Dissertation (MSc. Mathematics)The study of an inverse problems for elliptic partial differential equations has been put in place in order to provide a broad understanding to the existence of inverse problems for elliptic partial differential equations particularly Poisson's equation. The main objective is to describe diferent approaches to inverse problems for elliptic partial differential equations and in particular, the factorization method. Another aim of this work is to study a certain inverse problem for an elliptic partial differential equation and demonstrate the uniqueness and local existence of its solution using the derivation of the special integral equation. Inverse problem for elliptic partial differential equations play very important role in both theoretical part and applications. After the inversion of powerfully computers, more have been achieved through inverse problems for elliptic partial differential equations.