Localized boundary-domain integral equation formulation for mixed type problems

Copyright @ 2010 Walter de Gruyter GmbHSome modified direct localized boundary-domain integral equations (LBDIEs) systems associated with the mixed boundary value problem (BVP) for a scalar “Laplace” PDE with variable coefficient are formulated and analyzed. The main results established in the paper are the LBDIEs equivalence to the original variable-coefficient BVPs and the invertibility of the corresponding localized boundary-domain integral operators in appropriately chosen function spaces

Localized direct segregated boundary-domain integral equations for variable-coefficient transmission problems with interface crack

The full text of the published article can be accessed at the link belowSome transmission problems for scalar second order elliptic partial differential equations are considered in a bounded composite domain consisting of adjacent anisotropic subdomains having a common interface surface. The matrix of coefficients of the differential operator has a jump across the interface but in each of the adjacent subdomains is represented as the product of a constant matrix by a smooth variable scalar function. The Dirichlet or mixed type boundary conditions are prescribed on the exterior boundary of the composite domain, the Neumann conditions on the the interface crack surfaces and the transmission conditions on the rest of the interface. Employing the parametrix-based localized potential method, the transmission problems are reduced to the localized boundary-domain integral equations. The corresponding localized boundary-domain integral operators are investigated and their invertibility in appropriate function spaces is proved.This research was supported by EPSRC grant No. EP/H020497/1 and partly by the Georgian Technical University gran

Analysis of some localized boundary-domain integral equations

Some direct segregated localized boundary-domain integral equation (LBDIE) systems associated with the Dirichlet and Neumann boundary value problems (BVP) for a scalar "Laplace" PDE with variable coefficient are formulated and analysed. The parametrix is localized by multiplication with a radial localizing function. Mapping and jump properties of surface and volume integral potentials based on a localized parametrix and constituting the LBDIE systems are studied in a scale of Sobolev (Bessel potential) spaces. The main results established in the paper are the LBDIEs equivalence to the original variable-coefficient BVPs and the invertibility of the LBDIE operators in the corresponding Sobolev spaces

About analysis of some localized boundary-domain integral equations for a variable-coefficient BVPs

Some direct localized boundary-domain integral equations (LBDIEs) associated with the Dirichlet and Neumann boundary value problems for the "Laplace" linear differential equation with a variable coefficient are formulated. The LBDIEs are based on a parametrix localized by a cut-off function. Applying the theory of pseudo-differential operators, invertibility of the localized volume potentials is proved first. This allows then to prove solvability, solution uniqueness and equivalence of the LBDIEs to the original BVP, and investigate the LBDIE operator invertibility in appropriate Sobolev spaces

Thermoelastic Oscillations of Anisotropic Bodies

The generalized radiation conditions at infinity of Sommerfeld-Kupradze type are established in the theory of thermoelasticity of anisotropic bodies. Applying the potential method and the theory of pseudodifferential equations on manifolds the uniqueness and existence theorems of solutions to the basic three-dimensional exterior boundary value problems are proved and representation formulas of solutions by potential type integrals are obtained

Determining the Optimal Sowing Frequency and Sowing Norm of Cereal Crops

Anyone interested in growing crops ¬ knows that all crops need favorable conditions to get a good crop. In particular, the yield of cereal crops depends on many random factors, the complex consideration of which is associated with great difficulties. Optimal plant distribution and grain crop nutrition conditions can only be achieved during sowing. The main calculation values ​​of sowing of cereal crops are the determination of the optimal sowing frequency and the sowing norm per hectare. Through the nomogram given in the paper it is possible to determine the required number of plants in the open sowing trail depending on the distance between the given rows and the optimal plant frequency. There is also a nomogram defining the sowing norm, through which it is possible to determine the sowing norm of the desired crop by considering the absolute mass of the grain and its ability to emerge

Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, I: Equivalence and Invertibility

Copyright @ 2009 Rocky Mountain Mathematics ConsortiumA mixed (Dirichlet-Neumann) boundary value problem (BVP) for the "stationary heat transfer" partial differential equation with variable coefficient is reduced to some systems of nonstandard segregated direct parametrix-based boundary-domain integral equations (BDIEs). The BDIE systems contain integral operators defined on the domain under consideration as well as potential-type and pseudo-differential operators defined on oopen submanifolds of the boundary. It is shown that the BDIS systems are equivalent to the original mixed BVP, and the operators of the BDIE systems are invertible in appropriate Sobolev spaces

Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, II : Solution regularity and asymptotics

Copyright @ 2010 Rocky Mountain Mathematics ConsortiumMapping and invertibility properties of some parametrix-based surface and volume potentials are studied in Bessel-potential and Besov spaces. These results are then applied to derive regularit and asymptotics of the solution to a system of boundary-domain integral equations associated with a mixed BVP for a variable-coefficient PDE, in a vicinity of the curve of change of the boundary condition type.This work was supported by the International Joint Project Grant - 2005/R4

Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed bvps in exterior domains

This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2013 World Scientific Publishing.Direct segregated systems of boundary-domain integral equations are formulated for the mixed (Dirichlet–Neumann) boundary value problems for a scalar second-order divergent elliptic partial differential equation with a variable coefficient in an exterior three-dimensional domain. The boundary-domain integral equation system equivalence to the original boundary value problems and the Fredholm properties and invertibility of the corresponding boundary-domain integral operators are analyzed in weighted Sobolev spaces suitable for infinite domains. This analysis is based on the corresponding properties of the BVPs in weighted Sobolev spaces that are proved as well.The work was supported by the grant EP/H020497/1 \Mathematical analysis of localised boundary-domain integral equations for BVPs with variable coefficients" of the EPSRC, UK

Localized boundary-domain singular integral equations of the Robin type problem for self-adjoint second-order strongly elliptic PDE systems

Shota Rustaveli National Science Foundation of Georgia (SRNSF) (Grant number FR-18-126)