On the Numerical Approximation of Three-Dimensional Time Fractional Convection-Diffusion Equations

Abstract

In this paper, we present an efficient method for the numerical investigation of three-dimensional non-integer-order convection-diffusion equation (CDE) based on radial basis functions (RBFs) in localized form and Laplace transform (LT). In our numerical scheme, first we transform the given problem into Laplace space using Laplace transform. Then, the local radial basis function (LRBF) method is employed to approximate the solution of the transformed problem. Finally, we represent the solution as an integral along a smooth curve in the complex left half plane. The integral is then evaluated to high accuracy by a quadrature rule. The Laplace transform is used to avoid the classical time marching procedure. The radial basis functions are important tools for scattered data interpolation and for solving partial differential equations (PDEs) of integer and non-integer order. The LRBF and global radial basis function (GRBF) are used to produce sparse collocation matrices which resolve the issue of the sensitivity of shape parameter and ill conditioning of system matrices and reduce the computational cost. The application of Laplace transformation often leads to the solution in complex plane which cannot be generally inverted. In this work, improved Talbot’s method is utilized which is an efficient method for the numerical inversion of Laplace transform. The stability and convergence of the method are discussed. Two test problems are considered to validate the numerical scheme. The numerical results highlight the efficiency and accuracy of the proposed method.