Burkhan T. Kalimbetov

1. We consider a linear singularly perturbed problem (1) Lεy(t,ε)≡εdydt−A(t)y=h(t)eiβ(t)ε,y(0,ε)=y0, t∈[0,T] where y(t,ε)={y1(t,ε),…,yn(t,ε)} is an unknown vector function, A(t) is a matrix functio...

This article examines an integro-differential singularly perturbed initial value problem with exponential inhomogeneity and with a spectral value that turns zero at an isolated point in time. The problem belongs to the class of singularly perturbed equations with an unstable spectrum. An algorithm is proposed for constructing an asymptotic solution of the problem based on the regularized (according to Lomov’s method) asymptotics of the fundamental solution of the corresponding homogeneous integro-differential system.

The paper considers the Cauchy problem for a singularly perturbed integro-differential partial differential equation with a rapidly oscillating right-hand side. When considering such problems, it turned out that the existing technique for regularizing singularly perturbed equations is not effective and requires significant rethinking. The development of a new technique for constructing regularized asymptotic solutions for integro-differential equations with partial derivatives and exponential inhomogeneity constitutes the main content of this work. The problem was regularized and the normal and unique solvability of general iterative problems was proved. The asymptotic convergence of formal solutions is proved and a solution to the first iterative problem is constructed.

We consider a linear singularly perturbed differential system, one of the points of the spectrum of the limiting operator of which goes to zero on some discrete subset of the segment of the independent variable. The problem belongs to the class of problems with unstable spectrum. Previously, S.A. Lomov’s regularization method was used to construct asymptotic solutions of a similar system. However, it was applied in the case of absence of fast oscillations. The presence of the latter does not allow us to approximate the exact solution by a degenerate one, since the limit transition in the initial system when a small parameter tends to zero in a uniform metric is impossible. Therefore, when constructing the asymptotic solution, it is necessary to take into account the effects introduced into the asymptotics by fast oscillations. In developing the corresponding algorithm, one could use the ideas of the classical Lomov regularization method, but considering that its implementation requires numerous calculations (e.g., to construct the main term of the asymptotics in the simplest case of the second-order zero eigenvalue of the limit operator one has to solve three algebraic systems of order higher than the first), the authors considered it necessary to develop a more economical algorithm based on regularization by means of normal forms.

Singularly perturbed integro-differential equations with degenerate kernels are considered. It is shown that in the linear case these problems are always uniquely solvable with continuous coefficients, while nonlinear problems either have no real solutions at all or have several of them. For linear problems, the results of Bobojanova are refined; in particular, necessary and sufficient conditions are given for the existence of a finite limit of their solutions as the small parameter tends to zero and sufficient conditions under which the passage to the limit to the solution of the degenerate equation is possible.

The paper considers the parallel algorithm for solving the inverse problem of identifying the space-dependent right-hand part of a time-fractional diffusion equation. After discretization and approximation, the initial boundary problem is reduced to solving the systems of linear algebraic equations. For solving the inverse problem, the iterative conjugate gradient method is used. It requires solving the auxilliary initial boundary problem at each iterative step. On the basis of the parallel sweep method, a parallel algorithm is implemented for multicore processors. Numerical experiments were performed.

In this paper, the regularization method of S.A. Lomov is generalized to singularly perturbed integro-differential fractional order derivative equation with rapidly oscillating coefficients. The main purpose of the study is to reveal the influence of the integral term and rapidly oscillating coefficients on the asymptotics of the solution of the original problem. To study the influence of rapidly oscillating coefficients on the leading term of the asymptotics of solutions, we consider a simple case, i.e. the case of no resonance (when an entire linear combination of frequencies of a rapidly oscillating cosine does not coincide with the frequency of the spectrum of the limit operator).

In this paper, we consider a singularly perturbed integro-differential equation with a rapidly oscillating right-hand side, which includes an integral operator with a slowly varying kernel. Earlier, singularly perturbed differential and integro-differential equations with rapidly oscillating coefficients were considered. The main goal of this work is to generalize the Lomov's regularization method and to identify the rapidly oscillating right-hand side to the asymptotics of the solution to the original problem.

Singulyarno vozmushchennye integro-differentsial'nye sistemy s yadrami, zavisyashchimi ot resheniy differentsial'nykh uravneniy Сингулярно возмущённые интегро-дифференциальные системы с ядрами, зависящими от решений дифференциальных уравнений

We consider integro-differential equations (IDEs) with a rapidly oscillating inhomogeneity and with a Volterra-type integral operator whose kernels can contain both a classical rapidly decreasing exponential (the simplest case) and fundamental solutions of differential systems (the general case). Difficulty in constructing a regularized (according to S.A. Lomov) asymptotics in the general case is due to the complex asymptotic structure of the fundamental solution matrix (Cauchy matrix) of the homogeneous differential system. In the present paper, we first construct a regularized asymptotics of the Cauchy matrix, which is then used to construct a regularized asymptotics of the solution of the IDE.