In this project, we aim to study the method of layer potential and to apply it to solve for a boundary value problem. In particular, we are interested to apply the method to solve Laplace equation with appropriate boundary conditions. Potential theory is the study of solutions of Laplace’s equation and its corresponding Poisson equation. These equations arise naturally in the context of conservative vector fields which can be written as the gradient of a scalar function and hence the potential. The methods of layer potentials are described in an integral sense. The applications related to layer potential methods have been developed for differential equations in general. The solutions of our model equations are obtained in the form of integrals.