We establish necessary and sufficient conditions for the solvability of the linear boundary-value problem for a weakly singular integral equation and find the general form of the solution to this problem.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. L. Auer and C. S. Gardner, “Note on singular integral equations of the Kirkwood–Riseman type,” J. Chem. Phys.,23, 1545–1546 (1955).
C. Constanda and S. Potapenko, Integral Methods in Science and Engineering: Techniques and Applications, Birkh¨auser, Boston (2008).
É. Goursat, Cours D’Analyse Mathématique [Russian translation], Vol. 3, Part 2, GTTI Moscow (1934).
S. G. Mikhlin, Lectures on Linear Integral Equations [in Russian], GIFML, Moscow (1959).
F. G. Tricomi, Integral Equations [Russian translation], Inostr. Lit., Moscow (1960).
V. I. Smirnov, A Course of Higher Mathematics [in Russian], Vol. 4, Part 1, Nauka, Moscow (1974).
G. Vainikko and A. Pedas, “The properties of solutions of weakly singular integral equations,” ANZIAM J.,22, 419–430 (1981).
I. G. Graham, “Galerkin methods for second kind integral equations with singularities,” Math. Comp.,39, No. 160, 519–533 (1982).
C. Huang, T. Tang, and Z. Zhang, “Supergeometric convergence of spectral collocation methods for weakly singular Volterra and Fredholm integral equations with smooth solutions,” J. Comput. Math.,29, No. 6, 698–719 (2011).
G. R. Richter, “On weakly singular Fredholm integral equations with displacement kernels,” J. Math. Anal. Appl.,55, No. 1, 32–42 (1976).
E. A. Galperin, E. J. Kansa, A. Makroglou, and S. A. Nelson, “Variable transformations in the numerical solution of second kind Volterra integral equations with continuous and weakly singular kernels; extensions to Fredholm integral equations,” J. Comput. Appl. Math.,115, No. 1-2, 193–211 (2000).
Y. Cao, M. Huang, L. Liu, and Y. Xu, “Hybrid collocation methods for Fredholm integral equations with weakly singular kernels,” Appl. Numer. Math.,57, No. 5-7, 549–561 (2007).
A. Amosov, M. Ahues, and A. Largillier, “Superconvergence of some projection approximations for weakly singular integral equations using general grids,” SIAM J. Numer. Anal.,47, No. 1, 646–674 (2009).
O. Gonzalez and J. Li, “A convergence theorem for a class of Nystrom methods for weakly singular integral equations on surfaces in R3;” Math. Comp.,84, No. 292, 675–714 (2015).
N. I. Muskhelishvili, Singular Integral Equations [in Russian], Nauka, Moscow (1968).
N. O. Kozlova and V. A. Feruk, “Noetherian boundary-value problems for integral equations,” Nelin. Kolyv.,19, No. 1, 58–66 (2016); English translation:J. Math. Sci.,222, No. 3, 266–275 (2017).
O. A. Boichuk, N. O. Kozlova, and V. A. Feruk, “Weakly perturbed integral equations,” Nelin. Kolyv.,19, No. 2, 151–160 (2016); English translation: J.Math. Sci.,223, No. 3, 199–209 (2017).
A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, de Gruyter, Berlin (2004); 2nd edn. (2016).
W. R. Smythe, Static and Dynamic Electricity [Russian translation], Inostr. Lit., Moscow (1954).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Neliniini Kolyvannya, Vol. 22, No. 1, pp. 27–35, January–March, 2019.
Rights and permissions
About this article
Cite this article
Boichuk, O.A., Feruk, V.A. Linear Boundary-Value Problems for Weakly Singular Integral Equations. J Math Sci 247, 248–257 (2020). https://doi.org/10.1007/s10958-020-04800-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-04800-6