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The local discontinuous Galerkin method for 2D nonlinear time-fractional advection–diffusion equations

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Abstract

This paper presents a numerical solution of time-fractional nonlinear advection–diffusion equations (TFADEs) based on the local discontinuous Galerkin method. The trapezoidal quadrature scheme (TQS) for the fractional order part of TFADEs is investigated. In TQS, the fractional derivative is replaced by the Volterra integral equation which is computed by the trapezoidal quadrature formula. Then the local discontinuous Galerkin method has been applied for space-discretization in this scheme. Additionally, the stability and convergence analysis of the proposed method has been discussed. Finally some test problems have been investigated to confirm the validity and convergence of the proposed method.

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Acknowledgements

The authors are grateful to the reviewers for their comments and suggestions which have improved the paper.

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  1. Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran

    Jafar Eshaghi, Saeed Kazem & Hojjatollah Adibi

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  1. Jafar Eshaghi
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Appendix 1: Discrete Gronwall’s inequality

Appendix 1: Discrete Gronwall’s inequality

If \(x_i,~f_i,~g_i,~h_i\) are non-negative sequences which satisfy

$$\begin{aligned} x_n \le f_n + g_n \sum _{i=0}^{n} h_i x_i,\qquad n\ge 1, \end{aligned}$$

then we have

$$\begin{aligned} x_n \le f_n + g_n \sum _{i=0}^{n} h_i f_i \prod _{j=i+1}^{n} (h_j g_j + 1). \end{aligned}$$

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Eshaghi, J., Kazem, S. & Adibi, H. The local discontinuous Galerkin method for 2D nonlinear time-fractional advection–diffusion equations. Engineering with Computers 35, 1317–1332 (2019). https://doi.org/10.1007/s00366-018-0665-8

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