This paper research initial value problems for a system of linear integral-differential equations with partial derivative operators on the surface of a right circular cylinder. The main focus is on proving the existence, uniqueness, and properties of solutions for such Volterra-type integral-differential systems. These problems are widely encountered in the mathematical modeling of physical, biological, and other hereditary phenomena.

The research considers a system of linear (θ, ω) -periodic equations. The existence of a unique solution satisfying the initial conditions is analyzed using the system's resolvent operator. The solutions are examined within the class of ω -periodic functions with respect to the variable t.

The main result of the work is the presentation of integral expressions for solutions satisfying the initial conditions. Additionally, by utilizing the properties of resolvent operators, the conditions for the existence and uniqueness of solutions to the system are determined. The results obtained are of theoretical significance. The research methods also enable the effective resolution of other problems involving systems of integral-differential equations. Moreover, the universal properties of the methods and their adaptability to various physical and engineering problems have been demonstrated. The results of the work play a significant role in the development of the theory of integral-differential equations and serve as a basis for creating effective algorithms for their numerical solutions.

SARTABANOV ZH.A.

Doctor of Physical and Mathematical Sciences, Professor, K. Zhubanov Aktobe Regional University, Aktobe, Kazakhstan

Е-mail: sartabanov42@mail.ru; https://orcid.org/0000-0003-2601-2678

AITENOVA G.M.

Doctor of Philosophy (PhD), M.Utemisov West Kazakhstan University, Uralsk, Kazakhstan

Е-mail: gulsezim-88@mail.ru; https://orcid.org/0000-0002-4572-8252

KURMANGALIEV O.A.

Master's student, K. Zhubanov Aktobe Regional University, Aktobe, Kazakhstan

Е-mail: orynbek.kurmangaliev@mail.ru; https://orcid.org/0009-0004-0589-4055

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integro-differential equations, Volterra system, resolvent operator, periodicity, initial conditions, existence of solution, uniqueness of solution