A Fast Implicit Finite Difference Method for Fractional Advection-Dispersion Equations with Fractional Derivative Boundary Conditions
Fractional advection-dispersion equations, as generalizations of classical integer-order advection-dispersion equations, are used to model the transport of passive tracers carried by fluid flow in a porous medium. In this paper, we develop an implicit finite difference method for fractional advectio...
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2017-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2017/8716752 |
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| author | Taohua Liu Muzhou Hou |
| author_facet | Taohua Liu Muzhou Hou |
| author_sort | Taohua Liu |
| collection | DOAJ |
| description | Fractional advection-dispersion equations, as generalizations of classical integer-order advection-dispersion equations, are used to model the transport of passive tracers carried by fluid flow in a porous medium. In this paper, we develop an implicit finite difference method for fractional advection-dispersion equations with fractional derivative boundary conditions. First-order consistency, solvability, unconditional stability, and first-order convergence of the method are proven. Then, we present a fast iterative method for the implicit finite difference scheme, which only requires storage of O(K) and computational cost of O(KlogK). Traditionally, the Gaussian elimination method requires storage of O(K2) and computational cost of O(K3). Finally, the accuracy and efficiency of the method are checked with a numerical example. |
| format | Article |
| id | doaj-art-08b3175e8f1c4c2aba7bd0c8a1cc2baa |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
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| series | Advances in Mathematical Physics |
| spelling | doaj-art-08b3175e8f1c4c2aba7bd0c8a1cc2baa2025-02-03T01:28:16ZengWileyAdvances in Mathematical Physics1687-91201687-91392017-01-01201710.1155/2017/87167528716752A Fast Implicit Finite Difference Method for Fractional Advection-Dispersion Equations with Fractional Derivative Boundary ConditionsTaohua Liu0Muzhou Hou1School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, ChinaSchool of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, ChinaFractional advection-dispersion equations, as generalizations of classical integer-order advection-dispersion equations, are used to model the transport of passive tracers carried by fluid flow in a porous medium. In this paper, we develop an implicit finite difference method for fractional advection-dispersion equations with fractional derivative boundary conditions. First-order consistency, solvability, unconditional stability, and first-order convergence of the method are proven. Then, we present a fast iterative method for the implicit finite difference scheme, which only requires storage of O(K) and computational cost of O(KlogK). Traditionally, the Gaussian elimination method requires storage of O(K2) and computational cost of O(K3). Finally, the accuracy and efficiency of the method are checked with a numerical example.http://dx.doi.org/10.1155/2017/8716752 |
| spellingShingle | Taohua Liu Muzhou Hou A Fast Implicit Finite Difference Method for Fractional Advection-Dispersion Equations with Fractional Derivative Boundary Conditions Advances in Mathematical Physics |
| title | A Fast Implicit Finite Difference Method for Fractional Advection-Dispersion Equations with Fractional Derivative Boundary Conditions |
| title_full | A Fast Implicit Finite Difference Method for Fractional Advection-Dispersion Equations with Fractional Derivative Boundary Conditions |
| title_fullStr | A Fast Implicit Finite Difference Method for Fractional Advection-Dispersion Equations with Fractional Derivative Boundary Conditions |
| title_full_unstemmed | A Fast Implicit Finite Difference Method for Fractional Advection-Dispersion Equations with Fractional Derivative Boundary Conditions |
| title_short | A Fast Implicit Finite Difference Method for Fractional Advection-Dispersion Equations with Fractional Derivative Boundary Conditions |
| title_sort | fast implicit finite difference method for fractional advection dispersion equations with fractional derivative boundary conditions |
| url | http://dx.doi.org/10.1155/2017/8716752 |
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