Asymptotic integration of singularly perturbed problems

Lomov, S. A.; Eliseev, A. G.

doi:10.1070/rm1988v043n03abeh001752

Cited by 10 publications

(3 citation statements)

References 39 publications

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“…(13) and (14), we obtain equations for Sj' which are also special cases of E~s. (7) and (8). Therefore, from (9) we obtain the estimate ~Sj'(t) u <_ c(IIQ'(t)II 2 + llQ"(t)ll), which is valid for almost all t ~R.…”

Section: ' It) S (T) +S~ (T) M~"(t) =M -'~' (T) (M'~ (T) )'M-'; Itmentioning

confidence: 89%

“…Questions related to the construction of the asymptotes of particular solutions determined by a point %(t) of the discrete spectrum of the coefficient Q(t) both in the case of systems of scalar equations and in the case of abstract equations (cf. [5,8] and the literature cited therein) have been subjected to extensive study. The essential part of the results presented in here involves the approximate reduction of a boundary-value problem to an evolutionary-type problems; these results are not relaged to a decomposition of a solution with respect to the spectrum Q(t).…”

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confidence: 99%

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Operator analogs of VKB-type estimates and the solvability of boundary-value problems

Édel'shtein

1992

Math Notes

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Let us consider the differential equation d~y/dt2--k2Q(t)=F(t) (t~R) (1) in the Banach space B. It is assumed that q is a sufficiently smooth operator function 9 Q(t) are !bounded and uniformly invertible, and, moreover 9 there exists a continuous branch of the root Ql/2(t), such that for all t e R, the operators -Q1/2(t) generate contraction semigroups.It is proved that if the spectra of the limit operators Q(• do not contain negative numbers, ithen, for sufficiently large k, the equation will have a unique bounded solution for any ~ounded F(t). This result turns out to be stronger than those corollaries that may be obtained for this problem from the results on solvability given in [1][2][3].We also consider a boundary-value problem with explicitly stipulated conditions at infinitely!distant points:It is prqved that problem (i), (2) [regardless of whether the condition imposed on Q(• holds] fqr F E LI(R , B) is solvable, and a formula which constitutes ageneralization of the following estimate is established: G(t,s)=-(2k)-tQ-'~'(t)Q-'~'(s) exp(,sign(t-s)kI[ O'~(s)ds) (i+O(k-') )for the Green's function of the scalar problem of the form (1) 9 (2) (formula (3) is an obvious consequence of the VKB estimates for the solution of a homogeneous scalar equation; cf. [4 9 5]).In the operator case, however, in place of exponential functions in the expression for the principal term of the asymptote evolutionary operators of the following Cauchy problem are encoUntered:Asymptotic decomposition of abstract differential equations has been investigated previously in [2,6,7] for the Cauchy problem. Questions related to the construction of the asymptotes of particular solutions determined by a point %(t) of the discrete spectrum of the coefficient Q(t) both in the case of systems of scalar equations and in the case of abstract equations (cf. [5,8] and the literature cited therein) have been subjected to extensive study. The essential part of the results presented in here involves the approximate reduction of a boundary-value problem to an evolutionary-type problems; these results are not relaged to a decomposition of a solution with respect to the spectrum Q(t).Note that problems such as (i), (2) arise under natural conditions in the description i. of harmonic fields in waveguides [though Q(t) may then be unbounded] 9 and the reduction referred tO above asserts, speaking in terms of physics 9 that it is possible to take into ac-! --P Ros~ov State University.

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Section: ' It) S (T) +S~ (T) M~"(t) =M -'~' (T) (M'~ (T) )'M-'; Itmentioning

confidence: 89%

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confidence: 99%

Operator analogs of VKB-type estimates and the solvability of boundary-value problems

Édel'shtein

1992

Math Notes

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“…where ϕ i (t) is a smooth function (in the general case, complex) of a real variable t. To solve linear homogeneous equations, such singularities have been described by Liouville [5][6][7][8].…”

Section: Formalism Of Regularization Methodsmentioning

confidence: 99%

Regularized Solution of Singularly Perturbed Cauchy Problem in the Presence of Rational “Simple” Turning Point in Two-Dimensional Case

Eliseev

Ratnikova

2019

Axioms

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By Lomov’s S.A. regularization method, we constructed an asymptotic solution of the singularly perturbed Cauchy problem in a two-dimensional case in the case of violation of stability conditions of the limit-operator spectrum. In particular, the problem with a ”simple” turning point was considered, i.e., one eigenvalue vanishes for t = 0 and has the form t m / n a ( t ) (limit operator is discretely irreversible). The regularization method allows us to construct an asymptotic solution that is uniform over the entire segment [ 0 , T ] , and under additional conditions on the parameters of the singularly perturbed problem and its right-hand side, the exact solution.

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Direct Asymptotic Methods

Kuehn

2014

Applied Mathematical Sciences

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Asymptotic integration of singularly perturbed problems

Abstract: A formula for the lower bound on the deuteron D-state probability is derived. This bound is shown to be an improvement on that recently obtained by Klarsfeld, Martorell and Sprung.

Cited by 10 publications

References 39 publications

Operator analogs of VKB-type estimates and the solvability of boundary-value problems

Operator analogs of VKB-type estimates and the solvability of boundary-value problems

Regularized Solution of Singularly Perturbed Cauchy Problem in the Presence of Rational “Simple” Turning Point in Two-Dimensional Case

Direct Asymptotic Methods

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