Constructive theory of boundary value problems for linear integral-differential equations
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Keywords:
constructive theory, boundary value problems, linear integro-differential equations, the principle of immersionAbstract
The necessary and sufficient conditions for solvability of boundary value problems of the linear integral-differential equations at phase and integral constraints are obtained. A method for constructing the solution of the boundary value problem with constraints by constructing minimizing sequences is developed. The basis of the proposed method for solving the boundary value problem is the principle of immersion. The principle of immersion is created by building the solution of a class of Fredholm integral equations of the first kind. The principal difference of the proposed method is that the origin value problem at the beginning immersed to the controllability problem with fictitious controls of functional spaces, followed by reduction to the initial problem of optimal control. Solvability and construction of a solution of the boundary value problems are solved together by solving an optimization problem. Creating a general theory of boundary value problems for linear integral-differential equations with complex boundary conditions in the presence of the phase and integral constraints is a topical problem with applications in the natural sciences, economics and ecology.References
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[2] 2. Samoilenko A.M., Bojchuk O.A., Krivosheja S.A. Boundary value problems for systems of the linear integral-differential equations with degenerate kernel // Ucr. math. journ. 1996. v. 48. No 11. P. 1576–1579.
[3] 3. Dzhumabaev D.S., Bakirova Je.A. About properties of solvability of the linear twopoint boundary value problem for systems of the integral-differential equations // Diff.equat. 2013. v. 49, No 9. P. 1125-1-140.
[4] 4. Aisagaliev S.A. Constructive theory of the boundary value problems of integral-differential equations with phase constraints // Proceedings of the International scientific conference "Problems of mathematics and informatics in XXI century". Bulletin of Kyrgyz state national university. Ser. 3, No. 4, 2000. P. 127–133.
[5] 5. Aisagaliev S.A., Aisagaliev T.S. Methods of solution of the boundary value problems. – Almaty, Kazakh university, 2002. – 348 p.
[6] 6. Aisagaliev S.A. Control by some system of differential equations // Diff. equat. 1991, v. 27, No 9. p. 1475–1486.
[7] 7. Aisagaliev S.A., Belogurov A.P. Controllability and speed of the process described by parabolic equation with limited control // Siberian mathematical journal. – 2012. – v. 53, № 1. – P. 20-37.
[8] 8. Ajsagaliev S.A. General solution of a class integral equations // Mathematical journal. Institute of Mathematics MES RK. - 2005. - v. 5, № 4. - P. 7-13.
[9] 9. Aisagaliev S.A., Kabidoldanova A.A. Optimal control by linear systems with linear quality criteria and restrictions //Differential equations. – 2012. – v. 48, № 6. – P. 826-838.
[10] 10. Aisagaliev S.A., Kalimoldaev M.N. Constructive method for solution of the boundary value problems for ordinary differential equations // Diff. equat. 2015. v. 51. No 2. P. 147–160.
[11] 11. Kalman R. On general theory of control systems. Proceedings of the i-st congress IFAK, v. 2. Moscow, 1961, pp. 521-547.
[12] 12. Kolmogorov A.N., Fomin S.V. Elements of the function theory and functional analysis. – M.: Nauka, 1989. – 623 p.
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Aisagaliev, S. A., & Aisagalieva, S. S. (2015). Constructive theory of boundary value problems for linear integral-differential equations. Journal of Mathematics, Mechanics and Computer Science, 87(4), 3–26. Retrieved from https://bm.kaznu.kz/index.php/kaznu/article/view/280
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