Abstract
We justify the direct projection method for solving an integral equation with a logarithmic singularity in the kernel. The equation is treated as a mapping of one Hilbert space into another Hilbert space. The spaces are chosen from conditions ensuring the solution of a broad class of mathematical modeling problems with the use of a simple layer potential. The idea of the projection method is to choose finite-dimensional subspaces into which the exact solution and the right-hand side of the equation are projected. In this case, the problem of finding an approximate solution does not require computing the convolution of kernels. We prove an estimate for the solution error in the norm of the original operator equation.
This is a preview of subscription content, log in via an institution to check access.
References
Hönl, H., Maue, A., and Westphahl, K., Theorie der Beugung, in Flügge, S., Handbuch der Physik, Bd. XXV/1: Krystalloptik-Beugung, Berlin: Springer, 1961. Translated under the title Teoriya difraktsii, Moscow: Mir, 1964.
Zakharov, E.V. and Pimenov, Yu.V., Chislennyi analiz difraktsii radiovoln (Numerical Analysis of the Diffraction of Radio Waves), Moscow: Radio i Svyaz’, 1982.
Guseinov, E.A. and Il’inskii, A.S., Integral Equations of the First Kind with a Logarithmic Singularity in the Kernel and Their Application in Problems of Diffraction by Thin Screens, Zh. Vychisl. Mat. Mat. Fiz., 1987, vol. 27, no. 7, pp. 1050–1057.
Vorovich, I.I., Aleksandrov, V.M., and Babeshko, V.A., Neklassicheskie smeshannye zadachi teorii uprugosti (Nonclassical Mixed Problems of Elasticity Theory), Moscow: Nauka, 1974.
Khapaev, M.M., Jr., Numerical Inversion of Some Integral Operators of the First Kind, Differ. Uravn., 1981, vol. 17, no. 7, pp. 1328–1338.
Shtaerman, I.Ya., Kontaktnaya zadacha teorii uprugosti (Contact Problem of the Theory of Elasticity), Moscow, 1949.
Il’inskii, A.S. and Smirnov, Yu.G., Investigation of Mathematical Models of Microband Lines, in Metody matematicheskogo modelirovaniya, avtomatizatsiya obrabotki nablyudenii i ikh primeneniya (Mathematical Modeling Methods, Automation of the Analysis of Observations, and Their Applications), Moscow: Moskov. Gos. Univ., 1986, pp. 175–198.
Il’inskii, A.S. and Smirnov, Yu.G., Mathematical Modeling of the Process of Propagation of Electromagnetic Oscillations in a Slot Transmission Line, Zh. Vychisl. Mat. Mat. Fiz., 1987, vol. 27, no. 2, pp. 252–261.
Dmitriev, V.I. and Zakharov, V.E., The Numerical Solution of Certain Fredholm Integral Equations of First Kind, in Vychislitel’nye metody i programmirovanie (Computing Methods and Programming), Moscow: Moskov. Gos. Univ., 1968, pp. 49–54.
Kress, R., Linear Integral Equations, Appl. Math. Sci., vol. 82, New York, 1989.
Krasnosel’skii, M.A., Vainikko, G.N., and Zabreiko, P.P., Priblizhennye resheniya operatornykh uravnenii (Approximate Solution of Operator Equations), Moscow: Nauka, 1969.
Timan, A.F., Teoriya priblizheniya funktsii deistvitel’nogo peremennogo (Theory of Approximation of Functions of a Real Variable), Moscow: Gosudarstv. Izdat. Fiz.-Mat. Lit., 1960.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.S. Il’inskii, 2014, published in Differentsial’nye Uravneniya, 2014, Vol. 50, No. 9, pp. 1279–1283.
Rights and permissions
About this article
Cite this article
Il’inskii, A.S. Justification of the direct projection method for solving an integral equation of the potential theory. Diff Equat 50, 1267–1271 (2014). https://doi.org/10.1134/S0012266114090146
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266114090146