### 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