This thesis is an extension (and generalization) of the work done by Li and Zhu. All the numerical methods for solving partial differential equations (PDEs) can be categories : the domain type and the boundary type. In this thesis some boundary-type methods for solving PDEs such as boundary element (BEM) and Galerkin boundary node method (GBNM) are presented. The boudary methods have some advantages over domain type methods: • Only the boundary of the domain needs to be discretized. Especially in two dimensions where the boundary is just a curve this allows very simple data input and storage methods. • Exterior problems with unbounded domains but bounded boundaries are handled as easily as interior problems. • In some applications, the physically relevant data are needed not by the solution in the interior of the domain but rather by the boundary values of the solution or its derivatives. These data can be obtained directly from the solution of boundary integral equations, whereas boundary values obtained from domain methods solutions are in general not very accurate. • The solution in the interior of the domain is approximated with a rather high convergence rate and moreover, the same rate of convergence holds for all derivatives of any order of the solution in the domain. There are difficulties, however, if the solution has to be evaluated close to, but not on the boundary. Some main difficulties with these methods are the following: • Boundary integral equations require the explicit knowledge of a fundamental solution of the differential equation. This is available only for linear partial differential equations with constant or some specifically variable coefficients. Problems with inhomogeneities or nonlinear differential equations are in general not accessible by pure boundary methods. • The reason for the difficulty of the mathematical analysis is that boundary integral equations frequently are not ordinary Fredholm integral equations of the second kind. The of integral equations and their numerical solution concentrates on second kind integral equations with regular kernel, however. Boundary integral equations may be of the first kind, and the kernels are in general singular. If the singularities are not integrable, one has to regularize the integrals which are then defined in a distributional sense. The theoretical framework for such integral equations is the theory of pseudodifferential operators. This theory was developed 20 years ago and is now a within applied mathematics. In recent years, the meshless methods have attracted much attention for solving boundary value problems. Compared to the finite element method (FEM), this family of methods do not require a mesh to discretize the problem domain, and the approximate solutions are constructed entirely based on a cluster of scattered nodes. The idea of meshless has also been used in boundary integral equations (BIEs), such as the boundary node method (BNM) and the hybrid boundary node method. The two approaches are formulated by using the moving least-squares (MLS) method. Compared with the domain-type meshless methods, boundary-type meshless methods reduce the dimension of the original problem by one, thus they are attractive computational techniques for linear and exterior problems. However, they cannot accurately satisfy boundary conditions, since their shape functions constructed by the MLS approximation lack the delta function property. This problem becomes even more serious in the boundary-type meshless methods, since a large number of nodes on the boundary should be satisfied. Another boundary-type meshless method is the Galerkin boundary node method (GBNM). It combines the MLS approximation with a variational formulation of BIEs. The MLS scheme is implemented for construction of the trial and test functions of the variational form, thus the boundary of a problem domain can be discretized by a set of scattered nodes instead of elements. In this method boundary conditions can be enforced directly and easily and the resulting system matrices inherit the symmetry and positive definiteness of the variational problems.