Geodesic
Mean value theorem for a~system of differential equations for the stress tensor and pore pressure
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 7 (2014) no. 1, pp. 132-138.

Voir la notice de l'article provenant de la source Math-Net.Ru

A system of second-order differential equations for the stress tensor and pore pressure for the poroelasticity statics in the absence of mass forces and energy dissipation is obtained. The stress tensor is shown to be a biharmonic function. Integral mean value relations in explicit form for the obtained systems of differential equations are found.
Keywords: differential equation, poroelasticity statics, stress tensor. 
@article{JSFU_2014_7_1_a13,
     author = {Nasridin M. Zhabborov and Kholmatzhon Kh. Imomnazarov},
     title = {Mean value theorem for a~system of differential equations for the stress tensor and pore pressure},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {132--138},
     publisher = {mathdoc},
     volume = {7},
     number = {1},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2014_7_1_a13/}
}
TY  - JOUR
AU  - Nasridin M. Zhabborov
AU  - Kholmatzhon Kh. Imomnazarov
TI  - Mean value theorem for a~system of differential equations for the stress tensor and pore pressure
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2014
SP  - 132
EP  - 138
VL  - 7
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2014_7_1_a13/
LA  - en
ID  - JSFU_2014_7_1_a13
ER  - 
%0 Journal Article
%A Nasridin M. Zhabborov
%A Kholmatzhon Kh. Imomnazarov
%T Mean value theorem for a~system of differential equations for the stress tensor and pore pressure
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2014
%P 132-138
%V 7
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2014_7_1_a13/
%G en
%F JSFU_2014_7_1_a13
Nasridin M. Zhabborov; Kholmatzhon Kh. Imomnazarov. Mean value theorem for a~system of differential equations for the stress tensor and pore pressure. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 7 (2014) no. 1, pp. 132-138. http://geodesic.mathdoc.fr/item/JSFU_2014_7_1_a13/

[1] I. G. Dyad'kin, V. N. Starikov, “A possible way of economizing computer time in the solution of the Laplace equation by the Monte Carlo method”, Zh. vych. mat. i mat. fiziki, 5:5 (1965), 936–938 (in Russian) | MR

[2] B. S. Elepov et al., Solution of boundary value problems by the Monte-Carlo method, Nauka, Novosibirsk, 1980 (in Russian) | Zbl

[3] I. M. Sobol', Monte Carlo numerical methods, Nauka, Moscow, 1973 (in Russian) | MR

[4] K. K. Sabelfeld, Monte Carlo methods: in boundary value problems, Nauka, Novosibirsk, 1989 (in Russian) | MR

[5] R. Courant, Solving boundary value problems by the Monte Carlo method, Nauka, Novosibirsk, 1980 (in Russian)

[6] G. Aquaro, “Un teorema di media per le equaziono dell'elasticita”, Riv. Mat. Univ. Parma, 1 (1950), 419–424 | MR | Zbl

[7] J. L. Synge, “Upper and lower bounds for solutions of problems in elasticity”, Proc. Roy. Irish Acad., 53 (1950), 41–64 | MR | Zbl

[8] S. M. Ermakov, V. V. Nekrutkin, A. S. Sipin, Random processes for classical equations of mathematical physics, Nauka, Moscow, 1984 (in Russian) | MR | Zbl

[9] K. K. Sabelfeld, I. A. Shalimova, Spherical means for PDEs, VSP, Utrecht, 1997 | MR | Zbl

[10] J. B. Diaz, L. E. Payne, “On a mean value theorem, and its converse, for the displacements in the theory of elasticity”, Portugaliae Mathematica, 17:4 (1958), 123–126 | MR | Zbl

[11] J. H. Bramble, L. E. Payne, “Some converses of mean value theorems in the theory of elasticity”, J. Math. Anal. Appl., 10:3 (1965), 553–567 | DOI | MR | Zbl

[12] A. V. Bitsadze, Some classes of partial differential equations, Nauka, Moscow, 1981 (in Russian) | MR | Zbl

[13] V. V. Naumov, “Mean value theorems for the equation of harmonic oscillations of an elastic body”, Fluid dynamics, Coll. sci. proc. USSR Acad. sci. Sib. Branch, Institute of Hydrodynamics, 82, 1987, 147–153 | MR | Zbl

[14] Yu. M. Grigor'ev, “Mean value theorems for heterogeneous Helmholtz and Lame equations”, Fluid dynamics, Coll. sci. proc. USSR Acad. sci., SB RAS, Institute of Hydrodynamics, 113, 1988, 53–59 | MR

[15] Yu. M. Grigor'ev, V. V. Naumov, “On mean value theorems for the Helmholtz and Lame equations”, Dokl. Ross. akad. nauk, 362:1 (1998), 51–52 (in Russian) | MR

[16] E. V. Grachev, N. M. Zhabborov, Kh. Kh. Imomnazarov, “A concentrated force in an elastic porous half-space”, Dokl. Ross. akad. nauk, 391:3 (2003), 331–333 (in Russian)

[17] E. Grachev, Kh. Imomnazarov, N. Zhabborov, “One nonclassical problem for the statics equations of elastic-deformed porous media in a half-plane”, Applied Mathematics Letters, 17:1 (2004), 31–34 | DOI | MR | Zbl

[18] N. M. Zhabborov, Kh. Kh. Imomnazarov, Some initial boundary value problems of mechanics of two-velocity media, Tashkent, 2012 (in Russian)

[19] G. Bonnet, “Basic singular solutions for a poroelastic medium in the dynamic range”, J. Acoust. Soc. Am., 82 (1987), 1758–1762 | DOI

[20] S. Gorog, R. Panneton, “Atalla N, Mixed displacement-pressure formulation for acoustic anisotropic open porous media”, J. Appl. Phys., 82 (1997), 4192–4196 | DOI

[21] V. N. Dorovsky, “Continual theory of filtration”, Geologiya i Geofisika, 1989, no. 7, 39–45 (in Russian)

[22] V. N. Dorovsky, Yu. V. Perepechko, E. I. Romensky, “Wave processes in saturated porous elastically deformed media”, Combustion, Explosion and Shock Waves, 1993, no. 1, 100–111

[23] A. M. Blokhin, V. N. Dorovsky, Mathematical modelling in the theory of multivelocity continuum, Nova Science Publishers Inc., 1995 | MR

[24] B. G. Mikhailenko, B. M. Glinsky, Kh. Kh. Imomnazarov, V. V. Kovalevsky, L. E. Sobisevich, A. L. Sobisevich, M. S. Khairetdinov, “Active seismology methods in problems of Earth's deep structure monitoring”, Extreme Natural Phenomena and Catastrophes, v. 1, Estimation and Ways to Decrease Negative Consequences of Extreme Natural Phenomena, ed. A. O. Gliko, A. L. Sobisevich, Institute of Physics of the Earth RAS, Moscow, 2010, 89–130