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.

Haar wavelet based numerical technique for the solutions of fractional advection diffusion equations

“…FPDEs can be solved using a variety of numerical techniques, including the finite difference method, homotopy perturbation approach [21], generalized differential transform technique [24], Sinc-Legendre technique [28], discontinuous Galerkin technique [43], and variational iteration method [9]. Recently, several methods were proposed to develop the solutions of TFDEs, which include finite difference and finite volume schemes [14,29], Gegenbauer spectral method [11], B-spline scaling function for time-fractional convection-diffusion equations [2], and high-order numerical algorithms for TFPDEs [46], Finite difference method for fractional dispersion equations [36], extended cubic B-spline technique [37], Chebyshev collocation methods [31,35], and RBF-based local meshless method for fractional diffusion equations [13].…”

On Dirichlet Problem of Time-Fractional Advection-Diffusion Equation

“…During last decade, Many Mathematicians have studied numerical methods for different types of fractional partial differential equations involving time and/or space derivative such as time-fractional diffusion equations, space-time fractional diffusion equations, Cauchy reaction diffusion equations, Time Fractional Convection-Diffusion Equations, etc., see [2,3,10,11]. Kamran and et al provided a method for the numerical simulation of time-fractional diffusion equations.…”