Integral formulas of solutions of the basic linear differential equations of mathematical physics with variable factors

V. I. Gorbachev

Čebyševskij sbornik, Tome 18 (2017) no. 3, pp. 210-234.

Résumé

In paper initial-regional problems for linear differential equations are considered The mathematical physics (elliptic, hyperbolic and parabolic) with variables In the factors depending on coordinates and time. Such equations together with input datas we will be To name initial. The equations with variable factors describe processes in the composite Materials at which mechanical performances change or a saltus or it is continuous in Boundary region between phases. Many problems from various sections linear and nonlinear Mechanics are reduced to a solution of simple equations with variable factors. In case of periodic factors on coordinates one of popular modes of a solution of the equations The method of average of Bahvalova–Pobedri (MBP), based on representation of a solution is initial Problems in the form of an asimptotical series on degrees of the small geometrical parametre equal to the ratio Characteristic size of a mesh of periodicity to a characteristic size of a skew field. In this method the initial The boundary value problem is reduced to two recurrent sequences of problems. The first recurrent The sequence consists in determination of periodic solutions of auxiliary problems in a mesh Periodicity. The second sequence consists in a solution of initial-regional problems for the equation with In constant effective factors. These factors are after a solution on a mesh Periodicity of auxiliary problems. As base of a recursion in the second sequence in MBP serves Solution of a initial-regional problem for the equation with effective factors in definition range, Having the same form and it is exact with the same input datas, as an initial problem. Input datas in each of recurrent sequences on what or a pitch are only after that as the previous recurrent problems are solved all. In the present paper the new integral formulas are received, allowing to express a solution of the initial Problems for the equation with the variable factors depending on co-ordinates and time, through a solution The same problem for the equation with constant factors. The equation with constant factors Is called as the accompanying equations, and the problem according to accompanying a problem. In the kernel The integral formula the Green function and a difference of factors initial and accompanying enters The equations. By means of expansion of an accompanying solution in a many dimensional Taylor series from the integral Formulas equivalent representation of a solution of an initial problem in the form of a series on the various is received Derivative of a solution of an accompanying problem. Factors at derivatives are called as structural Functions. They are continuous functions of coordinates and time, converted in zero at Coincidence of initial and accompanying factors. For definition of structural functions it is constructed System of the recurrent equations. Through structural functions factors of the accompanying are defined The equations, coinciding in a periodic case with effective factors in MBP. Unlike Method of Bahvalova–Pobedri in the new approach it is necessary to solve one recurrent sequence of problems For determination of structural functions and once to solve a problem for a homogeneous skew field with the effective In performances.

Détail

Keywords: the equations of mathematical physics, integral formulas, average of differential equations, structural functions, effective factors.

Bibliographie

[1] Lomakin V. A., The theory of elasticity of nonhomogeneous bodies, Izdatel'stvo MGU imeni M. V. Lomonosova, M., 1976, 367 pp.

[2] Chapman S., Kauling T., The mathematical theory of non-uniform gases, University Press, Camdridge, 1952, 511 pp. | MR

[3] Bogolyubov N. N., Mitropolsky Ju. A., Asymptotic methods in the theory of nonlinear oscillations, Nauka, M., 1976, 412 pp.

[4] Marchenko V. A., Hruslov E. Ja., The regional problems in areas with fine-grained boundary, Naukova Dumka, Kiev, 1974, 416 pp.

[5] Berdichevsky V. L., Variational principles of a mechanics of a continuous medium, Nauka, M., 1983, 448 pp.

[6] Bahvalov N. C., Panasenko G. P., Averaging of processes in periodic Materials, Nauka, M., 1984, 352 pp.

[7] Pobedria B. E., Mechanics of composite materials, Izdatel'stvo MGU imeni M. V. Lomonosova, M., 1984, 336 pp.

[8] Sanchez-Palencia E., Non-Homogeneous Media and Vibration theory, Springer-Verlag, New York, 1980, 472 pp. | MR | Zbl

[9] Shkil N.I., Voronoi A.N., Lejfura V.N., Asymptotic methods in the differential And the integro-differential equations, Vishaya shkola, Kiev, 1985, 248 pp.

[10] Andrianov I. V,, Lesnichaja V. A., Manevich L. I., Method of an average in a statics and dynamics Ridge covers, Nauka, M., 1985, 221 pp.

[11] Oleynik O. A., Iosifjan G. A., Shamaev A. S., Mathematical problems of the theory it is strong Non-Homogeneous media, Izdatel'stvo MGU imeni M. V. Lomonosova, M., 1990, 311 pp.

[12] Kalamkarov A. L., Composite and reinforced elements of construction, John wiley Sons Ltd., Baffins Lane, Chechester, West Sussex PO19, England, 1992, 286 pp.

[13] Jikov V. V., Kozlov S. M., Oleynik O. A., Averaging of differential operators, Fizmatlit, M., 1993, 462 pp.

[14] Levinski T., Telega J. J., Plates, Laminates and shells. Asymptotic Analysis and Homogenization, World Scientific Publishing Co, N.J., 2000, 739 pp. | MR

[15] Bardzokas D. I., Zobnin A. I., Mathematical modeling of physical processes in Composite materials of periodic structure, Editorial URSS, M., 2003, 376 pp.

[16] Colpakov A. G., Composite materials and elements of structures with initial stress, Izdatel'stvo SO RAN, Novosibirsk, 2007, 254 pp.

[17] Bolshakov V. I., Andrianov I. V., Danishevsky V. V., Asymptotic methods of calculation of composite materials taking into account interior structure, Porogi, Dnepropetrovsk, 2008, 197 pp.

[18] Levin V. A., Lohin V. V., Zingerman K. M., “About one expedient of an estimation of the effective Performances of porous skew fields at final strains”, Izvestiya RAN. Mekhanika tverdogo tela, 1997, no. 4, 45–50

[19] Levin V. A., Lohin V. V., Zingerman K. M., “About construction of the effective defining Relations for porous elastic materials at final strains and their superposition”, Reports of the Russian Academy of Sciences, 382:4 (2002), 482–487

[20] Pobedria B. E., Numerical methods in the elasticity and plasticity theory, Izdatel'stvo MGU imeni M. V. Lomonosova, M., 1995, 366 pp.

[21] Zubchaninov V. G., Bases of the elasticity and plasticity theory, Vysshaya shkola, M., 1990, 368 pp.

[22] Morozov E. M., Levin V. A., Vershinin A. V., Strengths Analysis. FIDESIS in hands of the engineer, LENLAND, M., 2015, 408 pp.

[23] Arhipov G. I., Sadovnichy V. A., Chubarikov V. N., Lectures on the mathematical analysis, Visshaia Shkola, M., 1999, 695 pp.

[24] Gelfand I. M., Shilov G. E., Generalised functions and actions over them, Ed. 2, Fizmatgiz, M., 1959, 470 pp.

[25] Kech V., Teodoresku P., Introduction in the theory of generalised functions with applications in the technician, Mir, M., 1978, 518 pp.

[26] Sobolev S. L., The equations of mathematical physics, Nauka, M., 1966, 443 pp.

[27] Vladimirov V. S., Generalised functions in the mathematical physics, Nauka, M., 1976, 280 pp.

[28] Novatsky V., The elasticity theory, Mir, M., 1975, 872 pp.

[29] Gorbachev V. I., “Heat propagation in a nonuniform rod of variable cross section”, Moscow University Mechanics Bulletin, 72:2 (2017), 48–53 | DOI | Zbl

[30] Gorbachev V. I., “Integral formulas in electromagnetic elasticity of heterogeneous bodies. Application in the mechanics of composite materials”, Composites: Mechanics, Computations, Applications. An International J., 8:2 (2017), 147–170 | DOI

[31] Gorbachev V. I., “Natural frequencies of longitudinal oscillations for a nonuniform variable cross-section rod”, Moscow University Mechanics Bulletin, 71:1 (2016), 7–15 | DOI | Zbl

[32] Gorbachev V. I., “Integral formulae in the coupled problem of the thermoelasticity of an inhomogeneous body. application in the mechanics of composite materials”, Journal of Applied Mathematics and Mechanics, 78:2 (2014), 192–208 | DOI | MR

[33] Gorbachev V. I., Moskalenko O. B., “Stability of bars with variable rigidity compressed by a distributed force”, Moscow University Mechanics Bulletin, 67:1 (2012), 5–10 | DOI | MR | Zbl

[34] Gorbachev V. I., “Dynamic problems of composite mechanics”, Bulletin of the Russian Academy of Sciences: Physics, 75:1 (2011), 110–115 | DOI | Zbl

[35] Gorbachev V. I., “About oscillations in an inhomogeneous elastic body”, Elasticity and an unelasticity, Materials of the international scientific symposium on problems of the mechanics of deformable bodies, devoted to the 100 anniversary from the date of A.A. Ilyushin's birth (Moscow, on January, 20–21th, 2011), Izdatel'stvo MGU imeni M. V. Lomonosova, M., 2011, 319–326

[36] Gorbachev V. I., “Integral formulas in symmetric and asymmetric elasticity”, Moscow University Mechanics Bulletin, 64:6 (2009), 148–151 | DOI | MR

[37] Arkhangel'skyi A.F., Gorbachev V.I., “Effective characteristics of corrugated plates”, Mechanics of Solids, 42:3 (2007), 447–462 | DOI

[38] Gorbachev V. I., “Method of tensors of Green for a solution of boundary value problems of the theory of elasticity of inhomogeneous media”, The Computing mechanics of a deformable rigid body, 1991, no. 2, 61–76

[39] Brebbia C., Telles J., Vroubel L., Boundary elements techniques, Springer-Verlag, Berlin–Heidelberg–New York–Tokyo, 1984, 526 pp. | MR

[40] Olekhova L. V., Torsion of an inhomogeneous non-isotropic rod, A thesis The candidate of physical and mathematical sciences. Master's thesis, The Moscow State University of M.V. Lomonosova, Mehaniko-mathematical Faculty, 2009

[41] Yemelyanov A. N., Effective performances in moment elasticity theories, A thesis The candidate of physical and mathematical sciences. Master's thesis, The Moscow State University of M.V. Lomonosova, Mehaniko-mathematical Faculty, 2016