This thesis is an extension (and generalization) of the work done by Belytschko et al. (\cite{Belytschko} ) and the work done by Atluri and Zhu (\cite{MLPG } ). Traditionall y , partial differential equations (PDEs ) are solved by using numerical met hods , such as the finite difference method (FDM ) , the finite element method (FEM ) and the finite volume method (FVM ). They relied on the use of interlaced grids, elements or finite volumes as the underlying structures upon which to discretize governing PDEs , actually domain of the problem discretized into a mesh . Mesh generation has always posed challenges for computational scientists because of its time consuming and complication . N owadays , a new T hese methods do not have difficulti es of the FDM , the FEM and the FVM . C ompatibility of these methods with deformations is relatively easy , since it is necessary to add nodes in a part of domain for decreasing error. Meshless methods are used to set up a system of algebraic equations for the whole problem domain without mesh generation . These methods utilize a set of arbitrarily distributed nodes within the problem domain and on the domain boundaries to represent the problem domain and its boundaries . All of them reduce necessity to mesh generation , but most of these methods are not truly meshfree methods because they need a mesh for integration purpose . These methods are called pseudomeshfree methods . O ne of the most powerful meshless methods that is investigate d in this work is e lement free Galerki n (EFG ) method . This method employes the moving least squares (MLS ) approximation to create shape functions and uses a background cells for evaluation the integrals appeared in the its global weak form. T hus the EFG method is a pseudomeshfree method . The integration in the EFG is evaluated by a large number of integration points which are arranged in a cell structure. The major disadvantage of the EFG method is high computational cost due to number of integration points . It is to be noted that t he MLS shape functions do no t possess the Kroneker delta property. T herefore in methods based on MLS approximation , the Dirichlet boundary conditions is emposed by Lagrange multipliers or penalty method . Ano ther meshless method that i s focused on it, is the meshless local Petrov- Galerkin (MLPG ) method , on which no elements or background cells are necessary for interpolation or integration purpose, so this method is a truly meshless method . The MLPG works with a local weak form instead of a global weak form . T he weak form is formulated over all local subdomains . The MLPG is a concept rather than a method itself. It can use any meshfree approximations and any convenient test function for the solution process. Atluri and Shen examined six diff erent realizations of the MLPG concept by using six different test functions and called them MLPG1 to MLPG6 . In th is thesis, MLPG1 to MLPG6 are reviewed , but the numerical results is reported just for MLPG5 . The Dirichlet boundary conditions in MLPG5 is emposed directly .