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This article is cited in 6 scientific papers (total in 6 papers)
A singular integral equation with small parameter on a finite interval
V. Yu. Novokshenov
Abstract:
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.
Received: 15.02.1977
Citation:
V. Yu. Novokshenov, “A singular integral equation with small parameter on a finite interval”, Math. USSR-Sb., 34:4 (1978), 475–502
Linking options:
https://www.mathnet.ru/eng/sm2540https://doi.org/10.1070/SM1978v034n04ABEH001222 https://www.mathnet.ru/eng/sm/v147/i4/p543
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Abstract page: | 412 | Russian version PDF: | 124 | English version PDF: | 23 | References: | 65 |
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