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Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation. (English) Zbl 1186.65128
Authors’ abstract: A Cauchy problem for the time fractional advection-dispersion equation (TFADE) is investigated. Such a problem is obtained from the classical advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α(0<α1). We show that the Cauchy problem of TFADE is severely ill-posed and further apply a spectral regularization method to solve it based on the solution given by the Fourier method. The convergence estimate is obtained under a priori bound assumptions for the exact solution. Numerical examples are given to show the effectiveness of the proposed numerical method.
65M30Improperly posed problems (IVP of PDE, numerical methods)
35R11Fractional partial differential equations
35R25Improperly posed problems for PDE
65M12Stability and convergence of numerical methods (IVP of PDE)
65M70Spectral, collocation and related methods (IVP of PDE)
[1]Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. rep. 339, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[2]Podlubny, I.: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in science and engineering 198 (1999) · Zbl 0924.34008
[3]Scalas, E.; Gorenflo, R.; Mainardi, F.: Fractional calculus and continuous-time finance, Physica A 284, No. 1–4, 376-384 (2000)
[4]Catania, F.; Massabò, M.; Paladino, O.: Estimation of transport and kinetic parameters using analytical solutions of the 2D advection-dispersion-reaction model, Environmetrics 17, No. 2, 199-216 (2006)
[5]Khalifa, M. E.: Some analytical solutions for the advection-dispersion equation, Appl. math. Comput. 139, No. 2–3, 299-310 (2003) · Zbl 1102.76334 · doi:10.1016/S0096-3003(02)00181-9
[6]Chen, S.; Liu, F.: ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation, J. appl. Math. comput. 26, No. 1–2, 295-311 (2008) · Zbl 1146.76037 · doi:10.1007/s12190-007-0013-4
[7]Ervin, V. J.; Roop, J. P.: Variational formulation for the stationary fractional advection dispersion equation, Numer. methods partial differential equations 22, No. 3, 558-576 (2006) · Zbl 1095.65118 · doi:10.1002/num.20112
[8]Ervin, V. J.; Roop, J. P.: Variational solution of fractional advection dispersion equations on bounded domains in rd, Numer. methods partial differential equations 23, No. 2, 256-281 (2007) · Zbl 1117.65169 · doi:10.1002/num.20169
[9]Huang, F.; Liu, F.: The fundamental solution of the space–time fractional advection-dispersion equation, J. appl. Math. comput. 18, No. 1–2, 339-350 (2005) · Zbl 1086.35003 · doi:10.1007/BF02936577
[10]Huang, F.; Liu, F.: The time fractional diffusion equation and the advection-dispersion equation, Anziam j. 46, No. 3, 317-330 (2005) · Zbl 1072.35218 · doi:10.1017/S1446181100008282
[11]Liu, F.; Anh, V. V.; Turner, I.; Zhuang, P.: Time fractional advection-dispersion equation, J. appl. Math. comput. 13, No. 1–2, 233-245 (2003) · Zbl 1068.26006 · doi:10.1007/BF02936089
[12]Liu, Q.; Liu, F.; Turner, I.; Anh, V.: Approximation of the Lévy–Feller advection-dispersion process by random walk and finite difference method, J. comput. Phys. 222, No. 1, 57-70 (2007) · Zbl 1112.65006 · doi:10.1016/j.jcp.2006.06.005
[13]Lu, X. Z.: Finite difference method for a time-fractional advection-dispersion equation, J. fuzhou univ. Nat. sci. Ed. 32, No. 4, 423-426 (2004) · Zbl 1087.65588
[14]Meerschaert, M. M.; Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations, J. comput. Appl. math. 172, No. 1, 65-77 (2004) · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033
[15]Momani, S.; Odibat, Z.: Numerical solutions of the space–time fractional advection-dispersion equation, Numer. methods partial differential equations 24, No. 6, 1416-1429 (2008) · Zbl 1148.76044 · doi:10.1002/num.20324
[16]Roop, J. P.: Numerical approximation of a one-dimensional space fractional advection-dispersion equation with boundary layer, Comput. math. Appl. 56, No. 7, 1808-1819 (2008) · Zbl 1152.76430 · doi:10.1016/j.camwa.2008.04.025
[17]Yong, Z.; Benson, D. A.; Meerschaert, M. M.; Scheffler, H. P.: On using random walks to solve the space-fractional advection-dispersion equations, J. stat. Phys. 123, No. 1, 89-110 (2006) · Zbl 1092.82038 · doi:10.1007/s10955-006-9042-x
[18]Berntsson, F.: A spectral method for solving the sideways heat equation, Inverse problems 15, No. 4, 891-906 (1999) · Zbl 0934.35201 · doi:10.1088/0266-5611/15/4/305
[19]Fu, C. L.; Dou, F. F.; Feng, X. L.; Qian, Z.: A simple regularization method for stable analytic continuation, Inverse problems 24, 1-15 (2008) · Zbl 1160.30023 · doi:10.1088/0266-5611/24/6/065003
[20]Fu, C. L.; Xiong, X. T.; Qian, Z.: Fourier regularization for a backward heat equation, J. math. Anal. appl. 331, No. 1, 472-480 (2007) · Zbl 1146.35420 · doi:10.1016/j.jmaa.2006.08.040
[21]Fu, P.; Fu, C. L.; Xiong, X. T.; Li, H. F.: Two regularization methods and the order optimal error estimates for a sideways parabolic equation, Comput. math. Appl. 49, No. 5–6, 777-788 (2005) · Zbl 1077.80005 · doi:10.1016/j.camwa.2004.08.012
[22]Qian, A. L.; Wu, Y. J.: A computational method for a Cauchy problem of Laplace’s equation, Appl. math. Comput. 207, No. 2, 478-485 (2009) · Zbl 1160.65049 · doi:10.1016/j.amc.2008.10.063
[23]Murio, D. A.: Time fractional IHCP with Caputo fractional derivatives, Comput. math. Appl. 56, No. 9, 2371-2381 (2008) · Zbl 1165.65386 · doi:10.1016/j.camwa.2008.05.015
[24]Murio, D. A.: Implicit finite difference approximation for time fractional diffusion equations, Comput. math. Appl. 56, No. 4, 1138-1145 (2008) · Zbl 1155.65372 · doi:10.1016/j.camwa.2008.02.015