Geodesic
A singular integral equation with small parameter on a finite interval
Sbornik. Mathematics, Tome 34 (1978) no. 4, pp. 475-502 
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/
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     title = {A~singular integral equation with small parameter on a~finite interval},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[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