System Of Boundary-domain Integral Equations Research Articles

Boundary Domain Integral Equations for Variable Coefficient Mixed BVP in 2D Unbounded Domain

In this paper, the direct segregated Boundary Domain Integral Equations (BDIEs) for the Mixed Boundary Value Problems (MBVPs) for a scalar second order elliptic Partial Differential Equation (PDE) with variable coefficient in unbounded (exterior) 2D domain is considered. Otar Chkadua, Sergey Mikhailov and David Natroshvili formulated both the interior and exterior 3D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order elliptic PDE with a variable coefficients. On the other hand Sergey Mikhailov and Tamirat Temesgen formulated only the interior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. However, in this paper we formulated the exterior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. The aim of this work is to reduce the MBVPs to some direct segregated BDIEs with the use of an appropriate parametrix (Levi function). We examine the characteristics of corresponding parametrix-based integral volume and layer potentials in some weighted Sobolev spaces, as well as the unique solvability of BDIEs and their equivalence to the original MBVPs. This analysis is based on the corresponding properties of the MBVPs in weighted Sobolev spaces that are proved as well.

Analysis of two-operator boundary-domain integral equations for variable-coefficient mixed BVP in 2D with general right-hand side

The mixed (Dirichlet–Neumann) boundary value problem (BVP) for the linear second-order scalar elliptic differential equation with variable coefficients in a bounded two-dimensional domain is considered. The PDE on the right-hand side belongs to H−1(Ω) or H˜−1(Ω), when neither classical nor canonical conormal derivatives of solutions are well defined. The two-operator approach and appropriate parametrix (Levi function) are used to reduce this BVP to four systems of boundary-domain integral equations (BDIEs). Although the theory of BDIEs in 3D is well developed, the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the domain or on the associated Sobolev spaces to ensure the invertibility of corresponding parametrix-based integral layer potentials, and hence the unique solvability of BDIEs. The equivalence of the BDIE systems to the original BVP is shown. The invertibility of the associated operators is proved in the corresponding Sobolev spaces.

Boundary-Domain Integral Equations equivalent to an exterior mixed BVP for the variable-viscosity compressible Stokes PDEs

Two direct systems of Boundary-Domain Integral Equations, BDIEs, associated with a mixed boundary value problem for the stationary compressible Stokes system with variable viscosity coefficient in an exterior domain of $ \mathbb{R}^3 $ are derived. This is done by employing the Stokes surface and volume potentials based on an appropriate parametrix (Levi function) in the third Green identities for the velocity and pressure. Mapping properties of the potentials in weighted Sobolev spaces are analysed. Finally, the equivalence between the BDIE systems and the BVP is shown and the isomorphism of operators defined by the BDIE systems is proved.

On the existence and uniqueness of solution of boundary‐domain integral equations for the Dirichlet problem for the nonhomogeneous heat transfer equation defined on a 2D unbounded domain

A system of boundary‐domain integral equations (BDIEs) is obtained from the Dirichlet problem for the diffusion equation in nonhomogeneous media defined on an exterior two‐dimensional domain. We use a parametrix different from the one employed in Dufera and Mikhailov (2019). The system of BDIEs is formulated in terms of parametrix‐based surface and volume potentials whose mapping properties are analyzed in weighted Sobolev spaces. The system of BDIEs is shown to be equivalent to the original boundary value problem and uniquely solvable in appropriate weighted Sobolev spaces suitable for unbounded domains.

A new family of boundary‐domain integral equations for the Dirichlet problem of the diffusion equation in inhomogeneous media with H−1(Ω) source term on Lipschitz domains

The interior Dirichlet boundary value problem for the diffusion equation in nonhomogeneous media is reduced to a system of boundary‐domain integral equations (BDIEs) employing the parametrix obtained in Portillo (2019). We further extend the results obtained in Portillo (2019) for the mixed problem in a smooth domain with L2(Ω) right‐hand side to Lipschitz domains and partial differential equation (PDE) right‐hand side in the Sobolev space H−1(Ω), where neither the classical nor the canonical conormal derivatives are well‐defined. Equivalence between the system of BDIEs and the original BVP is proved along with their solvability and solution uniqueness in appropriate Sobolev spaces.

A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains

A system of Boundary-Domain Integral Equations is derived from the mixed (Dirichlet-Neumann) boundary value problem for the diffusion equation in inhomogeneous media defined on an unbounded domain. Boundary-domain integral equations are formulated in terms of parametrix-based potential type integral operators defined on the boundary and the domain. Mapping properties of parametrix-based potentials on weighted Sobolev spaces are analysed. Equivalence between the original boundary value problem and the system of BDIEs is shown. Uniqueness of solution of the BDIEs is proved using Fredholm Alternative and compactness arguments adapted to weigthed Sobolev spaces.

On the existence of solution of the boundary-domain integral equation system derived from the 2D Dirichlet problem for the diffusion equation with variable coefficient using a novel parametrix

ABSTRACT A system of boundary-domain integral equations is derived from the bidimensional Dirichlet problem for the diffusion equation with variable coefficient using a novel parametrix different from the one widely used in the literature by the authors Chkadua, Mikhailov and Natroshvili. Mapping properties of the surface and volume parametrix-based potential-type operators are analysed. Invertibility of the single layer potential is also studied in detail in appropriate Sobolev spaces. We show that the system of boundary-domain integral equations derived is equivalent to the Dirichlet problem prescribed and we prove the existence and uniqueness of solution in suitable Sobolev spaces of the system obtained by using arguments of compactness and Fredholm Alternative theory. A discussion of the possible applications of this new parametrix is included.

ANALYSIS OF DIRECT SEGREGATED BOUNDARY-DOMAIN INTEGRAL EQUATIONS FOR VARIABLE-COEFFICIENT MIXED BVPs IN EXTERIOR DOMAINS

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.

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

Mapping 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 regularity and asymptotics of the solution to a system of boundary-domain integral equations associated with a mixed BVP for a variablecoefficient PDE, in a vicinity of the curve of change of the boundary condition type.