Gedif, Tsegaye (PhD)Alemayehu, Mulugeta2018-07-172023-11-042018-07-172023-11-042011-01http://etd.aau.edu.et/handle/123456789/8917The complete set of this project focuses on solvability of the Dirichlet problem of the type _ Δ_ = u=f __ Ω _ = u _Ω and the Neumann problem of the type _ Δ_ = u __ Ω __ __ = ℎ __ _Ω . The explicit solutions of the problem in particular when _ = 0, in elliptic linear operator specifically in Laplace’s operator and also discuss the integral representation for the solution of the Dirichlet problem for harmonic functions in upper half space and in a ball in the whole of ℝ_. Keywords Ball, Dirichlet problem, half space, harmonic functions, integral representation, Neumann problemenDirichlet problemhalf spaceharmonic functionsintegral representationNeumann problemThesis