In this dissertation, a new local meshfree method is developed for the solution of linear and nonlinear partial differential equations (PDEs). The method is first introduced in a local weighted residual style to meet the linear PDEs with constant coefficients. In this method, the field variable in each local cloud of nodes is approximated as series of exponential basis functions satisfying the PDE. The compatibility of the solutions of neighboring clouds is then taken into account in a pointwise weighted residual expression defined on a set of intermediate points. The definition of the residual expression for the boundary intermediate points is correspondingly set to capture the boundary conditions. The final system of equations is then constructed through the minimization of the accumulative total weighted residual. In spite of multiple salient features, the mentioned method encounters some limitations mainly due to its structure, which is dependent on the basis functions satisfying the PDE. In order to overcome such deficiencies a new approximation technique has been devised to handle a wider range of applications including PDEs with variable coefficients. In this technique, the PDE is divided into average and remainder parts and the solution function is divided accordingly. Average and remainder parts of the solution are approximated by linear combinations of exponential and radial basis functions respectively and the relation between these two parts are defined so that to satisfy the PDE. Using this recent approximation technique, a direct formulation is also derived from the compatibility/boundary weighted residuals to develop a novel meshfree method named as the local extended direct method. Capabilities of this method are further enhanced by introducing a compatible iterative linearization algorithm to deal with non-linear problems. 2D elastoplastic problems are among the most important applicable non-linear problems that have been addressed in this thesis. The method is finally applied to some linear and nonlinear initial value problems. In this regard, a time marching algorithm has been formulated based on the Taylor series expansion of the dependent field variable with respect to time. Capabilities of the proposed method have been examined in each section of this dissertation by several numerical examples. Key Words Partial Differential Equations (PDE), Mesh-Free Methods, Boundary Value Problems (BVP), Initial Value Problems (IVP), Non-Linear Problems, Elastoplastic Problems, Exponential Basis Functions (EBFs), Radial Basis Functions (RBFs)