Abstract
A Crank–Nicolson finite difference scheme to solve a time variable order fractional mobile–immobile advection–dispersion equation is introduced and analyzed. Some a priori estimates of discrete \(L^2\)-norm with order of convergence \(O(\tau +h^2)\) are established on uniform grids where \(\tau \) and h are the steps sizes in time and space. Stability and convergence of the numerical solutions are presented in detail. Numerical examples are provided to verify the theoretical analysis.
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Acknowledgements
This work was supported in part by the National Natural Science Foundation of China under Grants 91130010, 11471194 and 11571115, by the National Science Foundation under Grant DMS-1216923,by the OSD/ARO MURI Grant W911NF-15-1-0562, by the National Science and technology major projects of China under Grants 2011ZX05052 and 2011ZX05011-004, and by Natural Science Foundation of Shandong Province of China under Grant ZR2011AM015.
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Liu, Z., Li, X. A Crank–Nicolson difference scheme for the time variable fractional mobile–immobile advection–dispersion equation. J. Appl. Math. Comput. 56, 391–410 (2018). https://doi.org/10.1007/s12190-016-1079-7
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DOI: https://doi.org/10.1007/s12190-016-1079-7
Keywords
- Variable order derivative
- Crank–Nicolson finite difference
- Mobile–immobile advection–dispersion equation
- Stability results
- Error estimates
- Numerical examples