Tsegaye AyeleEshetu Seid2025-09-122025-09-122024-09https://etd.aau.edu.et/handle/123456789/7444In this thesis, 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 coefficients in an unbounded (exterior) 2D domain is considered. In this thesis, we formulate the exterior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second-order divergent elliptic PDE with 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.en-USBoundary-Domain Integral EquationsVariable-Coefficient Mixed BVP in 2−DimensionalUnbounded DomainThesis