In this paper we apply the boundary elements method (BEM) and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of two-dimensional time-fractional partial differential equations (TFPDEs). The fractional derivative of problem is described in the Caputo sense. In BEM, the main equation deduces to solving the Helmholtz equation in each time step. Therefore, we should compute the domain integral in each time step. So, we presented an approach to compute the domain integral with no singularity. On the other hand the DRBEM has the flexibility of discretizing only the boundary of the computational domain and evaluates the solution at any required interior point. We employ the radial basis functions (RBFs) for interpolation of the inhomogeneous and time derivative terms. The proposed method is employed for solving some problems in two-dimensions on unit square and some other complex regions to demonstrate the efficiency of the proposed method.

This article describes a numerical method based on the dual reciprocity boundary elements method (DRBEM) for solving some well-known nonlinear parabolic partial differential equations (PDEs). The equations include the classic and generalized Fisher's equations, Allen-Cahn equation, Newell-Whithead equation, FitzHugh-Nagumo equation and generalized FitzHugh-Nagumo equation with time-dependent coefficients. The concept of the dual reciprocity is used to convert the domain integral to the boundary that leads to an integration free method. We employ the time stepping scheme to approximate the time derivative, and the linear radial basis functions (RBFs) are used as approximate functions in presented method. The nonlinear terms are treated iteratively within each time step. The developed formulation is verified in some numerical test examples. The results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme.