zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions. (English) Zbl 1227.65075

Fractional differential equations and their novel numerical schemes are investigated. These equations play an important role in applications in physics, chemistry, engineering, finance and other sciences. The model presented here is of one space dimension with fractional sub-diffusion equation with the Riemann-Liouville operator and with Neumann boundary conditions. For deriving new numerical schemes, the authors choose the methodology of order reduction and transform the original problem into an equivalent system of lower order differential equations. For this system a construction of a so-called box-type numerical scheme is done using classical finite differences. Due to the fact that the fractional derivative operator is nonlocal, the authors call it box-type scheme.

First, a new discrete inner product, L 2 norm, H 1 semi-norm and maximum norm are defined. Then the stability estimations for the numerical solution are proved, using the novel technique. Finally, under some smoothing conditions on the continuous solution, and for 0<α<1 , which represents the fractional derivative of the type 1-α, the error estimate of the type C(τ 2-α +h 2 ) in the maximum norm is proved for τ and h time and space grid mesh size, respectively. Two numerical test examples, which confirm the accuracy and efficiency of the proposed scheme, conclude the paper.

MSC:
65M06Finite difference methods (IVP of PDE)
65M15Error bounds (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
35R11Fractional partial differential equations
References:
[1]Schumer, R.; Benson, D. A.; Meerschaert, M. M.; Baeumer, B.: Multiscaling fractional advection-dispersion equations and their solutions, Water resour. Res. 39, 1022-1032 (2003)
[2]S. Westerlund, Causality, No. 940426, University of Kalmar, 1994.
[3]Blumen, A.; Zumofen, G.; Klafter, J.: Transport aspects in anomalous diffusion: Lévy walks, Phys. rev. A 40, 3964-3973 (1989)
[4]Meerschaert, M. M.; Benson, D.; Baeumer, B.: Operator Lévy motion and multiscaling anomalous diffusion, Phys. rev. E 63, 1112-1117 (2001)
[5]Meerschaert, M. M.; Scheffler, H. P.: Semistable Lévy motion, Frac. cale. Appl. anal. 5, 27-54 (2002) · Zbl 1032.60043
[6]R. Gorenflo, F. Mainardi, E. Scalas, M. Raberto, Fractional calculus and continuous-time finance. III. The diffusion limit, Mathematical Finance (Konstanz, 2000), Trends in Math., Birkhuser, Basel, 2001, pp. 171 – 180. · Zbl 1138.91444
[7]Raberto, M.; Scalas, E.; Mainardi, F.: Waiting-times and returns in high-frequency financial data: an empirical study, Physica A 314, 749-755 (2002) · Zbl 1001.91033 · doi:10.1016/S0378-4371(02)01048-8
[8]Diethelm, K.; Freed, A. D.: On the solution of nonlinear fractional order differential equations used in the modelling of viscoplasticity, Reaction engineering and molecular properties, 217-224 (1999)
[9]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
[10]Podlubny, I.: Fractional differential equations, (1999)
[11]Yuste, S.; Acedo, L.: An explicit finite difference method and a new Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. anal. 42, No. 5, 1862-1874 (2005) · Zbl 1119.65379 · doi:10.1137/030602666
[12]Yuste, S.: Weighted average finite difference methods for fractional diffusion equations, J. comput. Phys. 216, 264-274 (2006) · Zbl 1094.65085 · doi:10.1016/j.jcp.2005.12.006
[13]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 · doi:10.1016/j.jcp.2007.05.012
[14]Murio, D.: Implicit finite difference approximation for time fractional diffusion equations, Comput. math. Appl. 56, 1138-1145 (2008) · Zbl 1155.65372 · doi:10.1016/j.camwa.2008.02.015
[15]Cui, M.: Compact finite difference method for the fractional diffusion equation, J. comput. Phys. 228, 7792-7804 (2009) · Zbl 1179.65107 · doi:10.1016/j.jcp.2009.07.021
[16]Chen, C.; Liu, F.; Anh, V.; Turner, I.: Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation, SIAM J. Sci. comput. 32, 1740-1760 (2010) · Zbl 1217.26011 · doi:10.1137/090771715
[17]Chen, C.; Liu, F.; Burrage, K.: Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation, Appl. math. Comput. 198, 754-769 (2008) · Zbl 1144.65057 · doi:10.1016/j.amc.2007.09.020
[18]Deng, W.: Numerical algorithm for the time-fractional Fokker – Planck equation, J. comput. Phys. 227, No. 2, 1510-1522 (2007)
[19]Odibat, Z.; Momani, S.: Numerical solution of Fokker – Planck equation with space- and time-fractional derivatives, Phys. lett. A 369, 349-358 (2007) · Zbl 1209.65114 · doi:10.1016/j.physleta.2007.05.002
[20]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 · doi:10.1016/j.apm.2007.11.005
[21]Cuesta, E.; Lubich, C.; Palencia, C.: Convolution quadrature time discretization of fractional diffusion-wave equations, Math. comput. 75, No. 254, 673-696 (2006) · Zbl 1090.65147 · doi:10.1090/S0025-5718-06-01788-1
[22]Zhuang, P.; Liu, F.: Implicit difference approximation for the two-dimensional space-time fractional diffusion equation, J. appl. Math. comput. 25, No. 1 – 2, 269-282 (2007) · Zbl 1144.65090 · doi:10.1007/BF02832352
[23]Brunner, H.; Ling, L.; Yamamoto, M.: Numerical simulations of 2D fractional subdiffusion problems, J. comput. Phys. 229, 6613-6622 (2010) · Zbl 1197.65143 · doi:10.1016/j.jcp.2010.05.015
[24]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, No. 2, 1079-1095 (2008) · Zbl 1173.26006 · doi:10.1137/060673114
[25]Zhuang, P.; Liu, F.; Anh, V.; Turner, I.: Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process, IMA J. Appl. math. 74, 6445-6467 (2009) · Zbl 1187.35271 · doi:10.1093/imamat/hxp015
[26]Liu, F.; Yang, C.; Burrage, K.: Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term, J. comput. Appl. math. 231, 160-176 (2009) · Zbl 1170.65107 · doi:10.1016/j.cam.2009.02.013
[27]Oldham, K.; Spanier, J.: The fractional calculus: theory and applications of differentiation and integration to arbitrary order, Mathematics in science and engineering (1974)
[28]Sun, Z. Z.; Wu, X. N.: A fully discrete difference scheme for a diffusion-wave system, Appl. numer. Math. 56, 193-209 (2006) · Zbl 1094.65083 · doi:10.1016/j.apnum.2005.03.003
[29]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 · doi:10.1016/j.apm.2010.01.008
[30]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 · doi:10.1016/j.jcp.2010.10.007
[31]Langlands, T. A. M.; Henry, B. I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. comput. Phys. 205, 719-736 (2005) · Zbl 1072.65123 · doi:10.1016/j.jcp.2004.11.025
[32]Keller, H. B.: A new difference scheme for parabolic problems, Numerical solution of partial differential equation II (1971)
[33]Sun, Z. Z.: A second-order accurate linearized difference scheme for the two-dimensional Cahn – Hilliard equation, Math. comput. 64, 1463-1471 (1995) · Zbl 0847.65056 · doi:10.2307/2153365
[34]Sun, Z. Z.; Zhu, Y. L.: A second-order accurate difference scheme for the heat equation with concentrated capacity, Numer. math. 97, 379-395 (2004) · Zbl 1060.65097 · doi:10.1007/s00211-003-0462-0
[35]Sun, Z. Z.: The method of order reduction and its application to the numerical solutions of partial differential equations, (2009)