Numerical solutions for fractional reaction-diffusion equations

Computers and Mathematics with Applications

Volumn 55, Issue 10, 2008, Pages 2212-2226

  Baeumer, Boris ^a   Kovács, Mihály ^a   Meerschaert, Mark M ^b  

^a UNIVERSITY OF OTAGO   (New Zealand)

^b MICHIGAN STATE UNIVERSITY   (United States)

Author keywords

Fractional reaction diffusion; Infinitely divisible distributions; Invasive species; Operator semigroups; Operator splitting

Indexed keywords

COMPUTER SIMULATION; MATHEMATICAL MODELS; NUMERICAL METHODS; PROBLEM SOLVING;

FRACTIONAL REACTION DIFFUSION; INFINITELY DIVISIBLE DISTRIBUTIONS; INVASIVE SPECIES; OPERATOR SEMIGROUPS; OPERATOR SPLITTING;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 41449095257     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2007.11.012     Document Type: Article

References (73)

1. Britton N.F. Reaction-Diffusion Equations and Their Applications to Biology (1986), Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London
2. Cantrell R.S., and Cosner C. Spatial ecology via reaction-diffusion equations. Wiley Series in Mathematical and Computational Biology (2003), John Wiley & Sons Ltd., Chichester
3. Grindrod P. The Theory and Applications of Reaction-Diffusion Equations. 2nd edn. Oxford Applied Mathematics and Computing Science Series (1996), The Clarendon Press Oxford University Press, New York
4. Rothe F. Global solutions of reaction-diffusion systems. Lecture Notes in Mathematics vol. 1072 (1984), Springer-Verlag, Berlin
5. Smoller J. Shock waves and reaction-diffusion equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 2nd edn vol. 258 (1994), Springer-Verlag, New York
6. Murray J.D. Mathematical Biology. I,II. 3rd edn. Interdisciplinary Applied Mathematics vols. 17, 18 (2002), Springer-Verlag, New York
7. Neubert M., and Caswell H. Demography and dispersal: Calculation and sensitivity analysis of invasion speed for structured populations. Ecology 81 6 (2000) 1613-1628
8. Bachelier L.J.B. Théorie de la Spéculation (1900), Gauthier-Villars, Paris
9. Einstein A. Investigations on the Theory of the Brownian Movement (1956), Dover Publications Inc., New York (Edited with notes by R. Fürth, Translated by A. D. Cowper)
10. Sokolov I.M., and Klafter J. From diffusion to anomalous diffusion: A century after Einstein's Brownian motion. Chaos 15 2 (2005) 26-103
11. Metzler R., and Klafter J. The random walk's guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339 1 (2000)
12. Metzler R., and 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 31 (2004) R161-R208
13. Benson D.A., Wheatcraft S.W., and Meerschaert M.M. The fractional-order governing equation of Lévy motion. Water Resources Res. 36 (2000) 1413-1424
14. Chaves A. A fractional diffusion equation to describe Lévy flights. Phys. Lett. A 239 (1998) 13-16
15. Feller W. An Introduction to Probability Theory and Applications vols. I, II (1966), John Wiley and Sons
16. Meerschaert M.M., and Scheffler H.P. Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice (2001), Wiley Interscience, New York
17. Samorodnitsky G., and Taqqu M.S. Stable Non-Gaussian Random Processes (1994), Chapman & Hall/CRC
18. Meerschaert M.M., and Scheffler H.P. Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Probab. 41 3 (2004) 623-638
19. Meerschaert M.M., Benson D.A., and Baeumer B. Multidimensional advection and fractional dispersion. Phys. Rev. E 59 (1999) 5026-5028
20. Taylor S.J. The measure theory of random fractals. Math. Proc. Cambridge Philos. Soc. 100 (1986) 383-406
21. Meerschaert M.M., Benson D.A., and Baeumer B. Operator Lévy motion and multiscaling anomalous diffusion. Phys. Rev. E 63 (2001) 1112-1117
22. Schumer R., Benson D.A., Meerschaert M.M., and Baeumer B. Multiscaling fractional advection-dispersion equations and their solutions. Water Resources Res. 39 (2003) 1022-1032
23. Deng Z., Singh V.P., and Bengtsson L. Numerical solution of fractional advection-dispersion equation. J. Hydraulic Eng. 130 (2004) 422-431
24. Lynch V.E., Carreras B.A., del-Castillo-Negrete D., Ferreira-Mejias K.M., and Hicks H.R. Numerical methods for the solution of partial differential equations of fractional order. J. Comput. Phys. 192 (2003) 406-421
25. Meerschaert M.M., and Tadjeran C. Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56 1 (2006) 80-90
26. Meerschaert M.M., Scheffler H.P., and Tadjeran C. Finite difference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 211 (2006) 249-261
27. Meerschaert M.M., and Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172 1 (2004) 65-77
28. Tadjeran C., Meerschaert M.M., and Scheffler H.P. A second order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213 (2006) 205-213
29. Tadjeran C., and Meerschaert M.M. A second order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220 (2007) 813-823
30. F. Liu, V. Ahn, I. Turner, Numerical solution of the fractional advection-dispersion equation, 2002, Preprint
31. Liu F., Ahn V., Turner I., and Zhuang P. Numerical simulation for solute transport in fractal porous media. ANZIAM J. 45 E (2004) C461-C473
32. Liu F., Ahn V., and Turner I. Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 166 (2004) 209-219
33. d. Numer. Methods Partial Differential Equations 23 (2007) 256-281
34. Fix G.J., and Roop J.P. Least squares finite element solution of a fractional order two-point boundary value problem. Comput. Math. Appl. 48 (2004) 1017-1033
35. 2. J. Comput. Appl. Math. 193 (2005) 243-268
36. Zhang Y., Benson D.A., Meerschaert M.M., and Scheffler H.P. On using random walks to solve the space-fractional advection-dispersion equations. J. Statist. Phys. 123 (2006) 89-110
37. Baeumer B., Kovács M., and Meerschaert M.M. Fractional reproduction-dispersal equations and heavy tail dispersal kernels. Bull. Math. Biol. 69 (2007) 2281-2297
38. del Castillo-Negrete D., Carreras B.A., and Lynch V.E. Front dynamics in reaction-diffusion systems with levy flights: A fractional diffusion approach. Phys. Rev. Lett. 91 1 (2003) 018302
39. Bullock J.M., and Clarke R.T. Long distance seed dispersal by wind: Measuring and modelling the tail of the curve. Oecologia 124 4 (2000) 506-521
40. Clark J.S., Silman M., Kern R., Macklin E., and HilleRisLambers J. Seed dispersal near and far: Patterns across temperate and tropical forests. Ecology 80 5 (1999) 1475-1494
41. Clark J.S., Lewis M., and Horvath L. Invasion by extremes: Population spread with variation in dispersal and reproduction. The Amer. Naturalist 157 5 (2001) 537-554
42. Katul G.G., Porporato A., Nathan R., Siqueira M., Soons M.B., Poggi D., Horn H.S., and Levin S.A. Mechanistic analytical models for long-distance seed dispersal by wind. The Amer. Naturalist 166 (2005) 368-381
43. Klein E.K., Lavigne C., Picault H., Renard M., and Gouyon P.H. Pollen dispersal of oilseed rape: Estimation of the dispersal function and effects of field dimension. J. Appl. Ecology 43 10 (2006) 141-151
44. Paradis E., Baillie S.R., and Sutherland W.J. Modeling large-scale dispersal distances. Ecological Modelling 151 2-3 (2002) 279-292
45. Jacob N. Pseudo-differential operators and Markov processes. Mathematical Research vol. 94 (1996), Akademie Verlag, Berlin
46. Gerisch A., and Verwer J.G. Operator splitting and approximate factorization for taxis-diffusion-reaction models. Appl. Numer. Math. 42 (2002) 159-176
47. Csomós P., Faragó I., and Havasi Á. Weighted sequential splittings and their analysis. Comput. Math. Appl. 50 (2005) 1017-1031
48. 0-semigroups expression. Semigroup Forum 74 1 (2007) 125-139
49. Marchuk G.I. Some application of splitting-up methods to the solution of mathematical physics problems. Applik. Mat. 13 (1968) 103-132
50. Strang G. Accurate partial difference methods I: Linear Cauchy problems. Arch. Ration. Mech. Anal. 12 (1963) 392-402
51. Strang G. On the construction and comparison of difference schemes. SIAM. J. Numer. Anal 5 (1968) 506-517
52. Arendt W., Batty C., Hieber M., and Neubrander F. Vector-valued Laplace transforms and Cauchy problems. Monographs in Mathematics (2001), Birkhauser-Verlag, Berlin
53. Engel K.J., and Nagel R. One-parameter semigroups for linear evolution equations. Graduate Texts in Mathematics vol. 194 (2000), Springer-Verlag, New York
54. Hille E., and Phillips R.S. Functional Analysis and Semi-Groups (1974), American Mathematical Society, Providence, RI Third printing of the revised edition of 1957, vol. XXXI, American Mathematical Society Colloquium Publications
55. Pazy A. Semigroups of Linear Operators and Applications to Partial Differential equations. Applied Mathematical Sciences vol. 44 (1983), Springer-Verlag, New York
56. Brézis H., and Pazy A. Convergence and approximation of semigroups of nonlinear operators in Banach spaces. J. Funct. Anal. 9 (1972) 63-74
57. Cliff M., Goldstein J.A., and Wacker M. Positivity, Trotter products, and blow-up. Positivity 8 2 (2004) 187-208
58. Miyadera I., and Ôharu S. Approximation of semi-groups of nonlinear operators. Tôhoku Math. J. 22 2 (1970) 24-47
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 (1952) 343-369
61. Schilling R.L. Growth and Hölder conditions for sample paths of Feller processes. Probab. Theory Related Fields 112 (1998) 565-611
62. Sato K.-I. Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics vol. 68 (1999), Cambridge University Press
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., and Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley and Sons, New York
65. Samko S., Kilbas A., and Marichev O. Fractional Integrals and derivatives: Theory and Applications (1993), Gordon and Breach, London
66. Meerschaert M.M., and Scheffler H.P. Semistable Lévy Motion. Fract. Calc. Appl. Anal. 5 (2002) 27-54
67. Balakrishnan A.V. An operational calculus for infinitesimal generators of semigroups. Trans. Amer. Math. Soc. 91 (1959) 330-353
68. Balakrishnan A.V. Fractional powers of closed operators and the semigroups generated by them. Pacific J. Math. 10 (1960) 419-437
69. Arendt W., Grabosch A., Greiner G., Groh U., Lotz H.P., Moustakas U., Nagel R., Neubrander F., and Schlotterbeck U. One-parameter semigroups of positive operators. Lecture Notes in Mathematics vol. 1184 (1986), Springer-Verlag, Berlin
70. van Kooten G.C., and Blute E.H. The Economics of Nature. Managing Biological Assets (2000), Blackwell Publishers Inc.
71. Lockwood D.R., and Hastings A. The effects of dispersal patterns on marine reserves: Does the tail wag the dog?. Theoretical Population Biology 61 (2002) 297-309
72. Zolotarev V.M. One-dimensional stable distributions. Translations of Mathematical Monographs vol. 65 (1986), American Mathematical Society, Providence, RI
73. Nolan J.P. Numerical calculation of stable densities and distribution functions. Heavy tails and highly volatile phenomena. Comm. Statist. Stochastic Models 13 4 (1997) 759-774