Numerical approximation of a variable-order time fractional advection-reaction-diffusion model via shifted Gegenbauer polynomials

Yumei Chen; Jiajie Zhang; Chao Pan; Yumei Chen; Jiajie Zhang; Chao Pan

doi:10.3934/math.2022855

AIMS Mathematics

2022, Volume 7, Issue 8: 15612-15632. doi: 10.3934/math.2022855

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Research article

Numerical approximation of a variable-order time fractional advection-reaction-diffusion model via shifted Gegenbauer polynomials

1.
College of Mathematics Education, China West Normal University, Nanchong 637009, China
2.
School of Mathematics and Information, China West Normal University, Nanchong 637009, China

Received: 11 April 2022 Revised: 11 June 2022 Accepted: 16 June 2022 Published: 23 June 2022
MSC : 65N35, 34A08

The fractional advection-reaction-diffusion equation plays a key role in describing the processes of multiple species transported by a fluid. Different numerical methods have been proposed for the case of fixed-order derivatives, while there are no such methods for the generalization of variable-order cases. In this paper, a numerical treatment is given to solve a variable-order model with time fractional derivative defined in the Atangana-Baleanu-Caputo sense. By using shifted Gegenbauer cardinal function, this approach is based on the application of spectral collocation method and operator matrices. Then the desired problem is transformed into solving a nonlinear system, which can greatly simplifies the solution process. Numerical experiments are presented to illustrate the effectiveness and accuracy of the proposed method.
Citation: Yumei Chen, Jiajie Zhang, Chao Pan. Numerical approximation of a variable-order time fractional advection-reaction-diffusion model via shifted Gegenbauer polynomials[J]. AIMS Mathematics, 2022, 7(8): 15612-15632. doi: 10.3934/math.2022855

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Abstract

The fractional advection-reaction-diffusion equation plays a key role in describing the processes of multiple species transported by a fluid. Different numerical methods have been proposed for the case of fixed-order derivatives, while there are no such methods for the generalization of variable-order cases. In this paper, a numerical treatment is given to solve a variable-order model with time fractional derivative defined in the Atangana-Baleanu-Caputo sense. By using shifted Gegenbauer cardinal function, this approach is based on the application of spectral collocation method and operator matrices. Then the desired problem is transformed into solving a nonlinear system, which can greatly simplifies the solution process. Numerical experiments are presented to illustrate the effectiveness and accuracy of the proposed method.

References

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