메뉴 건너뛰기




Volumn 22, Issue 4, 2012, Pages

Finite difference schemes for variable-order time fractional diffusion equation

Author keywords

anomalous diffusion; Crank Nicholson scheme; explicit scheme; Fourier method; implicit scheme; Variable order fractional derivative

Indexed keywords

FINITE DIFFERENCE METHOD; NONLINEAR EQUATIONS; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; POROUS MATERIALS;

EID: 84861204912     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S021812741250085X     Document Type: Article
Times cited : (138)

References (48)
  • 1
    • 0020765202 scopus 로고
    • A theoretical basis for the application of fractional calculus to viscoelasticity
    • Bagley, R L & Torvik, P. J. [1983] "A theoretical basis for the application of fractional calculus to viscoelasticity," J. Rheol. 27, 201-210
    • (1983) J. Rheol. , vol.27 , pp. 201-210
    • Bagley, R.L.1    Torvik, P.J.2
  • 2
    • 0038103148 scopus 로고    scopus 로고
    • Transport in a flashing ratchet in the presence of anomalous diffusion
    • Bao, J D. [2003] "Transport in a flashing ratchet in the presence of anomalous diffusion," Phys. Lett. A 314, 203-208
    • (2003) Phys. Lett. A , vol.314 , pp. 203-208
    • Bao, J.D.1
  • 5
    • 0036506044 scopus 로고    scopus 로고
    • Discretization schemes for fractional-order differentiators and integrators
    • Chen, Y. Q & Moore, K. L. [2002] "Discretization schemes for fractional-order differentiators and integrators," IEEE Trans. Circ. Syst. I: Fund. Theoret. Appl. 49, 363-367
    • (2002) IEEE Trans. Circ. Syst. I: Fund. Theoret. Appl. , vol.49 , pp. 363-367
    • Chen, Y.Q.1    Moore, K.L.2
  • 6
    • 33745725683 scopus 로고    scopus 로고
    • A speculative study of 2/3-order fractional Laplacian modeling of turbulence: Some thoughts and conjectures
    • Chen, W. [2006] "A speculative study of 2/3-order fractional Laplacian modeling of turbulence: Some thoughts and conjectures," Chaos 16, 023126
    • (2006) Chaos , vol.16 , pp. 023126
    • Chen, W.1
  • 7
    • 36149001420 scopus 로고    scopus 로고
    • A Fourier method for the fractional diffusion equation describing sub-diffusion
    • Chen, C. M., Liu, F., Turner, I & Anh, V. [2007] "A Fourier method for the fractional diffusion equation describing sub-diffusion," J. Comput. Phys. 227, 886-897
    • (2007) J. Comput. Phys. , vol.227 , pp. 886-897
    • Chen, C.M.1    Liu, F.2    Turner, I.3    Anh, V.4
  • 8
    • 0346897985 scopus 로고    scopus 로고
    • Mechanics with variable-order differential operators
    • Coimbra, C. F. M. [2003] "Mechanics with variable-order differential operators," Ann. Phys. (Leipzig) Math. Sci. 12, 692-703
    • (2003) Ann. Phys. (Leipzig) Math. Sci. , vol.12 , pp. 692-703
    • Coimbra, C.F.M.1
  • 9
    • 69049086472 scopus 로고    scopus 로고
    • Compact finite difference method for the fractional diffusion equation
    • Cui, M. R. [2009] "Compact finite difference method for the fractional diffusion equation," J. Comput. Phys. 228, 7792-7804
    • (2009) J. Comput. Phys. , vol.228 , pp. 7792-7804
    • Cui, M.R.1
  • 10
    • 32844473927 scopus 로고    scopus 로고
    • Concentration-dependent diffusivity and anomalous diffusion: A magnetic resonance imaging study of water ingress in porous zeolite
    • de Azevedo, E. N., de Sousa, P. L., de Souza, R. E., Engelsberg, M., Miranda, M.deN.do N & Silva, M. A. [2006] "Concentration-dependent diffusivity and anomalous diffusion: A magnetic resonance imaging study of water ingress in porous zeolite," Phys. Rev. E 73, 011204
    • (2006) Phys. Rev. e , vol.73 , pp. 011204
    • De Azevedo, E.N.1    De Sousa, P.L.2    De Souza, R.E.3    Engelsberg, M.4    Miranda, M.5    Den Do, N.6    Silva, M.A.7
  • 11
    • 84880235040 scopus 로고    scopus 로고
    • Nonlinear phenomena, turbulence, transport: Fractional diffusion in plasma turbulence
    • del-Castillo-Negrete, D., Carreras, B. A & Lynch, V. E. [2004] "Nonlinear phenomena, turbulence, transport: Fractional diffusion in plasma turbulence," Phys. Plasmas 11, 363-367
    • (2004) Phys. Plasmas , vol.11 , pp. 363-367
    • Del-Castillo-Negrete, D.1    Carreras, B.A.2    Lynch, V.E.3
  • 12
    • 36149001762 scopus 로고    scopus 로고
    • Numerical algorithm for the time fractional Fokker-Planck equation
    • Deng, W. H. [2007] "Numerical algorithm for the time fractional Fokker-Planck equation," J. Comput. Phys. 227, 1510-1522
    • (2007) J. Comput. Phys. , vol.227 , pp. 1510-1522
    • Deng, W.H.1
  • 13
    • 4043121080 scopus 로고    scopus 로고
    • Detailed error analysis for a fractional Adams method
    • Diethelm, K., Ford, N. J & Freed, A. D. [2004] "Detailed error analysis for a fractional Adams method," Numer. Algor. 36, 31-52
    • (2004) Numer. Algor. , vol.36 , pp. 31-52
    • Diethelm, K.1    Ford, N.J.2    Freed, A.D.3
  • 14
    • 79954926078 scopus 로고    scopus 로고
    • On the numerical solution of spacetime fractional diffusion models
    • Hanert, E. [2011] "On the numerical solution of spacetime fractional diffusion models," Comput. Fluids 46, 33-39
    • (2011) Comput. Fluids , vol.46 , pp. 33-39
    • Hanert, E.1
  • 15
    • 23044492109 scopus 로고    scopus 로고
    • Application of differential operator with servo-order function in model of viscoelastic deformation process
    • Ingman, D & Suzdalnitsky, J. [2005] "Application of differential operator with servo-order function in model of viscoelastic deformation process," J. Eng. Mech. 131, 763-767
    • (2005) J. Eng. Mech. , vol.131 , pp. 763-767
    • Ingman, D.1    Suzdalnitsky, J.2
  • 16
    • 0043026707 scopus 로고    scopus 로고
    • Statistical physics of dynamic systems with variable memory
    • Kobelev, Ya. L., Kobelev, L. Ya & Klimontovich, Yu. L. [2003] "Statistical physics of dynamic systems with variable memory," Dokl. Phys. 48, 285-289
    • (2003) Dokl. Phys. , vol.48 , pp. 285-289
    • Kobelev Ya., L.1    Ya., K.L.2    Klimontovich Yu., L.3
  • 17
    • 17144427014 scopus 로고    scopus 로고
    • The accuracy and stability of an implicit solution method for the fractional diffusion equation
    • Langlands, T. A. M & Henry, B. I. [2005] "The accuracy and stability of an implicit solution method for the fractional diffusion equation," J. Comput. Phys. 205, 719-736
    • (2005) J. Comput. Phys. , vol.205 , pp. 719-736
    • Langlands, T.A.M.1    Henry, B.I.2
  • 18
    • 1842832060 scopus 로고    scopus 로고
    • Chaos in Chen's system with a fractional order
    • Li, C. P & Peng, G. J. [2004] "Chaos in Chen's system with a fractional order," Chaos Solit. Fract. 22, 443-450
    • (2004) Chaos Solit. Fract. , vol.22 , pp. 443-450
    • Li, C.P.1    Peng, G.J.2
  • 19
    • 69749102394 scopus 로고    scopus 로고
    • On the fractional Adams method
    • Li, C. P & Tao, C. X. [2009] "On the fractional Adams method," Comput. Math. Appl. 58, 1573-1588
    • (2009) Comput. Math. Appl. , vol.58 , pp. 1573-1588
    • Li, C.P.1    Tao, C.X.2
  • 20
    • 79952454978 scopus 로고    scopus 로고
    • Numerical approaches to fractional calculus and fractional ordinary differential equation
    • Li, C. P., Chen, A & Ye, J. J. [2011] "Numerical approaches to fractional calculus and fractional ordinary differential equation," J. Comput. Phys. 230, 3352-3368
    • (2011) J. Comput. Phys. , vol.230 , pp. 3352-3368
    • Li, C.P.1    Chen, A.2    Ye, J.J.3
  • 21
    • 38849098330 scopus 로고    scopus 로고
    • Sample path properties of fractional Riesz-Bessel field of variable order
    • Lim, S. C & Teo, L. P. [2008] "Sample path properties of fractional Riesz-Bessel field of variable order," J Math. Phys. 49, 013509
    • (2008) J Math. Phys. , vol.49 , pp. 013509
    • Lim, S.C.1    Teo, L.P.2
  • 22
    • 34547548712 scopus 로고    scopus 로고
    • Finite difference/spectral approximations for the time-fractional diffusion equation
    • Lin, Y & Xu, C. [2007] "Finite difference/spectral approximations for the time-fractional diffusion equation," J. Comput. Phys. 225, 1533-1552
    • (2007) J. Comput. Phys. , vol.225 , pp. 1533-1552
    • Lin, Y.1    Xu, C.2
  • 23
    • 67349098149 scopus 로고    scopus 로고
    • Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation
    • Lin, R., Liu, F., Anh, V & Turner, I. [2009] "Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation," Appl. Math. Comput. 212, 435-445
    • (2009) Appl. Math. Comput. , vol.212 , pp. 435-445
    • Lin, R.1    Liu, F.2    Anh, V.3    Turner, I.4
  • 24
    • 0036650957 scopus 로고    scopus 로고
    • Variable order and distributed order fractional operators
    • Lorenzo, C. F & Hartley, T. T. [2002] "Variable order and distributed order fractional operators," Nonlin. Dyn. 29, 57-98
    • (2002) Nonlin. Dyn. , vol.29 , pp. 57-98
    • Lorenzo, C.F.1    Hartley, T.T.2
  • 27
    • 38349041965 scopus 로고    scopus 로고
    • Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation
    • Magin, R. L., Abdullah, O., Baleanu, D & Zhou, X. J. [2008] "Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation," J. Magn. Reson. 190, 255-270
    • (2008) J. Magn. Reson. , vol.190 , pp. 255-270
    • Magin, R.L.1    Abdullah, O.2    Baleanu, D.3    Zhou, X.J.4
  • 28
    • 4444368867 scopus 로고    scopus 로고
    • Finite difference approximations for fractional advectiondispersion equations
    • Meerschaert, M. M & Tadjeran, C. [2004] "Finite difference approximations for fractional advectiondispersion equations," J. Comput. Appl. Math. 172, 65-77
    • (2004) J. Comput. Appl. Math. , vol.172 , pp. 65-77
    • Meerschaert, M.M.1    Tadjeran, C.2
  • 29
    • 0002641421 scopus 로고    scopus 로고
    • The random walk's guide to anomalous diffusion: A fractional dynamics approach
    • Metzler, R & Klafter, J. [2000] "The random walk's guide to anomalous diffusion: A fractional dynamics approach," Phys. Rep. 339, 1-77
    • (2000) Phys. Rep. , vol.339 , pp. 1-77
    • Metzler, R.1    Klafter, J.2
  • 30
    • 46049119633 scopus 로고    scopus 로고
    • Implicit finite difference approximation for time fractional diffusion equations
    • Murio, D. A. [2008] "Implicit finite difference approximation for time fractional diffusion equations," Comput. Math. Appl. 56, 1138-1145
    • (2008) Comput. Math. Appl. , vol.56 , pp. 1138-1145
    • Murio, D.A.1
  • 31
    • 52349084738 scopus 로고    scopus 로고
    • Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere
    • Pedro, H. T. C., Kobayashi, M. H., Pereira, J. M. C & Coimbra, C. F. M. [2008] "Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere," J. Vib. Contr. 14, 1659-1672
    • (2008) J. Vib. Contr. , vol.14 , pp. 1659-1672
    • Pedro, H.T.C.1    Kobayashi, M.H.2    Pereira, J.M.C.3    Coimbra, C.F.M.4
  • 32
    • 0032653954 scopus 로고    scopus 로고
    • Fractional-order systems and PI?D?-controllers
    • Podlubny, I. [1999] "Fractional-order systems and PI?D?-controllers," IEEE Trans. Autom. Contr. 44, 208-214
    • (1999) IEEE Trans. Autom. Contr. , vol.44 , pp. 208-214
    • Podlubny, I.1
  • 33
    • 61349186917 scopus 로고    scopus 로고
    • Matrix approach to discrete fractional calculus II: Partial fractional differential equations
    • Podlubny, I., Chechkin, A., Skovranek, T., Chen, Y. Q & Vinagre Jara, B. M. [2009] "Matrix approach to discrete fractional calculus II: Partial fractional differential equations," J. Comput. Phys. 228, 3137-3153
    • (2009) J. Comput. Phys. , vol.228 , pp. 3137-3153
    • Podlubny, I.1    Chechkin, A.2    Skovranek, T.3    Chen, Y.Q.4    Vinagre Jara, B.M.5
  • 34
    • 0011189482 scopus 로고
    • Fractional integration operator of variable order in the Holder space H?(x)
    • Ross, B & Samko, S. G. [1995] "Fractional integration operator of variable order in the Holder space H?(x)," Int. J. Math. Math. Sci. 18, 777-788
    • (1995) Int. J. Math. Math. Sci. , vol.18 , pp. 777-788
    • Ross, B.1    Samko, S.G.2
  • 35
    • 84948882036 scopus 로고
    • Integration and differentiation to a variable fractional order
    • Samko, S. G & Ross, B. [1993] "Integration and differentiation to a variable fractional order," Integr. Trans. Spec. Funct. 1, 277-300
    • (1993) Integr. Trans. Spec. Funct. , vol.1 , pp. 277-300
    • Samko, S.G.1    Ross, B.2
  • 36
    • 33751019968 scopus 로고    scopus 로고
    • Anomalous diffusion in Purkinje cell dendrites caused by spines
    • Santamaria, F., Wils, S., de Schutter, E & Augustine, G. J. [2006] "Anomalous diffusion in Purkinje cell dendrites caused by spines," Neuron 52, 635-648
    • (2006) Neuron , vol.52 , pp. 635-648
    • Santamaria, F.1    Wils, S.2    De Schutter, E.3    Augustine, G.J.4
  • 37
    • 0000902727 scopus 로고
    • Generalized viscoelastic models: Their fractional equations with solutions
    • Schiessel, H., Metzler, R., Blumen, A & Nonnenmacher, T. F. [1995] "Generalized viscoelastic models: Their fractional equations with solutions," J. Phys. A: Math. Gen. 28, 6567-6584
    • (1995) J. Phys. A: Math. Gen. , vol.28 , pp. 6567-6584
    • Schiessel, H.1    Metzler, R.2    Blumen, A.3    Nonnenmacher, T.F.4
  • 38
    • 0001691616 scopus 로고
    • Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow
    • Solomon, T. H., Weeks, E. R & Swinney, H. L. [1993] "Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow," Phys. Rev. Lett. 71, 3975-3978
    • (1993) Phys. Rev. Lett. , vol.71 , pp. 3975-3978
    • Solomon, T.H.1    Weeks, E.R.2    Swinney, H.L.3
  • 39
    • 68649098514 scopus 로고    scopus 로고
    • Variableorder fractional differential operators in anomalous diffusion modeling
    • Sun, H. G., Chen, W & Chen, Y. Q. [2009] "Variableorder fractional differential operators in anomalous diffusion modeling," Phys. A 388, 4586-4592
    • (2009) Phys. A , vol.388 , pp. 4586-4592
    • Sun, H.G.1    Chen, W.2    Chen, Y.Q.3
  • 40
    • 78049351606 scopus 로고    scopus 로고
    • Randomorder fractional differential equation models
    • Sun, H. G., Chen, Y. Q & Chen, W. [2011] "Randomorder fractional differential equation models," Sign. Process. 91, 525-530
    • (2011) Sign Process. , vol.91 , pp. 525-530
    • Sun, H.G.1    Chen, Y.Q.2    Chen, W.3
  • 41
    • 31744438550 scopus 로고    scopus 로고
    • A second order accurate numerical approximation for the fractional diffusion equation
    • Tadjeran, C., Meerschaert, M. M & Scheffler, H. P. [2006] "A second order accurate numerical approximation for the fractional diffusion equation," J. Comput. Phys. 213, 205-213
    • (2006) J. Comput. Phys. , vol.213 , pp. 205-213
    • Tadjeran, C.1    Meerschaert, M.M.2    Scheffler, H.P.3
  • 42
    • 74349105778 scopus 로고    scopus 로고
    • Variable order differential equations and diffusion with changing modes
    • Umarov, S & Steinberg, S. [2009] "Variable order differential equations and diffusion with changing modes," Zeitschrift fr Analysis und ihre Anwendungen 28, 431-450
    • (2009) Zeitschrift Fr Analysis und Ihre Anwendungen , vol.28 , pp. 431-450
    • Umarov, S.1    Steinberg, S.2
  • 43
    • 25444472344 scopus 로고    scopus 로고
    • An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations
    • Yuste, S. B & Acedo, L. [2005] "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) SIAM J. Numer. Anal. , vol.42 , pp. 1862-1874
    • Yuste, S.B.1    Acedo, L.2
  • 44
    • 33646128485 scopus 로고    scopus 로고
    • Weighted average finite difference methods for fractional diffusion equations
    • Yuste, S. B. [2006] "Weighted average finite difference methods for fractional diffusion equations," J. Comput. Phys. 216, 264-274
    • (2006) J. Comput. Phys. , vol.216 , pp. 264-274
    • Yuste, S.B.1
  • 45
    • 0036887936 scopus 로고    scopus 로고
    • Chaos, fractional kinetics, and anomalous transport
    • Zaslavsky, G. M. [2002] "Chaos, fractional kinetics, and anomalous transport," Phys. Rep. 371, 461-580
    • (2002) Phys. Rep. , vol.371 , pp. 461-580
    • Zaslavsky, G.M.1
  • 46
    • 68949098580 scopus 로고    scopus 로고
    • A finite difference method for fractional partial differential equation
    • Zhang, Y. [2009] "A finite difference method for fractional partial differential equation," Appl. Math. Comput. 215, 524-529
    • (2009) Appl. Math. Comput. , vol.215 , pp. 524-529
    • Zhang, Y.1
  • 47
    • 84867978055 scopus 로고    scopus 로고
    • Implicit difference approximation for the time fractional diffusion equation
    • Zhuang, P & Liu, F. [2006] "Implicit difference approximation for the time fractional diffusion equation," Appl. Math. Comput. 22, 87-99
    • (2006) Appl. Math. Comput. , vol.22 , pp. 87-99
    • Zhuang, P.1    Liu, F.2
  • 48
    • 84907893973 scopus 로고    scopus 로고
    • Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term
    • Zhuang, P., Liu, F., Anh, V & Turner, I. [2009] "Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term," SIAM J. Numer. Anal. 47, 1760-1781.
    • (2009) SIAM J. Numer. Anal. , vol.47 , pp. 1760-1781
    • Zhuang, P.1    Liu, F.2    Anh, V.3    Turner, I.4


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.