Zaripov S.K. A New Solution for One-Class Model Integro-Differential Equations of First Order with Singularity in the Kernel

https://doi.org/10.15688/mpcm.jvolsu.2017.4.6

Sarvar Kakhramonovich Zaripov
Candidate of Physical and Mathematical Sciences,
Senior Lecturer, Department of Mathematical Analysis and Function Theory,
Tajik National University
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Travel Rudaki, 17, 734025 Dushanbe, Republic of Tajikistan

Abstract. Integral representations of manifold solutions are obtained by arbitrary constants for one-class first-order model integro-differential equation with singularity in the kernel. First of all, the author highlights a new class of functions that at point x = a convert to zero with some asymptotic behavior. The solution of the given equation is found in this class. It is shown that the solution of studied equation is equivalent to the solution of such system of integro-differential equation in which one of the equations is differential equation, and the other is integral equation. In the case when B < 0, the solution of homogeneous equation depends on two arbitrary constants and  general solutions  of  non-homogeneous  equation,  as well  as  on  two  arbitrary  constants. When B > 0 the solution of homogeneous equation depends only on one arbitrary constant, and the general solution of non-homogeneous equation depends on one arbitrary constant too.
The analogue of Fredholm theorems it proposed for a given integro-differential equation. In addition, the singular differential operators are described, and the main properties of these operators are studied.
In the cases, when the solution of a given integro-differential equation depends on any arbitrary constant, a Koshi-type problems are investigated. For the investigation of Koshi-type problems, first of all, the property of the obtained solution is studied. It is shown that, when some conditions are fulfilled, the Koshi-type problems have only unique solution. All received results can be transferred for the equations with right singular point in the kernel.
The method of solution of a given singular integro-differential equation can be used for the solution of higher-order model and non-model integro-differential equations with singular coefficients. Moreover, the proposed method can be used for the solution of partial model and non-model integro-differential equations with singular coefficients.

Key  words:  model  integro-differential  equation,  boundary  singular  points,  integral representations, boundary problems, systems of integro-differential equations.