In this dissertation solution of partial differential equations (PDEs) with constant coefficients has been investigated using smooth basis functions in a meshless local form. These PDEs have a wide range of applications in engineering and scientific fields. First, a local meshless method based on previous researches has been described and the properties of the method are investigated through solving some numerical examples. In this method the solution domain is discretized using a set of points. The main deficiency of this method is the lack of accuracy in problems with an irregular distribution of nodes. A simple workaround has been devised to address a wide Also a new meshless local method has been developed for solution of PDEs with constant coefficients. In this method the procedure of solution begins by discretizing the solution domain using nodal points. A cloud, consisting of a set of adjacent nodes, is associated with each of these nodes. Solution of the differential equation in each cloud is divided into homogeneous and particular parts. Each of these parts is constructed as a series of exponential basis functions (EBFs) with constant coefficients. Basis functions of the homogeneous solution are calculated such that the homogeneous differential equation is exactly satisfied in the domain. A set of intermediate points are distributed in the domain and on the boundaries in order to constitute the system of equations. The equations inside the domain must be formed such that the continuity between the solutions of adjacent clouds is provided. A residual value is constructed at each intermediate point in the domain to guarantee the continuity between clouds. For imposing the boundary conditions a similar approach has been used. This approach has a good capability for satisfaction of the boundary conditions and conforms to the formulation used inside the domain. Selection of the homogeneous basis functions is one of the most important parameters affecting the quality of the solution. Suitable selection of these functions plays an important role in reducing the computational error and reaching appropriate accuracy in the final resuts. In the present research these basis functions are selected such that the sampling theorems conditions are satisfied. By using the sampling theorems the maximum frequency of the basis functions is limited. Finally the formulation of the proposed method has been applied to some important and useful PDEs in solid mechanics problems including Helmholtz, Poisson, elasticity and elastic wave equations in 2D domains. Several benchmark examples have been solved in each category of problems. The results of solved numerical examples show that the proposed method is capable for solving problems on domains with arbitrary shapes and boundary conditions and is not sensitive due to irregularity of the nodal points. The ability to solve the problems with rather high frequencies and problems with singular points near the boundary of the domain are some of the other features of this method. Although the proposed method of this dissertation has been used for solving some Keywords differential equations, exponential basis functions, meshless method, sampling theorems, solid mechanics