The traditional Mesh-based methods for solving Partial Differential Equations are based on mesh generation that is complicated and time consuming process especially for complex geometries. In last decades, meshless methods are introduced for solving the problems that are arisen from mesh dependency. Particle methods are one of the most familiar of these methods. SPH method was a prelude to these methods that was introduced by Lucky and Gingold and Monaghan in 1977. There were some problems in this method. First and foremost of these problem is that this method is inaccurate near the boundary of domain. In 1996, Han introduced Reproducing Kernel Particle methods (RKPM) that smooth out most of SPH problems. In recent years, RKPM has appealed to the masses. The main motive behind this prosperity is that this method is truly meshless and is easy to implement. Other reason for its popularity is accuracy at the points which locate near and on boundary. The outstanding features of RKP shape functions have compelled Scientifics to use these shape functions in other methods and create new methods. Flexibility of RKPM in using strong form or weak form of Partial Differential Equations is one the reasons that have made this method a popular and strong method. Han et al. (1999) justified convergency and stability of this method for weak form approach. In last years, strong form approach make some progress but there is no strong mathematical background for convergency and stability in this approach that confine this approach rather than weak form approach. Belley et al. (2009) introduced some discontinuous constant kernels to solve strong form of Partial Differential Equations. Solutions that are earned with this method are stable and accurate. Also Using of these kernels is easy in implementation. In this thesis we tried to introduce and investigate RKPM especially strong form approach. Then this method is used to solve some problems. For any kind of problems that is solved in this thesis, firs a single Partial Differential equation and then a system of Partial Differential Equations is investigated and solved. In chapter one, history of Meshless methods and particle methods are explained. In chapter two, SPH method is reviewed. In this chapter, we investigate some of the problems of this method and review some of the method to solve these problems. In chapter three, we discussed on RKP approximation and its theorems. Then weak form approach and its theorems are mentioned. In chapter four, we introduced some discontinuous constant kernels. Then we investigate their accuracy and approximate diverse functions to show their efficiency. In chapter five, two different kind of boundary layer problems are investigated and solved. For first of these problems we investigate Convection-Diffusions problem. Then MHD problem as a system of boundary layer problems is studied and solved. In chapter six elliptic problems are studied. Laplace problem as single equation and Isotropic Elasticity problem as a system of equations are studied and solved. In both of the last chapters, we obtain a stable implementation method for each of problems. Then we studied and solved diverse problems to show accuracy and stability of this method.