The BEM and DRBEM schemes for the numerical solution of the two-dimensional time-fractional diffusion-wave equations

Abstract

This paper explores the application of two methods, namely the boundary elements method (BEM) and the dual reciprocity boundary elements method (DRBEM), in solving two-dimensional time-fractional partial differential equations (TFPDEs) using the Caputo sense to describe the fractional derivative. In the BEM, the main equation simplifies to solving the Helmholtz equation at each time step, which requires computing the domain integral. To address this, we present an approach that allows computation of the domain integral without encountering singularities. In contrast, the DRBEM offers the advantage of discretizing only the boundary of the computational domain while still being able to evaluate the solution at any desired interior point. For the interpolation of the inhomogeneous and time derivative terms, we utilize radial basis functions (RBFs). To demonstrate the effectiveness of our proposed method, we apply it to solve various problems in both the unit square and more complex regions in two dimensions.