zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. (English) Zbl 1228.65190
Summary: We study the time-space fractional order (fractional for simplicity) nonlinear subdiffusion and superdiffusion equations, which can relate the matter flux vector to concentration gradient in the general sense, describing, for example, the phenomena of anomalous diffusion, fractional Brownian motion, and so on. The semi-discrete and fully discrete numerical approximations are both analyzed, where the Galerkin finite element method for the space Riemann-Liouville fractional derivative with order $1+\beta \in [1,2]$ and the finite difference scheme for the time Caputo derivative with order $\alpha \in (0,1)$ (for subdiffusion) and $(1,2)$ (for superdiffusion) are analyzed, respectively. Results on the existence and uniqueness of the weak solutions, the numerical stability, and the error estimates are presented. Numerical examples are included to confirm the theoretical analysis. During our simulations, an interesting diffusion phenomenon of particles is observed, that is, on average, the diffusion velocity for $0<\alpha <1$ is slower than that for $\alpha =1$, but the diffusion velocity for $1<\alpha <2$ is faster than that for $\alpha =1$. For the spatial diffusion, we have a similar observation.

65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
35R11Fractional partial differential equations
26A33Fractional derivatives and integrals (real functions)
45K05Integro-partial differential equations
65M12Stability and convergence of numerical methods (IVP of PDE)
Full Text: DOI
[1] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993) · Zbl 0789.26002
[2] Dithelm, K.: An algorithm for the numeicial solution of differential equations of fractional order, Electron. trans. Numer. anal. 5, 1-6 (1997) · Zbl 0890.65071 · emis:journals/ETNA/vol.5.1997/pp1-6.dir/pp1-6.html
[3] Dithelm, K.; Ford, N. J.; Freed, A. D.: A predictor--corrector approach for the numerical solution of fractional differential equations, Nonlinear dynam. 29, 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341
[4] Li, C. P.; Dao, X. H.: On the fractional Adams method, Comput. math. Appl. 58, 1573-1588 (2009) · Zbl 1189.65142
[5] Li, C. P.; Chen, A.; Ye, J. J.: Numerical approaches to fractional calculus and fractional ordinary differential equation, J. comput. Phys. 230, 3352-3368 (2011) · Zbl 1218.65070 · doi:10.1016/j.jcp.2011.01.030
[6] Arora, H. L.; Abddwahidi, F. I.: Solution of non-integer order differential equations via the Adomian decomposition method, Appl. math. Lett. 6, 21-23 (1993) · Zbl 0772.34009 · doi:10.1016/0893-9659(93)90140-I
[7] George, A. J.; Chakrabarti, A.: The Adomian method applied to some extraordinary differential equations, Appl. math. Lett. 8, 91-97 (1993) · Zbl 0828.65081 · doi:10.1016/0893-9659(95)00036-P
[8] Jafari, H.; Momani, S.: Solving fractional diffusion and wave equations by modified homotopy perturbation method, Phys. lett. A 370, 388-396 (2007) · Zbl 1209.65111 · doi:10.1016/j.physleta.2007.05.118
[9] Momani, S.; Odibat, Z.; Erturk, V. S.: Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation, Phys. lett. A 370, 379-387 (2007) · Zbl 1209.35066 · doi:10.1016/j.physleta.2007.05.083
[10] Li, C. P.; Wang, Y. H.: Numerical algorithm based on Adomian decomposition for fractional differential equations, Comput. math. Appl. 57, 1672-1681 (2009) · Zbl 1186.65110 · doi:10.1016/j.camwa.2009.03.079
[11] Liu, F.; Anh, V.; Turner, I.; Zhuang, P.: Time fractional advection dispersion equation, J. appl. Math. comput. 13, 233-245 (2003) · Zbl 1068.26006 · doi:10.1007/BF02936089
[12] Meerschaert, M. M.; Tadjeran, Charles: Finite difference approxiamtion for two-sided space-fractional partial differential equations, J. appl. Math. 56, 80-90 (2006) · Zbl 1086.65087 · doi:10.1016/j.apnum.2005.02.008
[13] Yuste, S. B.; Acedo, L.: An explicit finite difference method and a new von numann-type stability analysis for fractional diffusion equations, SIAM J. Numer. anal. 42, 1862-1874 (2005) · Zbl 1119.65379 · doi:10.1137/030602666
[14] Liang, J. S.; Chen, Y. Q.: Hybrid symbolic and numerical simulation studies of time-fractional order wave-diffusion systems, Internat. J. Control 79, 1462-1470 (2006) · Zbl 1125.65364 · doi:10.1080/00207170600726493
[15] Tadjeran, C.; Meerschaert, M. M.: A second order accurate numerical method for the two-dimensional fractional diffusion equation, J. comput. Phys. 220, 813-823 (2007) · Zbl 1113.65124 · doi:10.1016/j.jcp.2006.05.030
[16] Tavazoei, M. S.; Haeri, M.; Bolouki, S.; Siami, M.: Stability preservation analysis for frequency-based methods in numerical simulation of fractional order systems, SIAM J. Numer. anal. 47, 321-338 (2008) · Zbl 1203.26012 · doi:10.1137/080715949
[17] Zhuang, P.; Liu, F.; Anh, V.; Turner, I.: New solution and analytical techniques of the implicit numerical method for the sub-diffusion equation, SIAM J. Numer. anal. 46, 1079-1095 (2008) · Zbl 1173.26006 · doi:10.1137/060673114
[18] Li, X. J.; Xu, C. J.: A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. anal. 47, 2108-2131 (2009) · Zbl 1193.35243 · doi:10.1137/080718942
[19] Fix, G. J.; Roop, J. P.: Least squares finite-element solution of a fractional order two-point boundary value problem, Comput. math. Appl. 48, 1017-1033 (2004) · Zbl 1069.65094 · doi:10.1016/j.camwa.2004.10.003
[20] Roop, J. P.: Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2, J. comput. Appl. math. 193, 243-268 (2006) · Zbl 1092.65122 · doi:10.1016/j.cam.2005.06.005
[21] Ervin, V. J.; Roop, J. P.: Variational formulation for the stationary fractional advection dispersion equation, Numer. methods partial differential equations 22, 558-576 (2006) · Zbl 1095.65118 · doi:10.1002/num.20112
[22] Ervin, V. J.; Heuer, N.; Roop, J. P.: Numerical approximation of a time dependent nonlinear, space-fractional diffusion equation, SIAM J. Numer. anal. 45, 572-591 (2007) · Zbl 1141.65089 · doi:10.1137/050642757
[23] Zhang, H.; Liu, F.; Anh, V.: Galerkin finite element approximations of symmetric space-fractional partial differential equations, Appl. math. Comput. 217, 2534-2545 (2010) · Zbl 1206.65234 · doi:10.1016/j.amc.2010.07.066
[24] Zheng, Y. Y.; Li, C. P.; Zhao, Z. G.: A note on the finite element method for the space-fractional advection diffusion equation, Comput. math. Appl. 59, 1718-1726 (2010) · Zbl 1189.65288 · doi:10.1016/j.camwa.2009.08.071
[25] Zheng, Y. Y.; Li, C. P.; Zhao, Z. G.: A fully discrete discontinuous Galerkin method for nonlinear fractional Fokker--Planck equation, Math. probl. Eng. 2010 (2010) · Zbl 1202.65157 · doi:10.1155/2010/279038
[26] Q. Liu, F. Liu, I. Turner, V. Anh, Finite element approximation for the modified anomalous subdiffusion process, Appl. Math. Modelling (2011), in press (doi:10.1016/j.apm.2011.02.036). · Zbl 1221.65257
[27] Wyss, W.: The fractional diffusion equation, J. math. Phys. 27, 2782-2785 (1986) · Zbl 0632.35031 · doi:10.1063/1.527251
[28] Gurtin, M. E.; Pipkin, A. C.: A general theory of heat conduction with finite wave speeds, Arch. ration. Mech. anal. 31, 113-126 (1968) · Zbl 0164.12901 · doi:10.1007/BF00281373
[29] Chen, P. J.; Gurtin, M. E.: On second sound in materials with memory, Z. angew. Math. phys. 21, 232-241 (1970) · Zbl 0218.73007 · doi:10.1007/BF01590647
[30] Moodi, T. B.; Tait, R. J.: On the thermal transients with finite wave speeds, Acta mech. 50, 97-104 (1983) · Zbl 0526.73129 · doi:10.1007/BF01170443
[31] Norwood, F. R.: Transient thermal waves in the general theory of heat conduction with finite wave speeds, J. appl. Mech. 39, 673-676 (1972)
[32] Green, A. E.; Naghdi, P. M.: Thermoelasticity without energy dissipation, J. elasticity 31, 189-208 (1993) · Zbl 0784.73009 · doi:10.1007/BF00044969
[33] Chandrasekharaiah, D. S.: Thermoelasticity with second sound: a review, Appl. mech. Rev. 39, 355-376 (1986) · Zbl 0588.73006 · doi:10.1115/1.3143705
[34] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[35] Povstenko, Y. Z.: Two-dimensional axisymmetric stresses exerted by instantaneous pulses and sources of diffusion in an infinite space in a case of time-fractional diffusion equation, Int. J. Solids structure 44, 2324-2348 (2007) · Zbl 1121.74022 · doi:10.1016/j.ijsolstr.2006.07.008
[36] Benson, D. A.; Wheatcraft, S. W.; Meerschaeert, M. M.: The fractional order governing equations of Lévy motion, Water resour. Res. 36, 1413-1423 (2000)
[37] Shlesinger, M. F.; West, B. J.; Klafter, J.: Lévy dynamics of enhanced diffusion: application to turbulence, Phys. rev. Lett. 58, 1100-1103 (1987)
[38] Biler, P.; Woyczyński, W. A.: Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. math. 59, 845-869 (1998) · Zbl 0940.35035 · doi:10.1137/S0036139996313447
[39] Sugimoto, N.: Burgers equation with a fractional derivative: hereditary effects on nonlinear acoustic waves, J. fluid mech. 225, 631-653 (1991) · Zbl 0721.76011 · doi:10.1017/S0022112091002203
[40] S.C. Samko, A.A. Kilbas, O.I. Maxitchev, Integrals and derivatives of the fractional order and some of their applications, Nauka i Tekhnika, Minsk, 1987 (in Russian). · Zbl 0617.26004
[41] Caputo, M.: Linear models of dissipation whose Q is almost frequency independent, part II, Geophys. J. R. astron. Soc. 13, 529-539 (1963)
[42] Heymans, N.; Podlubny, I.: Physical interpretation of initial conditons for frational differenial equations with Riemann--Liouville fractionsl derivatives, Rheol. acta 45, 765-772 (2006)
[43] Li, C. P.; Deng, W. H.: Remarks on fractional derivatives, Appl. math. Comput. 187, 777-784 (2007) · Zbl 1125.26009 · doi:10.1016/j.amc.2006.08.163
[44] Li, C. P.; Dao, X. H.; Guo, P.: Fractional derivatives in complex plane, Nonlinear anal. TMA 71, 1857-1869 (2009) · Zbl 1173.26305
[45] Lin, Y.; Xu, C. J.: Finite difference/spectral approximations for the time-fractional diffusion equation, J. comput. Phys. 225, 1533-1552 (2007) · Zbl 1126.65121 · doi:10.1016/j.jcp.2007.02.001
[46] Adams, R. A.: Sobolev space, (1975) · Zbl 0314.46030
[47] Brenner, S. C.; Scott, L. R.: The mathematical theory of finite element methods, (1994) · Zbl 0804.65101
[48] Deng, W. H.: Short memory principal and a predictor--corrector approach for fractional differential equations, J. comput. Appl. math. 206, 174-188 (2007) · Zbl 1121.65128 · doi:10.1016/j.cam.2006.06.008