WELL-POSEDNESS OF A DIRICHLET PROBLEM FOR A HYPERBOLIC TYPE INTEGRO-DIFFERENTIAL EQUATION

  • A. Anikushyn Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
  • O. Zhyvolovych Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
Keywords: Integro-differential equation, hyperbolic equation, Volterra operator, well-posedness, a priori estimates

Abstract

In the paper we consider a Dirichlet problem for an integro-differential equation with Volterra type integral term. Proving a priori estimates for the differential and integral parts, we provide negative norms’ a priori estimates for the operator of the problem. Based on the latest, we formulate theorems regarding the well-posedness of the formulated boundary value problem.

References

Anikushin A. V., Nomirovskii D. A. Trajectory and final controllability of hyperbolic systems in different classes of generalized functions. Visnyk Kyiv. Univer., Fiz-Mat Series. 2008. No 3. P. 119–124.

Anikushin A. V. Generalized solvability of hyperbolic integro-differential equations. Visnyk Kyiv. Univer., Fiz-Mat Series, 2013. No 4. P. 60–65.

Anikushyn A. V., Hranishak H. M. On a weak solvability of a hyperbolic integro-differential equation. 5-th International Conference for Young Scientists on Differential Equations and Applications dedicated to Ya. B. Lopatynsky. Kyiv. 2016. P. 31–33

Lyashko S. I. Generalized optimal control of linear systems. K.: Naukova Dumka, 1998. 465 p.

Anikushin A. V. Generalized sovability of linear integro-differential elliptic equations. Visnyk Kyiv. Univer., Fiz-Mat Series, 2010. No 3. P. 163–168.

Anikyshin A. V., Hulianitsky A. L. Generalized solvability of parabolic integro-differential equations. Differential Equations. 2014. V. 50. No 1. P. 98–109.

Anikushin A. V. Optimal control of parabolic integro-differential equations. Journal on Numerical and Applied Mathematics. 2010. No 3. P. 3–16.

Anikushin A. V., Kosteeva L. O. A priori and generalized solvability of initial-boundary problem for one integro-differential equation. Journal on Numerical and Applied Mathematics. 2018. No 3. P. 2–14.

Anikushyn A. V. Weak solvability and optimal control for a class of partial integro-differential equations. International Conference on Differential Equations Dedicated to 110th Anniversary of Ya. B. Lopatynsky. Lviv. 2016. P. 13.

Anikyshin A. V., Hulianitsky A. L. Simulation and optimization of process described by linear integro-differential equations. Kyiv, Paperovy Zmiy, 2015. 134 p.

Lyashko S. I. Generalized optimal control of linear systems with distributed parameters. Boston–Dordrecht–London: Kluwer Academic Publishers, 2002. 466 p.

Published
2023-02-02
How to Cite
Anikushyn, A., & Zhyvolovych, O. (2023). WELL-POSEDNESS OF A DIRICHLET PROBLEM FOR A HYPERBOLIC TYPE INTEGRO-DIFFERENTIAL EQUATION. Journal of Numerical and Applied Mathematics, 1(2), 13-23. https://doi.org/10.17721/2706-9699.2022.2.02