The element-free Galerkin (EFG) method is a promising method for solving partial differential equations in which trial and test functions employed in the discretization process result from moving least-squares (MLS) approximation . A disadvantage of the EFG method is that the final algebraic equations system is sometimes ill-conditioned . Hence , in the MLS approximation , the ill-conditioned algebraic equations system should be considered . However , it is difficult to determine which algebraic equations system is ill-conditioned before the equation is solved . To overcome this problem , the improved moving least-squares (IMLS) approximation has been developed to obtain the approximation function . In the IMLS approximation , the orthogonal functions with respect to a weight function is used as the basis function . The algebraic equations system in this approximation is not ill-conditioned , and it can be solved without involving matrix inversion .Based on the IMLS approximation and the EFG method , an improved element-free Galerkin (IEFG) method has been developed . This method is one of the meshless methods that does not require any element connectivity data , and does not suffer much degradation in accuracy when nodal arrangements are very irregular . Hence , the IEFG method is somewhat more efficient than the conventional EFG method . In this thesis , we first prepare some definitions and theorems which will be used for the next chapters . Then , we have presented some tests related to the moving least-squares and improved moving least-squares approximations . Next various weight functions and theirs properties are investigated . We have constructed the weighted orthogonal basis functions with the aid of the Gram-Schmidt orthogonalization process . By replacing these basis functions with the traditional basis functions in the MLS approximation , IMLS approximation has been derived . We have compared these approximations by examples , and found that the IMLS approximation works well , and it is somewhat more efficient than the other one . Finally , we have expanded the IEFG method and implemented it on the modified equal width wave equation , Poisson and Schr?dinger equations in one and two dimensions . Motion of single , two and three solitary waves have been simulated . Some conserved quantities tested to deal with the validity of the methods . Keywords: Moving least-squares approximation , improved moving least-squares approximation , element-free Galerkin method , improved element-free Galerkin method , modified equal width wave equation , Schr?dinger equation.