The content of this dissertation is arranged in two parts. The purpose of the first part is to enhance the accuracy of the Global Exponential Basis Functions (EBFs) method for solving partial differential equations (PDEs) with constant coefficients on concave domains by using the multiplicative Schwarz method. As it has been demonstrated in solution of some important PDEs in solid mechanics including Poisson, Helmholtz, two-dimensional elasticity and bending of plates differential equations, application of this method simplifies the solving procedure and decreases the computation error. In the second part of this study, a new algorithm of GEBF has been proposed. In this algorithm, unlike the previous one, the solution accuracy is not dependent on the basis selection and the rank of the unknown matrix is higher. In this algorithm, the approximate solution of the homogeneous equation is expressed as a series, using source functions centered on some source points. Since each source function affects a few adjacent boundary points, the rank of the unknown matrix is approximately full and therefore, it is much easier to compute its inverse. Higher precision and accuracy on concave domains is another advantage of this method. To this aim, an appropriate algorithm has been proposed to solve the Helmholtz equation. Then, this algorithm has been extended to other important PDEs in structural engineering problems including the equations of advection-diffusion-absorption, two-dimensional elasticity, bending of plates and forced vibration of plates. The presented examples show that using the new pattern of EBFs method is adequately able to solve different problems on complicated domain shapes with various types of boundary conditions. Key Words The multiplicative Schwarz method, Exponential Basis Functions (EBFs) method, partial differential equations (PDEs), Helmholtz, advection-diffusion-absorption, elasticity, bending of plates, forced vibration of plates