### Abstract:

A classical solution of partial differential equation has to be sufficiently smooth i.e. (has to satisfy some differentiability and continuity assumptions that is in order to satisfy the given partial differential equation and it initial and boundary conditions. In higher dimensions, also the domain has to satisfy certain regularity conditions. Such smoothness or regularity conditions are often not satisfied in applications. Nevertheless, the processes, which are modelled with the partial differential equations, occur and there is obviously a solution. However, this solution will not possess the (regularity) properties of the classical solution and therefore one needs an extension of the notion of the solution. The aim of this thesis is, to present the notion of weak solution of partial differential equations. The main feature of the weak solution is that, we rewrite a given partial differential equation in a new form, the new formulation called weak formulation and the solution satisfied this formulation( not necessary smooth may not even be differentiable) called the weak solution of the given partial differential equations. The need for weak solution of partial differential equations comes from the fact that many if not most of the partial differential equations cannot be solved in the classical sense. In this thesis we consider the weak solution of linear partial differential equations .Our results we as follows. In chapter one, we present the main concepts from functional analysis needed in the following chapters. Chapter two is devoted to present the existence and uniqueness of weak solutions of elliptic partial differential equations in one dimension with several boundary conditions such as Dirichlet and Neumann boundary conditions. In chapter three we generalize the results in chapter two from one to N-dimension. In that chapter, we study elliptic problems with more complicated boundary conditions such as Robin boundary conditions. In the last chapter we study the existence and uniqueness of weak solutions of evolution partial differential equations. We recommend to analysis the weak solution of nonlinear of partial differential equations.

### Description:

A Dissertation Submitted to University of Gezira in Partial Fulfilment of the Requirements, For the Degree of Master of Science
In Mathematical Science, Department of Mathematical Science, Faculty of Mathematical and Computer Sciences, December, 2017