Regularized asymptotical solutions of integro-differential systems with spectral singularities

Abstract

In the work, the Cauchy problem is considered for a singularly perturbed system with zero points of the spectrum. It is proved normal solvability of general iterative problems. Unique solvability and convergence of formal solutions to the exact one are formulated.

Asymptotic solutions of integro-differential equations with partial derivatives and with rapidly varying kernel

“…The asymptotic convergence of the series (5) and 8is proved in the same way as the analogous statement in [1]. Let nowỹ(x, t, τ, ε) be an arbitrary function continuous with respect to (x, t, τ ) ∈ [0, 1] × [0, 1] × {Re a(x)} and having the asymptotic expansioñ…”

“…The problem of constructing a regularized asymptotic solution of the problem (1) [1] is posed. Earlier, in [3][4][5][6][7], systems for ordinary integro-differential equations were mainly considered. In contrast to these works, in this paper we consider an partial integro-differential equation.…”

Internal boundary layer for integral-differential equations with zero spectrum of the limit operator and rapidly changing kernel

“…Proof Let h(t, τ, σ) = h 0 (t) + p j=0 h j (t)e τ j + h m (t)e τm + h m+1 (t)σ. Define a solution of the system (14) in the form (9). Putting (9) into the system (14) and equating coefficients of e τ j , e τm , σ, and the free terms, we get the following systems:…”

“…At the end of the last century articles devoted to the study of singularly perturbed integral-differential systems with rapidly changing kernels began to appear [3,5,6,8,9,10,11,12,13,14,16,17]. Integral operators with rapidly decreasing kernel create in the structure of solutions of the original problem a new type of singularities by a small parameter, which describes the internal boundary layer.…”

Asymptotic Solutions of Scalar Integro-Differential Equations with Partial Derivatives and with Fast Oscillating Coefficients

“…The problem of constructing a regularized asymptotic solution [1] of the problem (1) is posed. Earlier, in [2], [3], [4], [5], [6], [7], systems for ordinary integro-differential equations were mainly considered. In this paper we consider an partial integro-differential equations.…”