A~singular integral equation with small parameter on a~finite interval

V. Yu. Novokshenov

Sbornik. Mathematics, Tome 34 (1978) no. 4, pp. 475-502

Résumé

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.