A novel finite volume method for the Riesz space distributed-order diffusion equation. (English) Zbl 1384.65059
Summary: In recent years, considerable attention has been devoted to distributed-order differential equations mainly because they appear to be more effective for modelling complex processes which obey a mixture of power laws or flexible variations in space. In this paper, we propose a novel finite volume method (FVM) for a distributed-order space-fractional diffusion equation (FDE). Firstly, we use the mid-point quadrature rule to transform the space distributed-order diffusion equation into a multi-term fractional equation. Secondly, the transformed multi-term fractional equation is solved by discretising in space using the finite volume method and then in time using the Crank-Nicolson scheme. Thirdly, we prove that the Crank-Nicolson scheme with FVM is unconditionally stable and convergent with second order accuracy in both time and space. Finally, two numerical examples are presented to show the effectiveness of the numerical method. These methods and techniques can also be used to solve other types of fractional partial differential equations.
MSC:
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
| 35K05 | Heat equation |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
Keywords:
distributed-order equation; finite volume method; Riesz fractional derivative; Crank-Nicolson scheme; stability; convergence; space-fractional diffusion equation; numerical exampleReferences:
| [1] | Guo, B. L.; Pu, X. K.; Huang, F. H., Fractional Partial Differential Equations and their Numerical Solutions (2015), Science Press: Science Press Beijing · Zbl 1335.35001 |
| [2] | Kilbas, A. A.; Srivastave, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier B.V.: Elsevier B.V. Amsterdam · Zbl 1092.45003 |
| [3] | Liu, F.; Anh, V.; Turner, I., Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166, 209-219 (2004) · Zbl 1036.82019 |
| [4] | Liu, F.; Zhuang, P.; Anh, V.; Turner, I.; Burrage, K., Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput., 191, 12-20 (2007) · Zbl 1193.76093 |
| [5] | Liu, F.; Zhuang, P.; Liu, Q., Numerical Methods of Fractional Partial Differential Equations and Applications (2015), Science Press: Science Press Beijing |
| [6] | Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010 |
| [7] | Yu, B.; Jiang, X. Y.; Xu, H. Y., A novel compact numerical method for solving the two-dimensional non-linear fractional reaction-subdiffusion equation, Numer. Algorithms, 68, 923-950 (2015) · Zbl 1314.65114 |
| [8] | Yu, B.; Jiang, X. Y.; Wang, C., Numerical algorithms to estimate relaxation parameters and Caputo fractional derivative for a fractional thermal wave model in spherical composite medium, Appl. Math. Comput., 274, 106-118 (2016) · Zbl 1410.80021 |
| [9] | Soklov, I. M.; Chechkin, A. V.; Klafter, J., Distributed-order fractional kinetics, Acta Phys. Polon. B, 35, 1323-1341 (2004) |
| [10] | Caputo, M., Distributed order differential equations modelling dielectric induction and diffuion, Fract. Calc. Appl. Anal., 4, 421-442 (2001) · Zbl 1042.34028 |
| [11] | Caputo, M., Diffusion with space memory modelled with distributed order space fractional differential equations, Ann. Geophys., 46, 223-234 (2003) |
| [12] | Chechkin, A. V.; Gorenflo, R.; Sokolov, I. M., Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E, 66, 046-129 (2002) |
| [13] | Jiao, Z.; Chen, Y.; Podlubny, I., Distributed-Order Dynamic Systems: Stability, Simulation, Applications and Perspectives (2012), Springer · Zbl 1401.93005 |
| [14] | Lorenzo, C. F.; Hartley, T. T., Variable order and distributed order fractional operators, Nonlinear Dynam., 29, 57-98 (2002) · Zbl 1018.93007 |
| [15] | Naber, M., Distributed order fractional subdiffusion, Fractals, 12, 23-32 (2004) · Zbl 1083.60066 |
| [16] | Umarov, S., Continuous time random walk models associated with distributed order diffusion equations, Fract. Calc. Appl. Anal., 18, 821-837 (2015) · Zbl 1319.60096 |
| [17] | Meerschaert, M. M.; Nane, E.; Vellaisamy, P., Distributed-order fractional diffusions on bounded domains, J. Math. Anal. Appl., 379, 216-228 (2011) · Zbl 1222.35204 |
| [18] | Gorenflo, R.; Luchko, Y.; Stojanović, M., Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density, Fract. Calc. Appl. Anal., 16, 297-316 (2013) · Zbl 1312.35179 |
| [19] | Luchko, Y., Boundary value problems for the generalized time-fractional diffusion equation of distributed order, Fract. Calc. Appl. Anal., 12, 409-422 (2009) · Zbl 1198.26012 |
| [20] | Atanacovic, T. M.; Pilipovic, S.; Zorica, D., Existence and calculation of the solution to the time distributed order diffusion equation, Phys. Scr., 014012 (2009) |
| [21] | Ye, H.; Liu, F.; Anh, V.; Turner, I., Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains, IMA J. Appl. Math., 80, 825-838 (2015) · Zbl 1337.65120 |
| [22] | Ye, H.; Liu, F.; Anh, V., Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains, J. Comput. Phys., 98, 652-660 (2015) · Zbl 1349.65353 |
| [23] | Hu, X.; Liu, F.; Anh, V.; Turner, I., A numerical investigation of the time distributed-order diffusion mode, ANZIAM J., 55, 464-478 (2013) · Zbl 1390.65133 |
| [24] | Hu, X.; Liu, F.; Turner, I.; Anh, V., An implicit numerical method for a new time distributed-order and two-sided space-fractional advection-dispersion equation, Numer. Algorithms, 72, 393-407 (2016) · Zbl 1343.65110 |
| [25] | Gao, G. H.; Sun, Z. Z., Two alternating direction implicit difference schemes with the extrapolation method for the two-dimensional distributed-order differential equations, Comput. Math. Appl., 69, 926-948 (2015) · Zbl 1443.65124 |
| [26] | Gao, G. H.; Sun, H. W.; Sun, Z. Z., Some high-order difference schemes for the distributed-order differential equations, J. Comput. Phys., 298, 337-359 (2015) · Zbl 1349.65296 |
| [27] | Ford, N. J.; Morgado, M. L.; Rebelo, M., An implicit finite difference approximation for the solution of the diffusion equation with distributed order in time, Electron. Trans. Numer. Anal., 44, 289-305 (2015) · Zbl 1330.65130 |
| [28] | Alikhanov, A. A., Numerical methods of solutions of boundary value problems for the multi-term variable-distributed order diffusion equation, Appl. Math. Comput., 268, 12-22 (2015) · Zbl 1410.65294 |
| [29] | Morgadoa, M. L.; Rebelo, M., Numerical approximation of distributed order reaction-diffusion equations, J. Comput. Appl. Math., 275, 216-227 (2015) · Zbl 1298.35242 |
| [30] | Wang, X.; Liu, F.; Chen, X., Novel second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations, Adv. Math. Phys. (2015) · Zbl 1380.65188 |
| [31] | Zhang, X.; Crawford, J. W.; Deeks, L. K.; Shutler, M. I.; Bengough, A. G.; Young, I. M., A mass balance based numerical method for the fractional advection-dispersion equation: theory and application, Water Resour. Res., 41, 1-10 (2005) |
| [33] | Samko, S. G.; Ross, B., Integration and differentiation to a variable fractional order, Integral Transform. Spec. Funct., 1, 277-300 (1993) · Zbl 0820.26003 |
| [34] | Saichev, A. I.; Zaslavsky, G. M., Fractional kinetic equations: solutions and applications, Chaos, 7, 753-764 (1997) · Zbl 0933.37029 |
| [35] | Ervin, V. J.; Roop, J. P., Variational solution of fractional advection dispersion equations on bounded domains in \(R^d\), Numer. Methods Partial Differential Equations, 23, 256-281 (2007) · Zbl 1117.65169 |
| [36] | Jiang, H.; Liu, F.; Turner, I.; Burrage, K., Analyticall solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain, Comput. Math. Appl., 64, 3377-3388 (2012) · Zbl 1268.35124 |
| [37] | Liu, F.; Meerschaert, M. M.; Mcgough, R. J.; Zhuang, P.; Liu, Q., Numerical methods for solving the multi-term time fractional wave-diffusion equation, Fract. Calc. Appl. Anal., 16, 9-25 (2013) · Zbl 1312.65138 |
| [38] | Feng, L. B.; Zhuang, P.; Liu, F.; Turner, I., Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation, Appl. Math. Comput., 257, 52-65 (2015) · Zbl 1339.65144 |
| [39] | Li, J.; Liu, F.; Feng, L.; Turner, I., A novel finite volume method for the Riesz space distributed-order advection-diffusion equation, Appl. Math. Model., 46, 536-553 (2017) · Zbl 1443.65162 |
| [40] | Liu, F.; Zhuang, P.; Turner, I.; Burrage, K.; Anh, V., A new fractional finite volume method for solving the fractional diffusion equation, Appl. Math. Model., 38, 3871-3878 (2014) · Zbl 1429.65213 |
| [41] | Zhuang, P.; Liu, F.; Turner, I.; Gu, Y. T., Finite volume and finite element methods for solving a one-dimensional space-fractional Boussinesq equation, Appl. Math. Model., 38, 3860-3870 (2014) · Zbl 1429.65233 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
