Numerical Solution of Advection–Diffusion Equation of Fractional Order Using Chebyshev Collocation Method

Abstract

This work presents a highly accurate method for the numerical solution of the advection–diffusion equation of fractional order. In our proposed method, we apply the Laplace transform to handle the time-fractional derivative and utilize the Chebyshev spectral collocation method for spatial discretization. The primary motivation for using the Laplace transform is its ability to avoid the classical time-stepping scheme and overcome the adverse effects of time steps on numerical accuracy and stability. Our method comprises three primary steps: (i) reducing the time-dependent equation to a time-independent equation via the Laplace transform, (ii) employing the Chebyshev spectral collocation method to approximate the solution of the transformed equation, and (iii) numerically inverting the Laplace transform. We discuss the convergence and stability of the method and assess its accuracy and efficiency by solving various problems in two dimensions.

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].…”