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Numerical solutions for fractional reaction-diffusion equations. (English) Zbl 1142.65422
Summary: Fractional diffusion equations are useful for applications in which a cloud of particles spreads faster than predicted by the classical equation. In a fractional diffusion equation, the second derivative in the spatial variable is replaced by a fractional derivative of order less than two. The resulting solutions spread faster than the classical solutions and may exhibit asymmetry, depending on the fractional derivative used. Fractional reaction-diffusion equations combine the fractional diffusion with a classical reaction term. In this paper, we develop a practical method for numerical solution of fractional reaction-diffusion equations, based on operator splitting. Then we present results of numerical simulations to illustrate the method, and investigate properties of numerical solutions. We also discuss applications to biology, where the reaction term models species growth and the diffusion term accounts for movements.
MSC:
65M99Numerical methods for IVP of PDE
35K57Reaction-diffusion equations
Software:
STABLE
References:
[1]Britton, N. F.: Reaction–diffusion equations and their applications to biology, (1986) · Zbl 0602.92001
[2]Cantrell, R. S.; Cosner, C.: Spatial ecology via reaction–diffusion equations, (2003)
[3]Grindrod, P.: The theory and applications of reaction–diffusion equations, Oxford applied mathematics and computing science series (1996) · Zbl 0867.35001
[4]Rothe, F.: Global solutions of reaction–diffusion systems, Lecture notes in mathematics 1072 (1984) · Zbl 0546.35003
[5]Smoller, J.: Shock waves and reaction–diffusion equations, Grundlehren der mathematischen wissenschaften [Fundamental principles of mathematical sciences] 258 (1994) · Zbl 0807.35002
[6]Murray, J. D.: Mathematical biology. I,II, Interdisciplinary applied mathematics 17, 18 (2002)
[7]Neubert, M.; Caswell, H.: Demography and dispersal: calculation and sensitivity analysis of invasion speed for structured populations, Ecology 81, No. 6, 1613-1628 (2000)
[8]Bachelier, L. J. B.: Théorie de la spéculation, (1900) · Zbl 31.0075.12
[9]Einstein, A.: Investigations on the theory of the Brownian movement, (1956) · Zbl 0071.41205
[10]Sokolov, I. M.; Klafter, J.: From diffusion to anomalous diffusion: A century after Einstein’s Brownian motion, Chaos 15, No. 2, 26-103 (2005) · Zbl 1080.82022 · doi:10.1063/1.1860472
[11]Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. rep. 339, No. 1 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[12]Metzler, R.; Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. phys. A 37, No. 31, R161-R208 (2004) · Zbl 1075.82018 · doi:10.1088/0305-4470/37/31/R01
[13]Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M.: The fractional-order governing equation of Lévy motion, Water resources res. 36, 1413-1424 (2000)
[14]Chaves, A.: A fractional diffusion equation to describe Lévy flights, Phys. lett. A 239, 13-16 (1998) · Zbl 1026.82524 · doi:10.1016/S0375-9601(97)00947-X
[15]Feller, W.: An introduction to probability theory and applications, An introduction to probability theory and applications , II (1966) · Zbl 0138.10207
[16]Meerschaert, M. M.; Scheffler, H. P.: Limit distributions for sums of independent random vectors: heavy tails in theory and practice, (2001)
[17]Samorodnitsky, G.; Taqqu, M. S.: Stable non-Gaussian random processes, (1994)
[18]Meerschaert, M. M.; Scheffler, H. P.: Limit theorems for continuous-time random walks with infinite mean waiting times, J. appl. Probab. 41, No. 3, 623-638 (2004) · Zbl 1065.60042 · doi:10.1239/jap/1091543414
[19]Meerschaert, M. M.; Benson, D. A.; Baeumer, B.: Multidimensional advection and fractional dispersion, Phys. rev. E 59, 5026-5028 (1999)
[20]Taylor, S. J.: The measure theory of random fractals, Math. proc. Cambridge philos. Soc. 100, 383-406 (1986) · Zbl 0622.60021 · doi:10.1017/S0305004100066160
[21]Meerschaert, M. M.; Benson, D. A.; Baeumer, B.: Operator Lévy motion and multiscaling anomalous diffusion, Phys. rev. E 63, 1112-1117 (2001)
[22]Schumer, R.; Benson, D. A.; Meerschaert, M. M.; Baeumer, B.: Multiscaling fractional advection–dispersion equations and their solutions, Water resources res. 39, 1022-1032 (2003)
[23]Deng, Z.; Singh, V. P.; Bengtsson, L.: Numerical solution of fractional advection-dispersion equation, J. hydraulic eng. 130, 422-431 (2004)
[24]Lynch, V. E.; Carreras, B. A.; Del-Castillo-Negrete, D.; Ferreira-Mejias, K. M.; Hicks, H. R.: Numerical methods for the solution of partial differential equations of fractional order, J. comput. Phys. 192, 406-421 (2003) · Zbl 1047.76075 · doi:10.1016/j.jcp.2003.07.008
[25]Meerschaert, M. M.; Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations, Appl. numer. Math. 56, No. 1, 80-90 (2006) · Zbl 1086.65087 · doi:10.1016/j.apnum.2005.02.008
[26]Meerschaert, M. M.; Scheffler, H. P.; Tadjeran, C.: Finite difference methods for two-dimensional fractional dispersion equation, J. comput. Phys. 211, 249-261 (2006) · Zbl 1085.65080 · doi:10.1016/j.jcp.2005.05.017
[27]Meerschaert, M. M.; Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations, J. comput. Appl. math. 172, No. 1, 65-77 (2004) · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033
[28]Tadjeran, C.; Meerschaert, M. M.; Scheffler, H. P.: A second order accurate numerical approximation for the fractional diffusion equation, J. comput. Phys. 213, 205-213 (2006) · Zbl 1089.65089 · doi:10.1016/j.jcp.2005.08.008
[29]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
[30]F. Liu, V. Ahn, I. Turner, Numerical solution of the fractional advection–dispersion equation, 2002, Preprint
[31]Liu, F.; Ahn, V.; Turner, I.; Zhuang, P.: Numerical simulation for solute transport in fractal porous media, Anziam j. 45, No. E, C461-C473 (2004) · Zbl 1123.76363 · doi:http://anziamj.austms.org.au/V45/CTAC2003/Liuf/
[32]Liu, F.; Ahn, V.; Turner, I.: Numerical solution of the space fractional Fokker–Planck equation, J. comput. Appl. math. 166, 209-219 (2004) · Zbl 1036.82019 · doi:10.1016/j.cam.2003.09.028
[33]Ervin, V. J.; Roop, J. P.: Variational solution of fractional advection dispersion equations on bounded domains in rd, Numer. methods partial differential equations 23, 256-281 (2007) · Zbl 1117.65169 · doi:10.1002/num.20169
[34]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
[35]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 (2005) · Zbl 1092.65122 · doi:10.1016/j.cam.2005.06.005
[36]Zhang, Y.; Benson, D. A.; Meerschaert, M. M.; Scheffler, H. P.: On using random walks to solve the space-fractional advection-dispersion equations, J. statist. Phys. 123, 89-110 (2006)
[37]Baeumer, B.; Kovács, M.; Meerschaert, M. M.: Fractional reproduction-dispersal equations and heavy tail dispersal kernels, Bull. math. Biol. 69, 2281-2297 (2007)
[38]Del Castillo-Negrete, D.; Carreras, B. A.; Lynch, V. E.: Front dynamics in reaction–diffusion systems with Lévy flights: A fractional diffusion approach, Phys. rev. Lett. 91, No. 1, 018302 (2003)
[39]Bullock, J. M.; Clarke, R. T.: Long distance seed dispersal by wind: measuring and modelling the tail of the curve, Oecologia 124, No. 4, 506-521 (2000)
[40]Clark, J. S.; Silman, M.; Kern, R.; Macklin, E.; Hillerislambers, J.: Seed dispersal near and far: patterns across temperate and tropical forests, Ecology 80, No. 5, 1475-1494 (1999)
[41]Clark, J. S.; Lewis, M.; Horvath, L.: Invasion by extremes: population spread with variation in dispersal and reproduction, The amer. Naturalist 157, No. 5, 537-554 (2001)
[42]Katul, G. G.; Porporato, A.; Nathan, R.; Siqueira, M.; Soons, M. B.; Poggi, D.; Horn, H. S.; Levin, S. A.: Mechanistic analytical models for long-distance seed dispersal by wind, The amer. Naturalist 166, 368-381 (2005)
[43]Klein, E. K.; Lavigne, C.; Picault, H.; Renard, M.; Gouyon, P. H.: Pollen dispersal of oilseed rape: estimation of the dispersal function and effects of field dimension, J. appl. Ecology 43, No. 10, 141-151 (2006)
[44]Paradis, E.; Baillie, S. R.; Sutherland, W. J.: Modeling large-scale dispersal distances, Ecological modelling 151, No. 2–3, 279-292 (2002)
[45]Jacob, N.: Pseudo-differential operators and Markov processes, Mathematical research 94 (1996) · Zbl 0860.60002
[46]Gerisch, A.; Verwer, J. G.: Operator splitting and approximate factorization for taxis-diffusion–reaction models, Appl. numer. Math. 42, 159-176 (2002) · Zbl 0998.65102 · doi:10.1016/S0168-9274(01)00148-9
[47]Csomós, P.; Faragó, I.; Havasi, Á: Weighted sequential splittings and their analysis, Comput. math. Appl. 50, 1017-1031 (2005) · Zbl 1086.65053 · doi:10.1016/j.camwa.2005.08.004
[48]Faragó, I.; Havasi, Á.: Consistency analysis of operator splitting methods for C0-semigroups expression, Semigroup forum 74, No. 1, 125-139 (2007) · Zbl 1125.47033 · doi:10.1007/s00233-006-0640-3
[49]Marchuk, G. I.: Some application of splitting-up methods to the solution of mathematical physics problems, Applik. mat. 13, 103-132 (1968) · Zbl 0159.44702
[50]Strang, G.: Accurate partial difference methods I: Linear Cauchy problems, Arch. ration. Mech. anal. 12, 392-402 (1963) · Zbl 0113.32303 · doi:10.1007/BF00281235
[51]Strang, G.: On the construction and comparison of difference schemes, SIAM J. Numer. anal 5, 506-517 (1968) · Zbl 0184.38503 · doi:10.1137/0705041
[52]Arendt, W.; Batty, C.; Hieber, M.; Neubrander, F.: Vector-valued Laplace transforms and Cauchy problems, (2001)
[53]Engel, K. J.; Nagel, R.: One-parameter semigroups for linear evolution equations, Graduate texts in mathematics 194 (2000)
[54]Hille, E.; Phillips, R. S.: Functional analysis and semi-groups, Functional analysis and semi-groups (1974)
[55]Pazy, A.: Semigroups of linear operators and applications to partial differential equations, Applied mathematical sciences 44 (1983) · Zbl 0516.47023
[56]Brézis, H.; Pazy, A.: Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. funct. Anal. 9, 63-74 (1972) · Zbl 0231.47036 · doi:10.1016/0022-1236(72)90014-6
[57]Cliff, M.; Goldstein, J. A.; Wacker, M.: Positivity, Trotter products, and blow-up, Positivity 8, No. 2, 187-208 (2004) · Zbl 1076.47047 · doi:10.1023/B:POST.0000042835.75168.22
[58]Miyadera, I.; Ôharu, S.: Approximation of semi-groups of nonlinear operators, Tôhoku math. J. 22, No. 2, 24-47 (1970) · Zbl 0195.15001 · doi:10.2748/tmj/1178242858
[59]B. Baeumer, M. Kovács, Subordinated groups of linear operators: properties via the transference principle and the related unbounded operational calculus, (2006) (submitted for publication)
[60]Phillips, R. S.: On the generation of semigroups of linear operators, Pacific J. Math. 2, 343-369 (1952) · Zbl 0047.11004
[61]Schilling, R. L.: Growth and hölder conditions for sample paths of Feller processes, Probab. theory related fields 112, 565-611 (1998) · Zbl 0930.60013 · doi:10.1007/s004400050201
[62]Sato, K. -I.: Lévy processes and infinitely divisible distributions, Cambridge studies in advanced mathematics 68 (1999) · Zbl 0973.60001
[63]B. Baeumer, M. Kovács, M.M. Meerschaert, Subordinated multiparameter groups of linear operators: Properties via the transference principle, Functional Analysis and Evolution Equations: Dedicated to Günter Lumer, Birkhäuser Basel, (2008) (in press)
[64]Miller, K.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[65]Samko, S.; Kilbas, A.; Marichev, O.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003
[66]Meerschaert, M. M.; Scheffler, H. P.: Semistable Lévy motion, Fract. calc. Appl. anal. 5, 27-54 (2002) · Zbl 1032.60043
[67]Balakrishnan, A. V.: An operational calculus for infinitesimal generators of semigroups, Trans. amer. Math. soc. 91, 330-353 (1959) · Zbl 0090.09701 · doi:10.2307/1993125
[68]Balakrishnan, A. V.: Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10, 419-437 (1960) · Zbl 0103.33502
[69]Arendt, W.; Grabosch, A.; Greiner, G.; Groh, U.; Lotz, H. P.; Moustakas, U.; Nagel, R.; Neubrander, F.; Schlotterbeck, U.: One-parameter semigroups of positive operators, Lecture notes in mathematics 1184 (1986)
[70]Van Kooten, G. C.; Blute, E. H.: The economics of nature. Managing biological assets, (2000)
[71]Lockwood, D. R.; Hastings, A.: The effects of dispersal patterns on marine reserves: does the tail wag the dog?, Theoretical population biology 61, 297-309 (2002)
[72]Zolotarev, V. M.: One-dimensional stable distributions, Translations of mathematical monographs 65 (1986) · Zbl 0589.60015
[73]Nolan, J. P.: Numerical calculation of stable densities and distribution functions. Heavy tails and highly volatile phenomena, Comm. statist. Stochastic models 13, No. 4, 759-774 (1997) · Zbl 0899.60012 · doi:10.1080/15326349708807450