×

An orthonormal basis in Sobolev-Slobodetskii spaces on an interval. (English. Russian original) Zbl 1087.45500

Differ. Equ. 41, No. 4, 594-597 (2005); translation from Differ. Uravn. 41, No. 4, 558-560 (2005).
Introduction: Numerous diffraction and elasticity problems can be reduced to the solution of the following integral equation with a logarithmic singularity in the kernel: \[ {1\over\pi}\int^1_{-1} u(t)\ln{1\over |t-\tau|} dt+ \int^1_{-1} u(t) M(t,\tau)\,dt= e(\tau),\quad -1\leq \tau\leq 1.\tag{1} \] There are quite a few papers dealing with the numerical solution of this equation. By expanding the unknown function as \(u(\tau)= \sum^{+\infty}_{n=1} c_n\varphi_n(\tau)\) in the function system \[ \varphi_n(\tau)= \begin{cases} {1\over\sqrt{\pi\ln 2}}{1\over \sqrt{1-\tau_2}}\quad &\text{for }n= 1,\\ \sqrt{{2(n-1)\over \pi}} {\cos[(n-1)\arccos(\tau)]\over \sqrt{1-\tau^2}}\quad &\text{for }n> 1,\end{cases} \] one can reduce the integral equation to an infinite system of the form \[ c_n+ \sum^{+\infty}_{m=1} c_m M_{mn}= c_n,\quad 1\leq n<+\infty.\tag{2} \] Under certain conditions on the kernel \(M(t,\tau)\), this is a Fredholm system of the second kind, and the approximate solution found by the truncation method converges to the exact solution. Likewise, the integro-differential equation \[ {1\over\pi} {\partial\over\partial\tau} \int^1_{-1} u(t){\partial\over\partial t}\ln{1\over |t-\tau|} dt+ \int^1_{-1} u(t) N(t,\tau)\,dt= e(\tau),\quad -1\leq \tau\leq 1,\tag{3} \] can be solved with the use of an expansion in the functions \[ \varphi_n(\tau)= \sqrt{{2\over\pi n}} \sin[n\arccos(\tau)],\quad n= 1,2,\dots\;. \] (1) and (3) can be included in the family of equations with a parameter.

MSC:

45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45J05 Integro-ordinary differential equations
45L05 Theoretical approximation of solutions to integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Vorovich, I.I., Aleksandrov, V.M., and Babeshko, V.A., Neklassicheskie smeshannye zadachi teorii uprugosti (Nonclassical Mixed Problems of Elasticity), Moscow, 1974.
[2] Nefedov, E.I., Radtsig, Yu.Yu., and Eminov, S.I., Doklady RAN, 1995, vol. 345, no.2, pp. 186–187.
[3] Eminov, S.I., Radiotekhnika i Elektronika, 1993, vol. 38, no.12, pp. 2160–2168.
[4] Lions, J.-L. and Magenes, E., Problemes aux limites non homogenes et applications, Paris: Dunod, 1968. Translated under the title Neodnorodnye granichnye zadachi i ikh prilozheniya, Moscow: Mir, 1971.
[5] Prudnikov, A.P., Brychkov, Yu.A., and Marichev, O.N., Integraly, ryady. Spetsial’nye funktsii (Integrals, Series. Special Functions), Moscow, 1983. · Zbl 0626.00033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.