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A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions. (English) Zbl 1242.65160
Summary: One-dimensional fractional anomalous sub-diffusion equations on an unbounded domain are considered. Beginning with the derivation of the exact artificial boundary conditions, the original problem on an unbounded domain is converted into mainly solving an initial-boundary value problem on a finite computational domain. The main contribution of our work, as compared with the previous work, lies in the reduction of fractional differential equations on an unbounded domain by using artificial boundary conditions and construction of the corresponding finite difference scheme with the help of the method of order reduction. The difficulty is the treatment of the Neumann condition on the artificial boundary, which involves the time-fractional derivative operator. The stability and convergence of the scheme are proven using the discrete energy method. Two numerical examples clarify the effectiveness and accuracy of the proposed method.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
35R11 Fractional partial differential equations
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[1] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004
[2] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[3] Kilbas, A.; Srivastava, H.; Trujillo, J., Theory and applications of fractional differential equations, (2006), Elsevier Science and Technology Boston · Zbl 1092.45003
[4] Schot, A.; Lenzi, M.K.; Evangelista, L.R.; Malacarne, L.C.; Mendes, R.S.; Lenzi, E.K., Fractional diffusion equation with an absorbent term and a linear external force: exact solution, Phys. lett. A, 366, 346-350, (2007)
[5] Agrawal, O.P., A general solution for a fourth-order fractional diffusion-wave equation defined in a bounded domain, Comput. struct., 79, 1479-1501, (2001)
[6] Agrawal, O.P., Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear dyn., 29, 145-155, (2002) · Zbl 1009.65085
[7] 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
[8] Abu-Saman, A.M.; Assaf, A.M., Stability and convergence of Crank-Nicolson method for fractional advection dispersion equation, Adv. appl. math. anal., 2, 117-125, (2007) · Zbl 1178.65145
[9] Tadjeran, C.; Meerschaert, M.M.; Scheffler, H.P., A second-order accurate numerical approximation for the fractional diffusion equation, J. comput. phys., 213, 205-213, (2006) · Zbl 1089.65089
[10] Wang, K.X.; Wang, H., A fast characteristic finite difference method for fractional advection-diffusion equations, Adv. water resour., 34, 810-816, (2011)
[11] Yuste, S.B.; Acedo, L., An explicit finite difference method and a new von-Neumann-type stability analysis for fractional diffusion equations, SIAM J. numer. anal., 42, 1862-1874, (2005) · Zbl 1119.65379
[12] Yuste, S.B., An explicit difference method for solving fractional diffusion and diffusion-wave equations in the Caputo form, J. comput. nonlinear dyn., 6, (2011)
[13] Murillo, J.Q.; Yuste, S.B., On three explicit difference schemes for fractional diffusion and diffusion-wave equations, Phys. scr., (2009)
[14] Meerschaert, M.M.; Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. comput. appl. math., 172, 65-77, (2004) · Zbl 1126.76346
[15] Lynch, V.; Carreras, B.; Castillo-Negrete, D.; Ferreira-Mejias, K.; Hicks, H., Numerical methods for the solution of partial differential equations of fractional order, J. comput. phys., 192, 406-421, (2003) · Zbl 1047.76075
[16] Chen, C.; Liu, F.; Turner, I.; Anh, V., A Fourier method for the fractional diffusion equation describing sub-diffusion, J. comput. phys., 227, 886-897, (2007) · Zbl 1165.65053
[17] Chen, S.; Liu, F.; Zhuang, P.; Anh, V., Finite difference approximations for the fractional Fokker-Planck equation, Appl. math. model., 33, 256-273, (2009) · Zbl 1167.65419
[18] Zhuang, P.; Liu, F., Implicit difference approximation for the time fractional diffusion equation, J. appl. math. comput., 22, 87-99, (2006) · Zbl 1140.65094
[19] Liu, F.; Zhuang, P.; Anh, V.; Turner, I., A fractional-order implicit difference approximation for the space-time fractional diffusion equation, Anziam j., 47, C48-C68, (2006)
[20] Zhuang, P.; Liu, F.; Anh, V.; Turner, I., New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. numer. anal., 46, 1079-1095, (2008) · Zbl 1173.26006
[21] Chen, C.; Liu, F.; Turner, I.; Anh, V., Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation, SIAM J. sci. comput., 32, 1740-1760, (2010) · Zbl 1217.26011
[22] Liu, Q.; Gu, Y.; Zhang, P.; Liu, F.; Nie, Y., An implicit RBF meshless approach for time fractional diffusion equations, Comput. mech., 48, 1-12, (2011) · Zbl 1377.76025
[23] Sun, Z.Z.; Wu, X., A fully discrete difference scheme for a diffusion-wave system, Appl. numer. math., 56, 193-209, (2006) · Zbl 1094.65083
[24] Langlands, T.; Henry, B., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. comput. phys., 205, 719-736, (2005) · Zbl 1072.65123
[25] Cui, M., Compact finite difference method for the fractional diffusion equation, J. comput. phys., 228, 7792-7804, (2009) · Zbl 1179.65107
[26] Du, R.; Cao, W.; Sun, Z.Z., A compact difference scheme for the fractional diffusion-wave equation, Appl. math. model., 34, 2998-3007, (2010) · Zbl 1201.65154
[27] Gao, G.H.; Sun, Z.Z., A compact finite difference scheme for the fractional sub-diffusion equations, J. comput. phys., 230, 586-595, (2011) · Zbl 1211.65112
[28] Roop, J.P., Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in \(\mathbb{R}^2\), J. comput. appl. math., 193, 243-268, (2006) · Zbl 1092.65122
[29] Deng, W., Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. numer. anal., 47, 204-226, (2008) · Zbl 1416.65344
[30] Gorenflo, R.; Mainardi, F.; Moretti, D.; Paradisi, P., Time fractional diffusion: A discrete random walk approach, Nonlinear dyn., 29, 129-143, (2002) · Zbl 1009.82016
[31] Li, X.; Xu, C., A space-time spectral method for the time fractional diffusion equation, SIAM J. numer. anal., 47, 2108-2131, (2009) · Zbl 1193.35243
[32] Li, X.; Xu, C., Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. comput. phys., 8, 1016-1051, (2010) · Zbl 1364.35424
[33] Odibat, Z., Rectangular decomposition method for fractional diffusion-wave equations, Appl. math. comput., 179, 92-97, (2006) · Zbl 1100.65125
[34] El-Sayed, A.; Behiry, S.; Raslan, W., Adomian’s decomposition method for solving an intermediate fractional advection-dispersion equation, Comput. math. appl., 59, 1759-1765, (2010) · Zbl 1189.35358
[35] Momani, S.; Odibat, Z., Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. lett. A, 365, 345-350, (2007) · Zbl 1203.65212
[36] Yildirim, A., Analytical approach to Fokker-Planck equation with space- and time-fractional derivatives by means of the homotopy perturbation method, J. King saud univ. (science), 22, 257-264, (2010)
[37] Kumar, P.; Agrawal, O.P., An approximate method for numerical solution of fractional differential equations, Signal process., 86, 2602-2610, (2006) · Zbl 1172.94436
[38] Jiang, W.; Lin, Y., Approximate solution of the fractional advection-dispersion equation, Comput. phys. commun., 181, 557-561, (2010) · Zbl 1210.65168
[39] Odibat, Z.; Momani, S., The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Comput. math. appl., 58, 2199-2208, (2009) · Zbl 1189.65254
[40] Lin, X.; Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. comput. phys., 225, 1533-1552, (2007) · Zbl 1126.65121
[41] Lin, Y.; Li, X.; Xu, C., Finite dfiference/specrtal approximations for the fractional cable equation, Math. comput., 80, 1369-1396, (2011) · Zbl 1220.78107
[42] Benchohra, M.; Hamidi, N., Fractional order differential inclusions on the half-line, Surv. math. appl., 5, 99-111, (2010) · Zbl 1413.34010
[43] Agarwal, R.P.; Benchohra, M.; Hamani, S.; Pinelas, S., Boundary value problems for differential equations involving Riemann-Liouville fractional derivative on the half-line, Dyn. contin. discrete impuls. syst. ser. A math. anal., 18, 235-244, (2011) · Zbl 1208.26012
[44] Arara, A.; Benchohra, M.; Hamidi, N.; Nieto, J.J., Fractional order differential equations on an unbounded domain, Nonlinear anal., 72, 580-586, (2010) · Zbl 1179.26015
[45] F. Huang, Analytical solution for the time-fractional telegraph equation, J. Appl. Math. 2009, Article ID: 890158, p. 9, doi: 10.1155/2009/890158. · Zbl 1190.35224
[46] Langlands, T.; Henry, B.; Wearne, S., Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions, J. math. biol., 59, 761-808, (2009) · Zbl 1232.92037
[47] Valkó, P.P.; Zhang, X.H., Finite domain anomalous spreading consistent with first and second laws, Commun. nonlinear sci. numer. simulat., 15, 3455-3470, (2010)
[48] Mainardi, F., The fundamental solutions for the fractional diffusion-wave equation, Appl. math. lett., 9, 23-28, (1996) · Zbl 0879.35036
[49] Gorenflo, R.; Luchko, Y.; Mainardi, F., Analytical properties and applications of the wright function, Fract. calc. appl. anal., 2, 383-414, (1999) · Zbl 1027.33006
[50] Han, H.D.; Wu, X.N., Approximation of infinite boundary condition and its application to finite element methods, J. comput. math., 3, 179-192, (1985) · Zbl 0579.65111
[51] Han, H.D.; Huang, Z.Y., A class of artificial boundary conditions for heat equation in unbounded domains, Comput. math. appl., 43, 889-900, (2002) · Zbl 0999.65086
[52] Wu, X.N.; Sun, Z.Z., Convergence of difference scheme for heat equation in unbounded domains using artificial boundary conditions, Appl. numer. math., 50, 261-277, (2004) · Zbl 1053.65074
[53] Han, H.D.; Huang, Z.Y., Exact and approximating boundary conditions for the parabolic problems on unbounded domains, Comput. math. appl., 44, 655-666, (2002) · Zbl 1030.35092
[54] Han, H.D.; Yin, D.S., Numerical solutions of parabolic problems on unbounded 3-d spatial domain, J. comput. math., 23, 449-462, (2005) · Zbl 1102.65091
[55] Han, H.D.; Jin, J.C.; Wu, X.N., A finite-difference method for the one-dimensional time-dependent Schrödinger equation on unbounded domain, Comput. math. appl., 50, 1345-1362, (2005) · Zbl 1092.65071
[56] Sun, Z.Z.; Wu, X., The stability and convergence of a difference scheme for the Schrödinger equation on an infinite domain by using artificial boundary conditions, J. comput. phys., 214, 209-223, (2006) · Zbl 1094.65088
[57] Sun, Z.Z., The stability and convergence of an explicit difference scheme for the Schrödinger equation on an infinite domain by using artificial boundary conditions, J. comput. phys., 219, 879-898, (2006) · Zbl 1175.65105
[58] Antoine, X., Artificial boundary conditions for one-dimensional cubic nonlinear Schrödinger equations, SIAM J. sci. comput., 43, 2272-2293, (2006) · Zbl 1109.35102
[59] Antoine, X.; Arnold, A.; Besse, C.; Ehrhardt, M.; Schädle, A., A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. comput. phys., 4, 729-796, (2008) · Zbl 1364.65178
[60] Nazarov, S.A.; Specovius-Neugebauer, M., Artificial boundary conditions for the Stokes and Navier-Stokes equations in domains that are layer-like at infinity, Z. anal. anwend., 27, 125-155, (2008) · Zbl 1144.35044
[61] Han, H.D.; Zhu, L.; Brunner, H.; Ma, J.T., The numerical solution of parabolic Volterra integro-differential equations on unbounded spatial domains, Appl. numer. math., 55, 83-99, (2005) · Zbl 1078.65126
[62] Han, H.D.; Zhu, L.; Brunner, H.; Ma, J.T., Artificial boundary conditions for parabolic Volterra integro-differential equations on unbounded two-dimensional domains, J. comput. appl. math., 197, 406-420, (2006) · Zbl 1104.65125
[63] Han, H.D.; Zhang, Z.W., An analysis of the finite-difference method for one-dimensional Klein-Gordon equation on unbounded domain, Appl. numer. math., 59, 1568-1583, (2009) · Zbl 1162.65377
[64] Dorodnicyn, L.W., Artificial boundary conditions for high-accuracy aeroacoustic algorithms, SIAM J. sci. comput., 32, 1950-1979, (2010) · Zbl 1225.35014
[65] Tsynkov, S.V., Numerical solution of problems on unbounded domains, A review, Appl. numer. math., 27, 465-532, (1998) · Zbl 0939.76077
[66] Han, H.D., The artificial boundary method - numerical solutions of partial differential equations on unbounded domains, (), 33-58 · Zbl 1097.65115
[67] Sun, Z.Z., The method of order reduction and its application to the numerical solutions of partial differential equations, (2009), Science Press Beijing
[68] Sun, Z.Z., Numerical methods of partial differential equations (in Chinese), (2005), Science Press Beijing
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