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)
Numerical simulations of 2D fractional subdiffusion problems. (English) Zbl 1197.65143
Summary: The growing number of applications of fractional derivatives in various fields of science and engineering indicates that there is a significant demand for better mathematical algorithms for models with real objects and processes. Currently, most algorithms are designed for 1D problems due to the memory effect in fractional derivatives. In this work, the 2D fractional subdiffusion problems are solved by an algorithm that couples an adaptive time stepping and adaptive spatial basis selection approach. The proposed algorithm is also used to simulate a subdiffusion-convection equation.
MSC:
65M70Spectral, collocation and related methods (IVP of PDE)
35K05Heat equation
35R11Fractional partial differential equations
References:
[1]Podlubny, I.: Fractional differential equations, Mathematics in science and engineering 198 (1999) · Zbl 0924.34008
[2]X. Li, C. Xu, Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral approximation, Commun. Comput. Phys. To appear, doi:10.4208/cicp.020709.221209a.
[3]Gorenflo, R.; Mainardi, F.; Moretti, D.; Paradisi, P.: Time fractional diffusion: a discrete random walk approach, Nonlinear dyn. 29, No. 1 – 4, 129-143 (2002) · Zbl 1009.82016 · doi:10.1023/A:1016547232119
[4]Gorenflo, R.; Mainardi, F.; Vivoli, A.: Continuous-time random walk and parametric subordination in fractional diffusion, Chaos solitons fractals 34, No. 1, 87-103 (2007) · Zbl 1142.82363 · doi:10.1016/j.chaos.2007.01.052
[5]Metzler, R.; Klafter, J.: Boundary value problems for fractional diffusion equations, Phys. A 278, No. 1 – 2, 107-125 (2000)
[6]Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. rep. 339, No. 1, 77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[7]Al-Khaled, K.; Momani, S.: An approximate solution for a fractional diffusion-wave equation using the decomposition method, Appl. math. Comput. 165, No. 2, 473-483 (2005) · Zbl 1071.65135 · doi:10.1016/j.amc.2004.06.026
[8]Cuesta, E.; Lubich, C.; Palencia, C.: Convolution quadrature time discretization of fractional diffusion-wave equations, Math. comp. 75, No. 254, 673-696 (2006) · Zbl 1090.65147 · doi:10.1090/S0025-5718-06-01788-1
[9]Cuesta, E.; Palencia, C.: A numerical method for an integro – differential equation with memory in Banach spaces: qualitative properties, SIAM J. Numer. anal. 41, No. 4, 1232-1241 (2003) · Zbl 1054.65131 · doi:10.1137/S0036142902402481
[10]Deng, W.: Numerical algorithm for the time-fractional Fokker – Planck equation, J. comput. Phys. 227, No. 2, 1510-1522 (2007)
[11]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, No. 2, 719-736 (2005) · Zbl 1072.65123 · doi:10.1016/j.jcp.2004.11.025
[12]Odibat, Z.; Momani, S.: Approximate solutions for boundary-value problems of time-fractional wave equation, Appl. math. Comput. 181, No. 1, 767-774 (2006) · Zbl 1148.65100 · doi:10.1016/j.amc.2006.02.004
[13]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
[14]Odibat, Z.; Momani, S.: Numerical methods for nonlinear partial differential equations of fractional-order, Appl. math. Model. 32, No. 1, 28-39 (2008) · Zbl 1133.65116 · doi:10.1016/j.apm.2006.10.025
[15]Shen, S.; Liu, F.; Anh, V.; Turner, I.: Detailed analysis of a conservative difference approximation for the time-fractional diffusion equation, J. appl. Math. comput. 22, No. 3, 1-19 (2006) · Zbl 1111.65115 · doi:10.1007/BF02832034
[16]Sun, Z.; Wu, X.: A fully discrete difference scheme for a diffusion-wave system, Appl. numer. Math. 56, No. 2, 193-209 (2006) · Zbl 1094.65083 · doi:10.1016/j.apnum.2005.03.003
[17]Yuste, S. B.: Weighted average finite difference methods for fractional diffusion equations, J. comput. Phys. 216, No. 1, 264-274 (2006) · Zbl 1094.65085 · doi:10.1016/j.jcp.2005.12.006
[18]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
[19]Diethelm, K.; Ford, N. J.; Freed, A. D.; Luchko, Y.: Algorithms for the fractional calculus: a selection of numerical methods, Comput. methods appl. Mech. eng. 194, No. 6 – 8, 743-773 (2005) · Zbl 1119.65352 · doi:10.1016/j.cma.2004.06.006
[20]Valkó, P. P.; Abate, J.: Numerical inversion of 2D Laplace transforms applied to fractional diffusion equations, Appl. numer. Math. 53, No. 1, 73-88 (2005) · Zbl 1060.65681 · doi:10.1016/j.apnum.2004.10.002
[21]Diethelm, K.; Freed, A. D.: An efficient algorithm for the evaluation of convolution integrals, Comput. math. Appl. 51, No. 1, 51-72 (2006) · Zbl 1093.65022 · doi:10.1016/j.camwa.2005.07.010
[22]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 (2005)
[23]Ford, N. J.; Simpson, A. C.: The numerical solution of fractional differential equations: speed versus accuracy, Numer. algorithms 26, No. 4, 333-346 (2001) · Zbl 0976.65062 · doi:10.1023/A:1016601312158
[24]Tang, T.; Trummer, M. R.: Boundary layer resolving pseudospectral methods for singular perturbation problems, SIAM J. Sci. comput. 17, No. 2, 430-438 (1996) · Zbl 0851.65058 · doi:10.1137/S1064827592234120
[25]Ling, L.; Trummer, M. R.: Multiquadratic collocation method with integral formulation for boundary layer problems, Comput. math. Appl. 48, No. 5 – 6, 927-941 (2004) · Zbl 1066.65080 · doi:10.1016/j.camwa.2003.06.010
[26]Ling, L.; Trummer, M. R.: Adaptive multiquadric collocation for boundary layer problems, J. comp. Appl. math. 188, No. 2, 256-282 (2006) · Zbl 1086.65078 · doi:10.1016/j.cam.2005.04.018
[27]Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional-order, Elec. transact. Numer. anal. 5, 1-6 (1997) · Zbl 0890.65071 · doi:emis:journals/ETNA/vol.5.1997/pp1-6.dir/pp1-6.html
[28]Brunner, H.: Collocation methods for Volterra integral and related functional differential equations, Cambridge monographs on applied and computational mathematics 15 (2004)
[29]Brunner, H.; Pedas, A.; Vainikko, G.: Piecewise polynomial collocation methods for linear Volterra integro – differential equations with weakly singular kernels, SIAM J. Numer. anal. 39, No. 3, 957-982 (2001) · Zbl 0998.65134 · doi:10.1137/S0036142900376560
[30]Ling, L.; Schaback, R.: An improved subspace selection algorithm for meshless collocation methods, Int. J. Numer. methods eng. 80, No. 13, 1623-1639 (2009) · Zbl 1183.65153 · doi:10.1002/nme.2674
[31]Ling, L.; Opfer, R.; Schaback, R.: Results on meshless collocation techniques, Eng. anal. Bound. elem. 30, No. 4, 247-253 (2006) · Zbl 1195.65177 · doi:10.1016/j.enganabound.2005.08.008
[32]Ling, L.; Schaback, R.: Stable and convergent unsymmetric meshless collocation methods, SIAM J. Numer. anal. 46, No. 3, 1097-1115 (2008) · Zbl 1167.65059 · doi:10.1137/06067300X
[33]Kwok, T. O.; Ling, L.: On convergence of a least-squares kansa’s method for the modified Helmholtz equations, Adv. appl. Math. mech. 1, No. 3, 367-382 (2009)
[34]Lee, C. -F.; Ling, L.; Schaback, R.: On convergent numerical algorithms for unsymmetric collocation, Adv. comput. Math. 30, No. 4, 339-354 (2009) · Zbl 1168.65419 · doi:10.1007/s10444-008-9071-x
[35]Luchko, Y.: Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Comput. math. Appl. 59, No. 5, 1766-1772 (2010) · Zbl 1189.35360 · doi:10.1016/j.camwa.2009.08.015
[36]K. Sakamoto, M. Yamamoto, Initial value/boundary-value problems for fractional diffusion-wave equations and applications to some inverse problems, (preprint). · Zbl 1219.35367 · doi:10.1016/j.jmaa.2011.04.058
[37]Seybold, H. J.; Hilfer, R.: Numerical results for the generalized Mittag – Leffler function, Fract. calc. Appl. anal. 8, No. 2, 127-140 (2005) · Zbl 1123.33018