A finite-difference method for a singularly perturbed delay integro-differential equation

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

We introduce a new norm (derived from Bombieri's norm for polynomials) on a class of functions on the complex plane. This norm is hilbertian, and can be viewed as a weighted $L_{2}$ norm (or a weighted $l_{2}$ norm). It allows us to give quantitative results of the following sort: If we solve $P(D)u = f$ (with boundary conditions), and if we modify $f$, how is the solution $u$ modified?

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Stability of the solutions of differential equations

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

Convergence analysis of fitted numerical method for a singularly perturbed nonlinear Volterra integro-differential equation with delay

This paper is concerned with a singularly perturbed integral system with rapidly varying kernels. In contrast to integral equations with slowly varying kernels, where regularization is carried out with respect to the spectrum of the "diagonal" kernel, the regularization problem in the case of rapidly varying kernels is solved in terms of the spectrum of a certain extended operator. The goal here is to construct this operator and develop an appropriate algorithm for asymptotic solutions.

http://iopscience.iop.org/article/10.1070/SM2001v192n08ABEH000586/pdf

Volterra integral equations with rapidly varying kernels and their asymptotic integration

We generalize the Lomov’s regularization method to partial integro-differential equations. It turns out that the procedure for regularization and the construction of a regularized asymptotic solution essentially depend on the type of the integral operator. The most difficult is the case, when the upper limit of the integral is not a variable of differentiation. In this paper, we consider its scalar option. For the integral operator with the upper limit coinciding with the variable of differentiation, we investigate the vector case. In both cases, we develop an algorithm for constructing a regularized asymptotic solution and carry out its full substantiation. Based on the analysis of the principal term of the asymptotic solution, we study the limit in solution of the original problem (with the small parameter tending to zero) and solve the so-called initialization problem about allocation of a class of input data, in which the passage to the limit takes place on the whole considered period of time, including the area of boundary layer.

A Generalization of the Regularization Method to the Singularly Perturbed Integro-Differential Equations With Partial Derivatives

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

Singularly Perturbed Integro-Differential Equations with Diagonal Degeneration of the Kernel in Reverse Time

Asymptotic integration of differential systems of equations with fast oscillating coefficients has been carried out by the Feschenko-Shkil-Nikolenko splitting method and the Lomov regularization method. Equations of this type are often encountered in study of various questions related to dynamic stability, to properties of media with a periodic structure and other applied problems. In the monograph by Yu.L. Daletski and M.G. Krein an asymptotic analysis is given for one of these problems the problem on parametric amplification. In the present paper, we generalize this problem to integro-differential equations, the differential part of which coincides with the parametric amplification problem. The main purpose of the research is to identify the influence of the integral term in the asymptotic behavior of the solution. It is considered the general case, i.e. the case of both the lack of resonance (when the integer linear combination of frequencies of the fast oscillating cosine does not coincide with the spectrum frequency of the limit operator), and its presence (when such coincidence takes place). The developed algorithm is obviously generalized to systems of equations with an arbitrary matrix of the differential part, Received: November 22, 2019 c © 2020 Academic Publications §Correspondence author 332 A.A. Bobodzhanov, B.T. Kalimbetov, V.F. Safonov the pure imaginary spectrum, and with an arbitrary number of fast oscillating coefficients (such as the cosine considered in the paper). AMS Subject Classification: 34K26, 45J05

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V. F. Safonov

Papers

Volterra integral equations with rapidly varying kernels and their asymptotic integration

A Generalization of the Regularization Method to the Singularly Perturbed Integro-Differential Equations With Partial Derivatives

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

Singularly Perturbed Integro-Differential Equations with Diagonal Degeneration of the Kernel in Reverse Time

Integro-differential problem about parametric amplification and its asymptotical integration