Hossein Hosseinzadeh

This thesis deals with the local meshless methods and numerical solutions of partial differential equations including differential equations of fractional distributed order. The latest version of meshless local Petrov-Galerkin method, MLPG7, is extended to solve time dependent PDEs. MLPG3 uses a modification of fundamental solution of Laplace operator as test function. In the new version, a new modified test function is proposed which in addition to itself, its first derivative also vanishes on boundaries of local subdomains while preserves the fundamental solution property at center of subdomain. Therefore, using the Green’s identities in local weak equations, a simple domain integral is obtained. Therefore, in addition to simplification of computations, the method becomes more stable due to domain integrals of unknown function. Convergence and stability analysis of the method are done both theoretically and numerically. Moreover, the numerical results obtained by the new method are compared with those obtained with some other well-known methods revealing the higher performance of the proposed method. Fractional differential equations of distributed order contain an integral over the order of differentiation which results in the complexity of its numerical treatments. A suitable discretization of this integral, coverts the equation into multi-order fractional differential equations. A local meshless method using strong form equation and the MLPG7 is used to efficiently solve distributed order fractional differential equations