Abstract
In this paper, we examine the basic boundary-value problems for a three-dimensional space filled with a solid body having double porosity and two nonintersecting spherical cavities. We search for a solution of the problem by using the representation of a general solution of a system of homogeneous differential equations of statics, which is expressed in terms of four harmonic and one metaharmonic functions. The solution of the problem is reduced to the investigation of an infinite system of linear algebraic equations. It is shown that the obtained system is of normal type. Solutions of the considered problems are obtained in the form of absolutely and uniformly convergent series.
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References
D. E. Beskos and E. C. Aifantis, “On the theory of consolidation with double porosity, II,” Int. J. Eng. Sci., 24, No. 11, 1697–1716 (1986).
L. Giorgashvili, “Solution of the basic boundary-value problems of stationary thermoelastic oscillations for domains bounded by spherical surfaces,” Georgian Math. J., 4, No. 5, 421–438 (1997).
L. Giorgashvili, G. Karseladze, and G. Sadunishvili, “Solution of a boundary-value problem of statics of two-component elastic mixtures for a space with two nonintersecting spherical cavities,” Mem. Differ. Equ. Math. Phys., 45, 85–115 (2008).
L. Giorgashvili and K. Skhvitaridze, “Motion of viscous incompressible fluid in infinite space bounded by two sphere surfaces,” in: Collection of Articles of the Faculty of Computational Mathematics and Cybernetics, Moscow State Univ., Moscow (2002), pp. 14–28.
A. N. Gruz’ and V. T. Golovchan, Diffraction of Elastic Waves in Multi-Connected Bodies [in Russian], Naukova Dumka, Kiev (1972).
E. A. Ivanov, Diffraction of Electromagnetic Waves on Two bodies [in Russian], Nauka i Mekhanika, Minsk (1968).
L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis [in Russian], Gos. Izd. Fiz.-Mat. Lit., Moscow–Leningrad (1962).
M. Y. Khaled, D. E. Beskos, and E. C. Aifantis, “On the theory of consolidation with double porosity, III. A finite-element formulation,” Int. J. Numer. Anal. Methods Geomech., 8, 101–123 (1984).
V. D. Kupradze, T. G. Gegelia, M. O. Basheleishvili, and T. V. Burchuladze, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity [in Russian], Nauka, Moscow (1976).
S. G. Mikhlin, Higher-Dimensional Singular Integrals and Integral Equations [in Russian], Gos. Izd. Fiz.-Mat. Lit., Moscow (1962).
P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York–Toronto–London (1953).
A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1966).
A. F. Ulitko, The Method of Eigenvector Functions in Three-Dimensional Problems of the Theory of Elasticity [in Russian], Naukova Dumka, Kiev (1979).
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 89, Differential Equations and Mathematical Physics, 2013.
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Giorgashvili, L., Karseladze, G. & Kharashvili, M. Basic Boundary-Value Problems for a Solid Body with Double Porosity and Two Nonintersecting Spherical Cavities. J Math Sci 206, 371–392 (2015). https://doi.org/10.1007/s10958-015-2318-4
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DOI: https://doi.org/10.1007/s10958-015-2318-4