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DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

Eigenfunction of the Laplace operator in +1-dimentional simplex

Вестник МГСУ 11/2014
  • Ovchintsev Mikhail Petrovich - Moscow State University of Civil Engineering (MGSU) Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Higher Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; Этот e-mail адрес защищен от спам-ботов, для его просмотра у Вас должен быть включен Javascript .
  • Sitnikova Elena Georgievna - Moscow State University of Civil Engineering (MGSU) Candidate of Physical and Mathematical Sciences, Professor, Department of Higher Mathematics, Moscow State University of Civil Engineering (MGSU), .

Страницы 68-73

In order to find eigenfunction of the Laplace operator in regular
n+1-dimensional simplex the barycentric coordinates are used. For obtaining this result we need some formulas of the analytical geometry. A similar result was obtained in the earlier papers of the author in a tetrahedron from
R
3 and in gipertetrahedron from
R
4. Let П be unlimited cylinder in the space
R
n, its cross-section with hyperplane has a special form. Let
L be a second order linear differential operator in divergence form, which is uniformly elliptic and η is its ellipticity constant. Let
u be a solution of the mixed boundary value problem in Π with homogeneous Dirichlet and Neumann data on the boundary of the cylinder. In some cases the eigenfunction of the Laplace operator allows us to continue this solution from the cylinder Π to the whole space
R
n with the same ellipticity constant. The obtained result allows us to get a number of various theorems on the solution growth for mixed boundary value problem for linear differential uniformly elliptical equation of the second order, given in unlimited cylinder with special cross-section. In addition we consider
n-1-dimensional hill tetrahedron and the eigenfunction for an elliptic operator with constant coefficients in it.

DOI: 10.22227/1997-0935.2014.11.68-73

Библиографический список
  1. Sitnikova E.G. Sobstvennaya funktsiya operatora Laplasa v gipertetraedre [Eigenfunction of the Laplace Operator in the Tetrahedron]. Integratsiya, partnerstvo i innovatsii v stroitel’noy nauke i obrazovanii : sbornik trudov Mezhdunaridnoy nauchnoy konferentsii [Integration, Partnership and Innovations in Construction Science and Education : Collection of Works of International Scientific Conference]. Moscow, MGSU, 2011, pp. 755—758. (In Russian).
  2. Sitnikova E.G. Neskol’ko teorem tipa Fragmena-Lindelefa dlya ellipticheskogo uravneniya vtorogo poryadka [Several Theorems of Phragmen-Lindelof Type for the Second Order Differential Equation]. Voprosy matematiki i mekhaniki sploshnykh sred : sbornik nauchnykh trudov [Problems of Continuum Mathematics and Mechanics: Collection of Works]. Moscow, MGSU Publ., 1984, pp. 98—104. (In Russian).
  3. Sitnikova E.G. Sobstvennaya funktsiya operatora Laplasa v tetraedre [Eigenfunction of the Laplace Operator in the Tetrahedron]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2011, no. 4, pp. 80—82. (In Russian).
  4. Mikhaylov V.P. Differentsial’nye uravneniya v chastnykh proizvodnykh [Differential Equations in Partial Derivatives]. Moscow, Nauka Publ., 1976, 391 p. (In Russian).
  5. Mikhlin S.G. Kurs matematicheskoy fiziki [Course in Mathematical Physics]. Moscow, Nauka Publ., 1968, 576 p. (In Russian).
  6. Lazutkin V.F. Ob asimptotike sobstvennykh funktsiy operatora Laplasa [On Asymptotics of Eigenfunctions of the Laplace Operator]. Doklady AN SSSR [Reports of the Academy of Sciences of the USSR]. 1971, vol. 200, no. 6, pp. 1277—1279. (In Russian).
  7. Lazutkin V.F. Sobstvennye funktsii s zadannoy kaustikoy [Eigenfunctions with Preassigned Caustic Curve]. Zhurnal vychislitel’noy matematiki i matematicheskoy fiziki [Computational Mathematics and Mathematical Physics]. 1970, vol. 10, no. 2, pp. 352—373. (In Russian).
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Boundary value problemfor multidimensional fractional advection-dispersion equation

Вестник МГСУ 5/2015
  • Khasambiev Mokhammad Vakhaevich - Moscow State University of Civil Engineering (MGSU) postgraduate student, Department of Higher Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; Этот e-mail адрес защищен от спам-ботов, для его просмотра у Вас должен быть включен Javascript .

Страницы 35-43

In recent time there is a very great interest in the study of differential equations of fractional order, in which the unknown function is under the symbol of fractional derivative. It is due to the development of the theory of fractional integro-differential theory and application of it in different fields.The fractional integrals and derivatives of fractional integro-differential equations are widely used in modern investigations of theoretical physics, mechanics, and applied mathematics. The fractional calculus is a very powerful tool for describing physical systems, which have a memory and are non-local. Many processes in complex systems have nonlocality and long-time memory. Fractional integral operators and fractional differential operators allow describing some of these properties. The use of the fractional calculus will be helpful for obtaining the dynamical models, in which integro-differential operators describe power long-time memory by time and coordinates, and three-dimensional nonlocality for complex medium and processes.Differential equations of fractional order appear when we use fractal conception in physics of the condensed medium. The transfer, described by the operator with fractional derivatives at a long distance from the sources, leads to other behavior of relatively small concentrations as compared with classic diffusion. This fact redefines the existing ideas about safety, based on the ideas on exponential velocity of damping. Fractional calculus in the fractal theory and the systems with memory have the same importance as the classic analysis in mechanics of continuous medium.In recent years, the application of fractional derivatives for describing and studying the physical processes of stochastic transfer is very popular too. Many problems of filtration of liquids in fractal (high porous) medium lead to the need to study boundary value problems for partial differential equations in fractional order.In this paper the authors first considered the boundary value problem for stationary equation for mass transfer in super-diffusion conditions and abnormal advection. Then the solution of the problem is explicitly given. The solution is obtained by the Fourier’s method.The obtained results will be useful in liquid filtration theory in fractal medium and for modeling the temperature variations in the heated bar.

DOI: 10.22227/1997-0935.2015.5.35-43

Библиографический список
  1. Nakhushev A.M. Drobnoe ischislenie i ego primenenie [Fractional Calculation and its Application]. Moscow, Fizmatlit Publ., 2003, 272 p. (In Russian)
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  3. Aleroev T.S., Kirane M., Malik S.A. Determination of a Source Term for a Time Fractional Diffusion Equation with an Integral Type Over-Determining Condition. Electronic Journal of Differential Equations. 2013, vol. 2013, no. 270, pp. 1—16.
  4. Al-Refai M., Luchko Y. Maximum Principle for the Multi-Term Time-Fractional Diffusion Equations with the Riemann-Liouville Fractional Derivatives. Applied Mathematics and Computation. April 2015, vol. 257, no. 15, pp. 40—51. DOI: http://dx.doi.org/10.2478/s13540-014-0181-5.
  5. Zhao K., Gong P. Existence of Positive Solutions for a Class of Higher-Order Caputo Fractional Differential Equation. Qualitative Theory of Dynamical Systems. April 2015, vol. 14, no. 1, pp. 157—171. DOI: http://dx.doi.org/10.1007/s12346-014-0121-0.
  6. Chen T., Liu W., Liu J. Solvability of Periodic Boundary Value Problem for Fractional p-Laplacian Equation. Applied Mathematics and Computation. 1 October 2014, vol. 244, pp. 422—431.
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Continuation of the solution of an elliptic equation and mathematical tesselations

Вестник МГСУ 10/2014
  • Ovchintsev Mikhail Petrovich - Moscow State University of Civil Engineering (MGSU) Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Higher Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; Этот e-mail адрес защищен от спам-ботов, для его просмотра у Вас должен быть включен Javascript .
  • Sitnikova Elena Georgievna - Moscow State University of Civil Engineering (MGSU) Candidate of Physical and Mathematical Sciences, Professor, Department of Higher Mathematics, Moscow State University of Civil Engineering (MGSU), .

Страницы 48-53

In the following article the authors continue investigating elliptical equation. Let P be an unlimited cylinder in the space R3, the cross-section of which is a regular dodecagon. The authors have previously estimated linear self-conjugate uniformly elliptic equation of second order in the cylinder and obtained theorems on the growth of the solution in bounded domain. In order to prove the theorems we have to continue solving the differential equation and its coefficients for the whole space Rn.
Let L be a second order linear differential operator in a divergence form which is uniformly elliptic and h is its ellipticity constant. Let u be a solution of the mixed boundary value problem in P for the equation Lu=0 (u>0) with homogeneous Dirichlet and Neumann data on the boundary of the cylinder.
In this paper the solution for mixed boundary value problem is continued from the cylinder to the whole space R3.
The solution of the mixed problem has connection with the notion of the mathematical tessellation. This tessellation is a sum of nonintersecting regular dodecagons and triangles filling the whole space R2

DOI: 10.22227/1997-0935.2014.10.48-53

Библиографический список
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