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)
Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. (English) Zbl 1171.65101

The authors construct an explicit finite difference scheme for the approximate solution of the nonlinear diffusion equation of fractional order

u(x,t) t=B(x,t) x R α(x,t) u(x,t)+f(u,x,t),X a <X<X b ,0<t<T

with the initial and boundary conditions of usual form. The derivative of fractional order is considered in the generalized sense of Riesz. The approximate scheme can be written in matrix form

U j+1 =P j U j +B j +F j ·

The convergence and stability of this scheme are proved and some numerical examples are presented.

MSC:
65R20Integral equations (numerical methods)
45K05Integro-partial differential equations
45G10Nonsingular nonlinear integral equations
35K57Reaction-diffusion equations
26A33Fractional derivatives and integrals (real functions)
65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
References:
[1]B. Baeumer, M. Kovacs, M.M. Meerschaert, Fractional reaction – diffusion equation for species growth and dispersal, J. Math. Biology (2007). Available from: lt;http://www.maths.otago.ac.nz/ mcubed/JMBseed.pdfgt;.
[2]Chen, Chang-Ming; Liu, F.; Turner, I.; Anh, V.: 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
[3]Davis, L. C.: Model of magnetorheological elastomers, J. appl. Phys. 85, No. 6, 3342-3351 (1999)
[4]Del Castillo-Negrete, D.; Carreras, B. A.; Lynch, V. E.: Front dynamics in reaction – diffusion systems with Lévy flghts: a fractional diffusion approach, Phys. rev. Lett. 91, No. 1, 018302 (2003)
[5]Evans, K. P.; Jacob, N.: Feller semigroups obtained by variable order subordination, Rev. mat. Complut. 20, No. 2, 293-307 (2007) · Zbl 1153.47033
[6]Glockle, W. G.; Nonnenmacher, T. F.: A fractional calculus approach to self-similar protein dynamics, Biophys. J. 68, 46-53 (1995)
[7]Henry, B. I.; Wearne, S. L.: Fractional reaction – diffusion, Physica A 276, 448-455 (2000)
[8]Klass, D. L.; Martinek, T. W.: Electroviscous fluids. I. rheological properties, J. appl. Phys. 38, No. 1, 67-74 (1967)
[9]Leopold, H. G.: Embedding of function spaces of variable order of differentiation, Czech math. J. 49, 633-644 (1999) · Zbl 1008.46015 · doi:10.1023/A:1022483721944
[10]Lin, R.; Liu, F.: Fractional high order methods for the nonlinear fractional ordinary differential equation, Nonlinear anal. 6, 856-869 (2006) · Zbl 1118.65079 · doi:10.1016/j.na.2005.12.027
[11]Liu, F.; Anh, V.; Turne, I.: Numerical solution of space fractional Fokker – Planck equation, J. comput. Appl. math. 166, 209-219 (2004) · Zbl 1036.82019 · doi:10.1016/j.cam.2003.09.028
[12]Liu, F.; Anh, V.; Turner, I.; Zhuang, P.: Numerical simulation for solute transport in fractal porous media, Anziam j. 45, No. E, 461-473 (2004)
[13]Liu, Q.; Liu, F.; Turner, I.; Anh, V.: Approximation of the Lévy – Feller advection – dispersion process by random walk and finite difference method, J. phys. Comput. 222, 57-70 (2007) · Zbl 1112.65006 · doi:10.1016/j.jcp.2006.06.005
[14]Liu, F.; Shen, S.; Anh, V.; Turner, I.: Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation, Anziam j. 46, No. E, 488-504 (2005)
[15]C.F. Lorenzo, T.T. Hartley, Initialization, conceptualization and application in the generalized fractional calculus, NASA/TP-1998-208-208415, 1999.
[16]Lorenzo, C. F.; Hartley, T. T.: Variable-order and distributed order fractional operators, Nonlinear dyn. 29, 57-98 (2002) · Zbl 1018.93007 · doi:10.1023/A:1016586905654
[17]Meerschaert, M.; Tadjeran, C.: Finite difference approximations for fractional advection – dispersion flow equations, J. comput. Appl. math. 172, 65-77 (2004) · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033
[18]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[19]Oldham, K. B.; Spanier, J.: The fractional calculus, (1974)
[20]Podlubny, I.: Fractional differential equations, (1999)
[21]Roop, J. P.: Computational aspects of FEM approximation of fractional advection dispersion equation on bounded domains in R2, J. comput. Appl. math. 193, No. 1, 243-268 (2006) · Zbl 1092.65122 · doi:10.1016/j.cam.2005.06.005
[22]Ruiz-Medina, M. D.; Anh, V. V.; Angulo, J. M.: Fractional generalized random fields of variable order, Stochastic anal. Appl. 22, No. 2, 775-799 (2004) · Zbl 1069.60040 · doi:10.1081/SAP-120030456
[23]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003
[24]Seki, K.; Wojcik, M.; Tachiya, M.: Fractional reaction – diffusion equation, J. chem. Phys. 119, 2165-2174 (2003)
[25]Shen, S.; Liu, F.: Error analysis of an explicit finite difference approximation for the space fractional diffusion, Anziam j. 46, No. E, 871-887 (2005)
[26]Shiga, T.: Deformation and viscoelastic behavior of polymer gel in electric fields, Proceedings of the Japanese Academy, series B, phys. Biol. sci. 74, 6-11 (1998)
[27]Yu, Q.; Liu, F.; Anh, V.; Turner, I.: Solving linear and nonlinear space – time fractional reaction – diffusion equations by Adomian decomposition method, Int. J. Numer. meth. Eng. 74, 138-158 (2008) · Zbl 1159.76367 · doi:10.1002/nme.2165
[28]Zhuang, P.; Liu, F.: Implicit difference approximation for the time fractional diffusion equation, J. appl. Math. comput. 22, No. 3, 87-99 (2006) · Zbl 1140.65094 · doi:10.1007/BF02832039
[29]Zhuang, P.; Liu, F.; Anh, V.; Turner, I.: New solution and analytical techniques of the implicit numerical methods for the anomalous sub-diffusion equation, SIAM J. Numer. anal. 46, No. 2, 1079-1095 (2008) · Zbl 1173.26006 · doi:10.1137/060673114