In this paper are considered the questions of unique solvability and redefinitions of a nonlocal inverse problem for the Fredholm integro-differential equation of the second order with degenerate kernel, integral condition, and spectral parameter. Calculations of the value of the spectral parameter are reduced to the solve of trigonometric equations. Systems of algebraic equations are obtained. The singularities that arose in determining arbitrary constants are studied. A criterion for unique solvability of the problem is established and the corresponding theorem is proved.

On Inverse Boundary Value Problem for a Fredholm Integro-Differential Equation with Degenerate Kernel and Spectral Parameter

The problems of solvability and construction of solutions of a nonlocal boundary value problem for the second-order Fredholm integro-differential equation with degenerate kernel, integral conditions, spectral parameters and reflecting deviation are considered. Using the method of the degenerate kernel, the boundary value problem is integrated as an ordinary differential equation. When we define the arbitrary integration constants there are possible five cases with respect to the first spectral parameter. Calculated values of the spectral parameter for each case. Further, the problem is reduced to solving systems of linear algebraic equations. Irregular values of the second spectral parameter are determined. At irregular values of the second spectral parameter the Fredholm determinant is degenerate. Other values of the second spectral parameter, for which the Fredholm determinant does not degenerate, are called regular values. Taking the values of the first spectral parameter into account for regular values of the second spectral parameter the corresponding solutions were constructed for each of five cases. The stability of the solution of the boundary value problem for given values in integral conditions is proved. The conditions under which the solution of the boundary value problem will be small are studied. For the irregular values of the second spectral parameter each of the five cases is checked separately. The orthogonality conditions are used. Cases are determined in which the problem has an infinite number of solutions and these solutions are constructed. For other cases, the absence of nontrivial solutions of the problem is proved.

Spectral Features of the Solving of a Fredholm Homogeneous Integro-Differential Equation with Integral Conditions and Reflecting Deviation

The problems of the existence and construction of solutions of a nonlocal boundary value problem for the homogeneous second-order Fredholm integrodifferential equation with a degenerate kernel and with two spectral parameters are considered. The singularities arising from the definition of arbitrary (unknown) constants are studied. The values of the spectral parameters are calculated and the solvability of the boundary value problem is established. The corresponding theorems are proven. Meaningful examples are provided.

On the Solvability of a Boundary Value Problem for the Ordinary Fredholm Integrodifferential Equation with a Degenerate Kernel

The paper deals with extending the Lomov regularization method to classes of singularly perturbed Fredholm-type integro-differential systems, which have not so far been studied. In these the limiting operator is discretely noninvertible. Such systems are commonly known as problems with unstable spectrum. Separating out the essential singularities in the solutions to these problems presents great difficulties. The principal one is to give an adequate description of the singularities induced by 'instability points' of the spectrum. A methodology for separating singularities by using normal forms is developed. It is applied to the above type of systems and is substantiated in these systems. Bibliography: 10 titles.

https://iopscience.iop.org/article/10.1070/SM2013v204n07ABEH004327/pdf

The method of normal forms for singularly perturbed systems of Fredholm integro-differential equations with rapidly varying kernels

In the study of various issues related to dynamic stability, with the properties of media with a periodic structure, in the study of other applied problems, one has to deal with differential equations with rapidly oscillating coefficients. Asymptotic integration of differential systems of equations with such coefficients was carried out by the splitting method and the regularization method. In this paper, a system of integrodifferential equations is considered. The main objective of the study is to identify the influence of the integral term on the asymptotics of the solution to the original problem. The case of the absence of resonance is considered, i.e. the case when the integer linear combination of frequencies of the rapidly oscillating coefficient does not coincide with the frequency of the spectrum of the limit operator.

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Integro-differentiated singularly perturbed equations with fast oscillating coefficients

The Lomov regularization method [1] is generalized to integro-partial differential equations. It turns out that the regularization procedure essentially depends on the type of integral operator. The case in which the upper limit of the integral is not the differentiation variable is the most difficult one. It is not considered in the present paper. Only the case in which the upper limit of the integral operator coincides with the differentiation variable is studied. For such problems, an algorithm for constructing regularized asymptotics is developed.

Regularized asymptotic solutions of the initial problem for the system of integro-partial differential equations

We consider a singularly perturbed Fredholm integro-differential equation with a rapidly varying kernel. We derive an algorithmfor constructing regularized asymptotic solutions. It is shown that, given a rapidly decreasing multiplier of the kernel, the original problem does no involve the spectrum (i.e., it is solvable for any right-hand side).

A. A. Bobodzhanov

Papers

Regularized asymptotic solutions of the initial problem for the system of integro-partial differential equations

“Splashes” in Fredholm integro-differential equations with rapidly varying kernels