Unique solvability of boundary value problem for functional differential equations with involution
Инволюциялы функционалды-дифференциалдық теңдеулер үшiн шеттiк есептiң бiрмәндi шешiлiмдiлiгi
Однозначная разрешимость краевой задачи для функционально-дифференциальных уравнений с инволюцией
Nazarova K.Zh. Usmanov K.I.
2021E.A.Buketov Karaganda State University Publish House
Bulletin of the Karaganda University. Mathematics Series
2021#103Issue 368 - 75 pp.
In this paper, we consider a boundary value problem for systems of Fredholm type integral-differential equations with involutive transformation, containing derivative of the required function on the right-hand side under the integral sign. Applying properties of an involutive transformation, original boundary value problem is reduced to a boundary value problem for systems of integral-differential equations, containing derivative of the required function on the right side under the integral sign. Assuming existence of resolvent of the integral equation with respect to the kernel K̃2(t, s) (this is the kernel of the integral equation that contains the derivative of the desired function) and using properties of the resolvent, integral-differential equation with a derivative on the right-hand side is reduced to a Fredholm type integral-differential equation, in which there is no derivative of the desired function on the right side of the equation. Further, the obtained boundary value problem is solved by the parametrization method created by Professor D. Dzhumabaev. Based on this method, the problem is reduced to solving a special Cauchy problem with respect to the introduced new functions and to solving systems of linear algebraic equations with respect to the introduced parameters. An algorithm to find a solution is proposed. As is known, in contrast to the Cauchy problem for ordinary differential equations, the special Cauchy problem for systems of integral-differential equations is not always solvable. Necessary conditions for unique solvability of the special Cauchy problem were established. By using results obtained by Professor D. Dzhumabaev, necessary and sufficient conditions for the unique solvability of the original problem were established.
boundary value conditions , integral equation , involution , parametrization method , resolvent , Special Cauchy Problem , system of integral-differential equations , unique solvability
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Khoja Akhmet Yassawi International Kazakh-Turkish University, Turkistan, Kazakhstan
Khoja Akhmet Yassawi International Kazakh-Turkish University
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