Geodesic
A singular integral equation with small parameter on a finite interval
Sbornik. Mathematics, Tome 34 (1978) no. 4, pp. 475-502 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The asymptotic properties of the following singular integral equation are investigated in the paper: \begin{equation} \int_0^1\biggl[\frac1{x-t}+a(x-t,\varepsilon)\biggr]u_\varepsilon(t)\,dt =f(t), \end{equation} where $\varepsilon>0$ is a small parameter and $f(x)\in C^\infty[0,1]$. Equation (1) is regarded as a boundary value problem for a one-dimensional elliptic pseudodifferential operator wtih piecewise smooth symbol. A typical example of the symbol is the function $\widetilde a(\lambda,\varepsilon)=\pi i\operatorname{sign}\lambda[1+e^{-\varepsilon|\lambda|}]$, which corresponds to an equation in the theory of dislocations. The asymptotic expansion of the solution of equation (1) contains functions of boundary layer type that depend on the variables $\xi=\frac x\varepsilon$ and $\eta=\frac{1-x}\varepsilon$ and decrease powerlike at infinity. The matching of the boundary layer expansion with the exterior expansion (in the variable $x$) is carried out by means of a special two-scaled representation of the integrals of form (1), in which the function $u_\varepsilon(x)$ is replaced by its asymptotic series. Bibliography: 10 titles. 
@article{SM_1978_34_4_a4,
     author = {V. Yu. Novokshenov},
     title = {A~singular integral equation with small parameter on a~finite interval},
     journal = {Sbornik. Mathematics},
     pages = {475--502},
     year = {1978},
     volume = {34},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1978_34_4_a4/}
}
TY  - JOUR
AU  - V. Yu. Novokshenov
TI  - A singular integral equation with small parameter on a finite interval
JO  - Sbornik. Mathematics
PY  - 1978
SP  - 475
EP  - 502
VL  - 34
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_1978_34_4_a4/
LA  - en
ID  - SM_1978_34_4_a4
ER  - 
%0 Journal Article
%A V. Yu. Novokshenov
%T A singular integral equation with small parameter on a finite interval
%J Sbornik. Mathematics
%D 1978
%P 475-502
%V 34
%N 4
%U http://geodesic.mathdoc.fr/item/SM_1978_34_4_a4/
%G en
%F SM_1978_34_4_a4
V. Yu. Novokshenov. A singular integral equation with small parameter on a finite interval. Sbornik. Mathematics, Tome 34 (1978) no. 4, pp. 475-502. http://geodesic.mathdoc.fr/item/SM_1978_34_4_a4/

[1] A. S. Demidov, “Asimptotika resheniya kraevoi zadachi dlya ellipticheskogo psevdodifferentsialnogo uravneniya s malym parametrom pri starshem operatore”, Trudy Mosk. matem. ob-va, XXXII (1975), 119–147 | MR

[2] G. I. Eskin, “Asimptotika reshenii ellipticheskikh psevdodifferentsialnykh uravnenii s malym parametrom”, DAN SSSR, 211:3 (1973), 547–550 | MR | Zbl

[3] Dzh. Khirt, I. Lote, Teoriya dislokatsii, “Atomizdat”, Moskva, 1972

[4] V. Yu. Novokshenov, “Asimptotika resheniya singulyarnogo integralnogo uravneniya s malym parametrom”, Matem. sb., 100(142) (1976), 455–475 | Zbl

[5] M. I. Vishik, G. I. Eskin, “Uravneniya v svertkakh v ogranichennykh oblastyakh”, Uspekhi matem. nauk, XX:3(123) (1965), 89–151

[6] N. I. Muskhelishvili, Singulyarnye integralnye uravneniya, izd-vo “Nauka”, Moskva, 1968 | MR

[7] E. Ya. Riekstynsh, Asimptoticheskie razlozheniya integralov, izd-vo “Zinatne”, Riga, 1974

[8] A. N. Panchenkov, Teoriya potentsiala uskorenii, izd-vo “Nauka”, Novosibirsk, 1975

[9] A. N. Tikhonov, A. A. Samarskii, “Asimptoticheskoe razlozhenie integralov s medlenno ubyvayuschim yadrom”, DAN SSSR, 126:1 (1959), 26–29 | MR | Zbl

[10] A. N. Tikhonov, A. A. Samarskii, A. A. Arsenev, “Ob odnom metode asimptoticheskikh otsenok integralov”, ZhVM i MF, 12:4 (1972), 1005–1012 | MR | Zbl