R ecently , many new applications in engineering and science are governed by the fractional partial differential equations (FPDEs) . If a fractional derivative is included in the governing equation , the equation will be usually called the fractional partial differential equation (FPDE) . We consider a fractional advection-diffusion equation (FADE) with the Caputo derivative among different types of fractional derivative . At present , most of the fractional differential equations are solved numerically using the finite difference method (FDM) , finite element method (FEM) , and spectral approximation . These methods lead to inherited issues including : difficulty in handling a complicated problem domain , difficulty in handling irregular nodal distribution and low accuracy. In the recent years , the developments of the meshless methods have been more attractive . This kind of methods use a set of nodes scattered within the problem domain . Therefore , they have many advantages over the conventional numerical methods . At present , there are many meshless methods such as the element-free Galerkin (EFG) method , meshless local Petrov–Galerkin (MLPG) method , reproducing kernel particle method (RKPM) and so on . In a MLPG method , it can be used one of the meshfree approximations and any convenient test function for the solution process . Atluri and Zu have been examined six diffrent realizations of the MLPG concept in 1998 . The MLPG method works with a local weak form instead of a global weak which formulated over all local subdomains . Schaback and Mirzaei could develop a MLPG method by using the generalized moving least squares (GMLS) and called it direct Meshless local Petrov–Galerkin (DMLPG) method . In this method the numerical integrations over the moving least squares (MLS) shape functions have been replaced by more accurate and cheaper integrations over polynomials . Several approximation have been used for construction of the meshless shape functions , such as the MLS approximation and the moving Kriging interpolation (MKI) . The MLS is accurate and stable for arbitrarily distributed nodes but it has not the Kronecker delta property . The Kriging interpolation has the Kronecker delta property as well as the consistency property. In this thesis , we develop MLPG2\\ 5 and DMLPG2\\ 5 methods for the numerical simulation of the FADE . For solving FADE by MLPG2; the test function is the Dirac Delta function . So at first a finite difference scheme for temporal variable is proposed where the stability and convergence analysis are proved , then MLS and MKI shape functions are developed for trial functions . Then a new ?nite differences scheme is considered to deal with the Caputo fractional derivative . Then for solving the FADE using the MLPG5 method , the local weak forms of the equation are constructed . For this purpose , the Heaviside test functions and MLS trial function are employed . At the end , The FADE is solved by DMLPG2 and DMLPG5 methods by a discretization similar to the methods of MLPG2 and MLPG5 . In almost all of numerical examples , both regular and random Halton points are tested . Finally these methods are compared with the methods of MLPG2 and MLPG5 and we conclude that for solving such problems , the DMLPG method is faster and more accurate than the MLPG method