrix functions that are solutions to four auxiliary Cauchy problems. Using solutions to the auxiliary problems in boundary conditions, we obtain the values of the unknown function at the loading points. This is enough to get the desired solution. The paper describes the application of the method using the example of solving a test model problem.
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| first_indexed | 2024-04-24T16:28:15Z |
| format | Article |
| id | doaj.art-80ae6e5f02d543a985f6be35e879f859 |
| institution | Directory Open Access Journal |
| issn | 2518-7929 2663-5011 |
| language | English |
| last_indexed | 2024-04-24T16:28:15Z |
| publishDate | 2024-03-01 |
| publisher | Academician Ye.A. Buketov Karaganda University |
| record_format | Article |
| series | Қарағанды университетінің хабаршысы. Математика сериясы |
| spelling | doaj.art-80ae6e5f02d543a985f6be35e879f8592024-03-30T14:47:14ZengAcademician Ye.A. Buketov Karaganda UniversityҚарағанды университетінің хабаршысы. Математика сериясы2518-79292663-50112024-03-01113110.31489/2024m1/5-20On the class of pointwise and integrally loaded differential equationsK.R. Aida-zadeV.M. Abdullayev We investigate a system of linear ordinary differential equations containing point and integral loadings with nonlocal boundary conditions. Boundary conditions include integral and point values of the unknown function. An essential feature of the problem is that the kernels of the integral terms in the differential equations depend only on the integration variable. It is shown that similar problems arise during feedback control of objects with both lumped and distributed parameters during point and integral measurements of the current state for the controllable object. The problem statement considered in the paper generalizes a lot of previously studied problems regarding loaded differential equations with nonlocal boundary conditions. By introducing auxiliary parameters, we obtain necessary conditions for the existence and uniqueness of a solution to the problem under consideration. To solve the problem numerically, we propose to use a representation of the solution to the original problem, which includes four matrix functions that are solutions to four auxiliary Cauchy problems. Using solutions to the auxiliary problems in boundary conditions, we obtain the values of the unknown function at the loading points. This is enough to get the desired solution. The paper describes the application of the method using the example of solving a test model problem. https://mathematics-vestnik.ksu.kz/index.php/mathematics-vestnik/article/view/680integro-differential equationsystem of loaded equationsintegral conditionsconditions of existence and uniqueness |
| spellingShingle | K.R. Aida-zade V.M. Abdullayev On the class of pointwise and integrally loaded differential equations Қарағанды университетінің хабаршысы. Математика сериясы integro-differential equation system of loaded equations integral conditions conditions of existence and uniqueness |
| title | On the class of pointwise and integrally loaded differential equations |
| title_full | On the class of pointwise and integrally loaded differential equations |
| title_fullStr | On the class of pointwise and integrally loaded differential equations |
| title_full_unstemmed | On the class of pointwise and integrally loaded differential equations |
| title_short | On the class of pointwise and integrally loaded differential equations |
| title_sort | on the class of pointwise and integrally loaded differential equations |
| topic | integro-differential equation system of loaded equations integral conditions conditions of existence and uniqueness |
| url | https://mathematics-vestnik.ksu.kz/index.php/mathematics-vestnik/article/view/680 |
| work_keys_str_mv | AT kraidazade ontheclassofpointwiseandintegrallyloadeddifferentialequations AT vmabdullayev ontheclassofpointwiseandintegrallyloadeddifferentialequations |