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 $v(\mathbf{x},t)$ 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.

65M60, 65N30, 65N15

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@Article{AAMM-11-197, author = {Yi , Shichao and Sun , Hongguang}, title = {A Hybrided Trapezoidal-Difference Scheme for Nonlinear Time-Fractional Fourth-Order Advection-Dispersion Equation Based on Chebyshev Spectral Collocation Method}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2019}, volume = {11}, number = {1}, pages = {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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2018-0045}, url = {http://global-sci.org/intro/article_detail/aamm/12927.html} }
TY - JOUR T1 - A Hybrided Trapezoidal-Difference Scheme for Nonlinear Time-Fractional Fourth-Order Advection-Dispersion Equation Based on Chebyshev Spectral Collocation Method AU - Yi , Shichao AU - Sun , Hongguang JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 197 EP - 215 PY - 2019 DA - 2019/01 SN - 11 DO - http://doi.org/10.4208/aamm.OA-2018-0045 UR - https://global-sci.org/intro/article_detail/aamm/12927.html KW - Trapezoidal-difference scheme, time-fractional order, variable coefficient fourth-order advection-dispersion equation, Chebyshev spectral collocation method, nonlinearity. AB -

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.

Shichao Yi & Hongguang Sun. (2020). A Hybrided Trapezoidal-Difference Scheme for Nonlinear Time-Fractional Fourth-Order Advection-Dispersion Equation Based on Chebyshev Spectral Collocation Method. Advances in Applied Mathematics and Mechanics. 11 (1). 197-215. doi:10.4208/aamm.OA-2018-0045
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