Volume 11, Issue 1
A Hybrided Trapezoidal-Difference Scheme for Nonlinear Time-Fractional Fourth-Order Advection-Dispersion Equation Based on Chebyshev Spectral Collocation Method

Adv. Appl. Math. Mech., 11 (2019), pp. 197-215.

Published online: 2019-01

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• 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 $v(\mathbf{x},t)$ are introduced in each time step, the constraints of traditional plans considering the non-integer time situation of $u(\mathbf{x},t)$  is no longer considered. The stability and solvability are proved with  detailed  proofs and the precise describe 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 are consistent with the theoretical value $3-\alpha$ order for different $\alpha$ under infinite norm.

• Keywords

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