Analytical and numerical solutions of a one-dimensional fractional-in-space diffusion equation in a composite medium. (English) Zbl 1193.65168

Summary: A one-dimensional space fractional diffusion equation in a composite medium consisting of two layers in contact is studied both analytically and numerically. Since domain decomposition is the only approach available to solve this problem, we at first investigate analytical and numerical strategies for a composite medium with the same fractal dimension in each layer to ascertain which domain decomposition approach is the most accurate and consistent with a global solution methodology, which is available in this case.
We utilise a matrix representation of the fractional-in-space operator to generate a system of linear ordinary differential equations with the matrix raised to the same fractional exponent. We show that the global and domain decomposition numerical strategies for this problem produce simulation results that are in good agreement with their analytic counterparts and conclude that the domain decomposition that imposes the Neumann condition at the interface produces the most consistent results. Finally, we carry this finding to study the composite problem with different fractal dimensions, where we again favourably compare analytic and numerical solutions. The resulting method can be naturally extended to space fractional diffusion in a composite medium consisting of more than two layers.


65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
Full Text: DOI


[1] Anh, V.V.; Leonenko, N.N., Spectral analysis of fractional kinetic equations with random data, J. stat. phys., 104, 349-1387, (2001) · Zbl 1034.82044
[2] Anh, V.V.; Leonenko, N.N., Renormalization and homogenization of fractional diffusion equations with random data, Probab. theor. relat. field, 12, 381-408, (2002) · Zbl 1031.60043
[3] Anh, V.V.; Lau, K.S.; Yu, Z.G., Multifractal characterisation of complete genomes, J. phys. A: math. gen., 34, 7127-7139, (2001) · Zbl 0989.92019
[4] Ayache, A.; Lévy Véhel, J., The generalized multifractional Brownian motion, Stat. infer. stoch. process., 3, 7-18, (2000) · Zbl 0979.60023
[5] Bass, R.F., Uniqueness in law for pure jump type Markov processes, Probab. theor. relat. field, 79, 271-287, (1988) · Zbl 0664.60080
[6] Benassi, A.; Jaffard, S.; Roux, D., Elliptic Gaussian random processes, Rev. mat. iberoam., 13, 19-90, (1997) · Zbl 0880.60053
[7] Benson, D.A.; Wheatcraft, S.W.; Meerschaert, M.M., Application of a fractional advection – dispersion equation, Water resour. res., 36, 6, 1403-1412, (2000)
[8] Benson, D.A.; Wheatcraft, S.W.; Meerschaert, M.M., The fractional-order governing equation of levy motion, Water resour. res., 36, 6, 1413-1423, (2000)
[9] Brenan, K.E.; Campbell, S.L.; Petzold, L.R., Numerical solution of initial-value problems in differential – algebraic equations, (1989), North-Holland New York · Zbl 0699.65057
[10] R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. 1, Wiley, 1991. ISBN: 978-0-471-50439-9. · Zbl 0788.00012
[11] Falconer, K., Techniques in fractal geometry, (1997), Wiley · Zbl 0869.28003
[12] Frisch, U., Turbulence, (1995), Cambridge University Press · Zbl 0727.76064
[13] Gorenflo, R.; Mainardi, F., Random walk models for space fractional diffusion processes, Fract. calc. appl. anal., 1, 167-191, (1998) · Zbl 0946.60039
[14] Gupta, V.K.; Waymire, E.C., A statistical analysis of mesoscale rainfall as a random cascade, J. appl. meteorol., 32, 251-267, (1993)
[15] Harte, D., Multifractals: theory and applications, (2001), Chapman & Hall · Zbl 1016.62111
[16] Ilic, M.; Liu, F.; Turner, I.; Anh, V., Numerical approximation of a fractional-in-space diffusion equation, Fract. calc. appl. anal., 8, 3, 323-341, (2005) · Zbl 1126.26009
[17] Ilic, M.; Liu, F.; Turner, I.; Anh, V., Numerical approximation of a fractional-in-space diffusion equation (II)—with nonhomogeneous boundary conditions, Fract. calc. appl. anal., 9, 4, 333-349, (2006) · Zbl 1132.35507
[18] Jacob, N., A class of Feller semigroups generated by pseudodifferential operators, Math. Z., 215, 151-166, (1994) · Zbl 0795.35154
[19] Jacob, N.; Leopold, H.G., Pseudodifferential operators with variable order of differentiation generating Feller semigroup, Integr. eq. oper. theor., 17, 544-553, (1993) · Zbl 0793.35139
[20] Jaffard, S., The multifractal nature of Lévy processes, Probab. theor. relat. field, 114, 207-227, (1999) · Zbl 0947.60039
[21] Kahane, J.P.; Peyrière, J., Sur sertaines martingales de benoit Mandelbrot, Adv. math., 22, 131-145, (1976) · Zbl 0349.60051
[22] Kikuchi, K.; Negoro, A., On Markov processes generated by pseudodifferentail operator of variable order, Osaka J. math., 34, 319-335, (1997) · Zbl 0913.60062
[23] Komatsu, T., On stable-like processes, (), 210-219 · Zbl 0977.47038
[24] Liu, F.; Anh, V.; Turner, I.; Bajracharya, K.; Huxley, W.; Su, N., A finite volume simulation model for saturated-unsaturated flow and application to gooburrum, bundaberg, queensland, Australia, Appl. math. model., 29, 852-870, (2005)
[25] Liu, F.; Anh, V.; Turner, I., Numerical solution of the space fractional fokker – planck equation, J. comput. appl. math., 166, 209-219, (2004) · Zbl 1036.82019
[26] Liu, F.; Anh, V.; Turner, I.; Zhuang, P., Numerical simulation for solute transport in fractal porous media, Anziam j., 45, 461-473, (2004) · Zbl 1123.76363
[27] Liu, F.; Zhuang, P.; Anh, V.; Turner, I.; Burrage, K., Stability and convergence of the difference methods for the space – time fractional advection – diffusion equation, Appl. math. comput., 191, 12-20, (2007) · Zbl 1193.76093
[28] Liu, Q.; Liu, F.; Turner, I.; Anh, V., Approximation of the levy – feller advection – dispersion process by random walk and finite difference method, J. phys. comput., 222, 57-70, (2007) · Zbl 1112.65006
[29] Mainardi, F.; Luchko, Yu.; Pagnini, G., The fundanental solution of the space – time fractional diffusion equation, Fract. calc. appl. anal., 4, 2, 153-192, (2001) · Zbl 1054.35156
[30] Mandelbrot, B., Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier, J. fluid mech., 62, 331-358, (1974) · Zbl 0289.76031
[31] B. Mandelbrot, A. Fisher, L. Calvet, A multifractal model of asset returns, Cowles Foundation Discussion Paper 1164, 1997.
[32] Meerschaert, M.; Tadjeran, C., Finite difference approximations for fractional advection – dispersion flow equations, J. comput. appl. math., 172, 65-77, (2004) · Zbl 1126.76346
[33] R. Metzler, J. Krafter, The random walk’s guide to anomalous diffusion: a fractional dynamic approach, Phys. Rep. 339 (2000) 1-72.
[34] Metzler, R.; Krafter, J., The restaurant at the end of the random walk: recent development in the description of anomalous transport by fractional dynamics, J. phys. A: math. gen., 37, R161-R208, (2004) · Zbl 1075.82018
[35] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), John Wiley New York · Zbl 0789.26002
[36] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York/London · Zbl 0428.26004
[37] Özşik, M.N., Heat conduction, (1980), Wiley
[38] Podlubny, I., Fractional differential equations, (1999), Academic Press · Zbl 0918.34010
[39] Quarteroni, A.; Valli, A., Domain decomposition methods for partial differential equations, (1999), Oxford University Press Oxford · Zbl 0931.65118
[40] Riedi, R.H., Multifractal processes, (), 625-716 · Zbl 1060.28008
[41] Riedi, R.H.; Crouse, M.S.; Ribeiro, V.J.; Baraniuk, R.G., A multifractal wavelet model with applications to network traffic, IEEE trans. inform. theor., 67, 130-146, (1999)
[42] Roop, J.P., Computational aspects of FEM approximation of fractional advection – dispersion equation on bounded domains in \(\mathtt{R} {}^2\), J. comput. appl. math., 193, 1, 243-268, (2006) · Zbl 1092.65122
[43] M.D. Ruiz-Medina, V.V. Anh, J.M. Angulo, Multifractional Markov processes in heterogeneous domains, Stoch. Anal. Appl., 2010, in press. · Zbl 1236.60041
[44] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach Science Publishers USA · Zbl 0818.26003
[45] Shen, S.; Liu, F., Error analysis of an explicit finite difference approximation for the space fractional diffusion, Anziam j., 46, 871-887, (2005)
[46] Trefethen, L.N.; Weideman, J.A.C.; Schmelzer, T., Talbot quadratures and rational approximations, BIT numer. math., 46, 653-670, (2006) · Zbl 1103.65030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.