Analysis Of Direct Boundary-Domain Integral Equations For Variable Coefficient Neumann Bvp In Plane

Abstract

In this thesis, analysis of direct boundary-domain integral equations for the scalar second order “stationary heat transfer” elliptic partial differential equation with variable coefficient subjected to Neumann boundary condition in plane were investigated. Using an appropriate parametrix in Green’s identity, the considered Neumann boundary value problem is reduced to two different systems of Boundary-Domain Integral Equations (BDIEs). Although the theory of BDIEs in 3D is well developed, the theory of BDIEs in plane need a special consideration due to their different equivalence properties. Consequently, we need to set conditions on the domain or on the spaces to insure the invertibility of the corresponding parametrix-based integral layer potentials and hence the unique solvability of BDIEs. The equivalence of the obtained BDIE systems to the original BVPs, BDIE systems solvability, solution uniqueness/non-uniqueness, as well as Fredholm property and invertibility of the BDIE operators for the Neumann BVP were analyzed.