Asymptotic behavior of the solution of the integral boundary value problem for singularly perturbed integro-differential equations

The work is devoted to clarifying asymptotic with respect to a small parameter behavior of the solution of the integral boundary value problem for singularly perturbed linear integro-differential equation. We study the boundary value problem for singularly perturbed integro-differential equations with the phenomena of the so-called boundary jumps, when the fast solution variable becomes unbounded at both boundaries. The exceptions of the qualitative influence of integral terms on the asymptotic behavior of the solutions for singularly perturbed integro-differential equations are shown. The presence of integral terms will significantly change the degenerate equation: the solution of the assumed singularly perturbed integro-differential equation does not tend to the solution of the usual degenerate equation, obtained from the supposed equation with the zero value of a small parameter and will tend to solve a specially modified degenerate integrodifferential equation with an additional term called the jump of the integral term. Boundary and initial functions are defined; their existence and uniqueness are proved. On the basis of the constructed boundary and initial functions are obtained analytical formula and asymptotic estimates of the solution for the integral boundary value problem. It is established that the solution of the considered boundary value problem at the ends of a given segment has the phenomena of boundary jumps of the same orders. A modified degenerate boundary value problem is constructed, to the solution of which approaches the solution of assumed singularly perturbed integral boundary value problem. The value of the jump of integral terms is found. An example was made based on the initial results.