A Hybrided Trapezoidal-Difference Scheme for Nonlinear Time-Fractional Fourth-Order Advection-Dispersion Equation Based on Chebyshev Spectral Collocation Method

A Hybrided Trapezoidal-Difference Scheme for Nonlinear Time-Fractional Fourth-Order Advection-Dispersion Equation  Based on Chebyshev Spectral Collocation Method

Year:    2019

Author:    Shichao Yi, Hongguang Sun

Advances in Applied Mathematics and Mechanics, Vol. 11 (2019), Iss. 1 : pp. 197–215

Abstract

In this paper, we firstly present a novel simple method based on a Picard integral type formulation for the nonlinear multi-dimensional variable coefficient fourth-order advection-dispersion equation with the time fractional derivative order $\alpha\in (1,2)$. A new unknown function $v(\mathbf{x},t)=\partial u(\mathbf{x},t)/\partial t$ is introduced and $u(\mathbf{x},t)$ is recovered using the trapezoidal formula. As a result of the variable $v(\mathbf{x},t)$ is introduced in each time step, the constraints of traditional plans considering the non-integer time situation of $u(\mathbf{x},t)$ are no longer considered. The stability and solvability are proved with  detailed  proofs and the precise description of error estimates is derived. Further, Chebyshev spectral collocation method supports accurate and efficient variable coefficient model with variable coefficients. Several numerical results are obtained and analyzed in multi-dimensional spatial domains and numerical convergence order is consistent with the theoretical value $3-\alpha$ order for different $\alpha$ under infinite norm.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.OA-2018-0045

Advances in Applied Mathematics and Mechanics, Vol. 11 (2019), Iss. 1 : pp. 197–215

Published online:    2019-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    19

Keywords:    Trapezoidal-difference scheme time-fractional order variable coefficient fourth-order advection-dispersion equation Chebyshev spectral collocation method nonlinearity.

Author Details

Shichao Yi

Hongguang Sun