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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2024. Vol 49

To the Solution of Loaded Differential Equations with Nonlocal Conditions

Author(s)
Vagif M. Abdullayev1,2,3

1Azerbaijan State Oil and Industry University, Baku, Azerbaijan

2Institute of Control Systems of Ministry of Science and Education of Republic of Azerbaijan, Baku, Azerbaijan

3Western Caspian University, Baku, Azerbaijan

Abstract
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 of 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.
About the Authors
Vagif M. Abdullayev, Dr. Sci. (Phys.Math.), Prof., Azerbaijan State Oil and Industry University and Senior Researcher of Institute of Control Systems of Ministry of Science and Education of Republic of Azerbaijan; employee of the Scientific and Innovation Center of the Western Caspian University, Baku, AZ1141, Azerbaijan, vaqif_ab@rambler.ru
For citation

Abdullayev V.M. To the Solution of Loaded Differential Equations with Nonlocal Conditions. The Bulletin of Irkutsk State University. Series Mathematics, 2024, vol. 49, pp. 45–62. (in Russian)

https://doi.org/10.26516/1997-7670.2024.49.45

Keywords
integro-differential equation, loaded equation, multipoint condition, integral condition, nonlocal condition, fundamental matrix of solution, existence and uniqueness condition
UDC
517.624
MSC
34B10, 45J05
DOI
https://doi.org/10.26516/1997-7670.2024.49.45
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