Abstract
A method of solving paired integral equations that appear in considerations of mixed problems of elasticity and thermoelasticity theory is given, with the help of generalized integral Weber transforms. The paired equations are reduced to an integral Fredholm equation of the second kind on the semiaxis, which have a discontinuous kernel, or to Fredholm equations of the second kind on a finite interval and infinite systems of linear algebraic equations, which are normal in the sense of Poincare-Koch. As an example, contact problems for an inhomogeneous fiber with a cavity are considered. If the fiber is bonded with the elastic half-space, then a second appproach is realized, which is based on a reduction to an equation with a self-adjoint operator, for which some method of sequential iteractions and the Bubnov-Galerkin method are justified.
Similar content being viewed by others
References
R. Bellman and K. Kuk, Differential-Difference Equations [Russian translation], Mir, Moscow (1967).
P. P. Zabreiko, Integral Equations [in Russian], Nauka, Moscow (1968).
L. V. Kantorovich, “Functional analysis and applied mathematics,” Uspekhi Mat. Nauk,3, No. 6, 89–185 (1948).
M. A. Krasnosel'skii, G. M. Vainikko, P. P. Zabreiko, et al., An Approximation Solution of Operator Equations [in Russian], Nauka, Moscow (1969).
P. Ya. Malits, On some decompositions of an arbitrary function into an integral of cylindrical functions and its application to elasticity theory. Stability and Strength of Constructions. Dnepropetrovsk: B.i., 93–99 (1973).
P. Ya. Malits, “A contact problem for a semispace with a reinforced cavity,” Dinam. Systemy,3, 39–44 (1984).
P. Ya. Malits and A. K. Privarnikov, An application of a transform of the Weber type to solution of problems of the elasticity theory for stratified media with cylindrical openings, Strength and Plasticity Problems, Dnepropetrovsk, B.I., 124–136 (1971).
F. Riesz and B. Szökefalvi-Nagy, Functional Analysis, Ungar, New York (1955).
R. P. Srivastav, “A pair of dual integral equations involving Bessel function of the first and the second kind,” Proc. Edinb. Math. Soc.14, No. 2, 25–36 (1964).
Author information
Authors and Affiliations
Additional information
Translated from Dinamicheskie Sistemy, No. 7, pp. 95–102, 1988.
Rights and permissions
About this article
Cite this article
Malits, P.Y. On the solution of mixed problems for a fiber with a vertical circular cylindrical cavity. J Math Sci 65, 1564–1569 (1993). https://doi.org/10.1007/BF01097665
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01097665