On the solutions of time-fractional reaction-diffusion equations

Communications in Nonlinear Science and Numerical Simulation

Volumn 15, Issue 12, 2010, Pages 3847-3854

  Rida, S Z ^a   El Sayed, A M A ^b   Arafa, A A M ^a  

^a SOUTH VALLEY UNIVERSITY   (Egypt)

^b Alexandria University Faculty Of Science   (Egypt)

Author keywords

Differential transform method; Fractional calculus; Reaction diffusion

Indexed keywords

DIFFERENTIATION (CALCULUS); DIFFUSION IN LIQUIDS; LINEAR EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS;

DIFFERENTIAL TRANSFORM METHOD; FRACTIONAL CALCULUS; FRACTIONAL REACTIONS; NEW APPLICATIONS; REACTION DIFFUSION EQUATIONS; REACTION-DIFFUSION;

CALCULATIONS;

EID: 77952744746     PISSN: 10075704     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cnsns.2010.02.007     Document Type: Article

References (64)

1. Murray J.D. Mathematical biology (2003), Springer, New York
2. Kuramoto Y. Chemical oscillations waves and turbulence (2003), Dover Publications, Inc., Mineola, New York
3. Wilhelmsson H., and Lazzaro E. Reaction-diffusion problems in the physics of hot plasmas (2001), Institute of Physics Publishing, Bristol and Philadelphia
4. Hundsdorfer W., and Verwer J.G. Numerical solution of time dependent advection-diffusion-reaction equations (2003), Springer, Berlin
5. Perc M. Spatial coherence resonance in excitable media. Phys Rev E 72 (2005) 016207
6. Perc M. Stochastic resonance on excitable small-world networks via a pacemaker. Phys Rev E 76 (2007) 066203
7. Perc M. Effects of small-world connectivity on noise-induced temporal and spatial order in neural media. Chaos Soliton Fract 31 2 (2007) 280-291
8. Gosaka M., Marhla M., and Perc M. Spatial coherence resonance in excitable biochemical media induced by internal noise. Biophys Chem 128 2-3 (2007) 210-214
9. Kenkre VM, Kuperman MN. Applicability of the Fisher equation to bacterial population dynamics. Phys Rev E 2003;67:051921.
10. Bar M., Gottschalk N., Eiswirth M., and Ertl G. Spiral waves in a surface reaction: model calculations. J Chem Phys 100 (1994) 1202-1214
11. Barkley D. A model for fast computer simulation of excitable media. Physica D 49 (1991) 61-70
12. Karma A. Meandering transition in two-dimensional excitable media. Phys Rev Lett 65 (1990) 2824-2828
13. Keener J.P. A geometrical theory for spiral waves in excitable media. SIAM J Appl Math 46 (1986) 1039-1056
14. Keener J.P. Mathematical physiology, interdisciplinary applied mathematics (1998), Springer, New York
15. Otani N. A primary mechanism for spiral wave meandering. Chaos 12 (2002) 829-842
16. Tyson J. What everyone should know about the Belousov-Zhabotinsky reaction?. Lecture notes in biomathematics vol. 100 (1994), Springer, Berlin 569-587
17. Fenton F., Cherry E., Hastings H.M., and Evans S.J. Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity. Chaos 12 (2002) 852-892
18. Krinsky V., and Pumir A. Models of defibrillation of cardiac tissue. Chaos 8 (1998) 188-203
19. Ramos JI. Numerical methods for one-dimensional reaction-diffusion equations arising in combustion theory. In: Chawla TC, editor. Annual review of numerical fluid mechanics and heat transfer, vol. 1. New York: Hemisphere; 1987. p. 150-261 [Chapter 4].
20. Ramos J.I. Finite element methods for one-dimensional flame propagation problems. In: Chung T.J. (Ed). Numerical modeling in combustion (1993), Taylor & Francis, Washington (DC) 3-131 [Chapter 1]
21. Chen Z., Gumel A.B., and Mickens R.E. Nonstandard discretizations of the generalized Nagumo reaction-diffusion equation. Numer Meth Partial Differ Eqns 19 (2003) 363-379
22. Mickens R.E. A best finite-difference scheme for the Fisher equation. Numer Meth Partial Differ Eqns 10 (1994) 581-585
23. Onyejekwe O.O. Green element procedures for transport accompanied by nonlinear reaction. Int J Therm Sci 42 (2003) 813-820
24. Uddin R. Comparison of the nodal integral method and nonstandard finite-difference schemes for the Fisher equation. SIAM J Sci Comput 22 (2001) 1926-1942
25. Ramos J.I. On diffusive and exponentially fitted techniques. Appl Math Comput 103 (1999) 69-96
26. Henry B.I., Langlands T.A.M., and Wearne S.L. Turing pattern formation in fractional activator-inhibitor systems. Phys Rev E 72 (2005) 026101
27. Henry B.I., and Wearne S.L. Fractional reaction-diffusion. Physica A 276 (2000) 448-455
28. Henry B.I., and Wearne S.L. Existence of Turing instabilities in a two-species fractional reaction-diffusion system. Siam J Appl Math 62 3 (2002) 870-887
29. Vlad M.O., and Ross J. Systematic derivation of reaction-diffusion equations with distributed delays and relations to fractional reaction-diffusion equations and hyperbolic transport equations: application to the theory of Neolithic transition. Phys Rev E 66 (2002) 061908
30. Seki K., Wojcik M., and Tachiya M. Fractional reaction-diffusion equation. J Chem Phys 119 (2003) 2165
31. Gafiychuk V., and Datsko B. Pattern formation in a fractional reaction-diffusion system. Phys A Stat Mech Appl 365 (2006) 300-306
32. Saxena RK, Mathai AM, Haubold HJ. Fractional reaction-diffusion equations. math.CA/0604473 v1 21; April 2006.
33. Gafiychuk V., and Datsko B. Stability analysis and oscillatory structures in time-fractional reaction-diffusion systems. Phys Rev E 75 (2007) 055201-1-1-055201-1-4
34. Weitzner H., and Zaslavsky G.M. Some applications of fractional equations. Commun Nonlinear Sci Numer Simulat 8 (2003) 273-281
35. Varea C., and Barrio R.A. Travelling Turing patterns with anomalous diffusion. J Phys Condens Matter 16 (2004) 5081-5090
36. Havlin S., and Ben-Avraham D. Adv Phys 36 (1987) 695
37. Sokolov I.M., Schmidt M.G.W., and Sagués F. Reaction-subdiffusion equations. Phys Rev E 73 (2006) 031102
38. Henry B.I., Langlands T.A.M., and Wearne S.L. Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations. Phys Rev E 74 (2006) 031116
39. Yuste S.B., and Acedo L. Some exact results for the trapping of subdiffusive particles in one dimension. Physica A 336 (2004) 334-346
40. Lewandowska K.D., and Kosztolowicz T. Numerical study of subdiffusion equation. Acta Phys Polon B 38 (2006) 1847
41. Lewandowsk K.D. Numerical study of subdiffusion equation. Acta Phys Polon B 38 (2007) 1847-1854
42. Bouchaud J.-P., and Georges A. Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys Rep 195 (1990) 127-293
43. Romero A.H., and Sancho J.M. Brownian motion in short range random potentials. Phys Rev E 58 (1998) 2833-2837
44. Klemm A., Metzler R., and Kimmich R. Diffusion on random-site percolation clusters: theory and NMR microscopy experiments with model objects. Phys Rev E 65 (2002) 021112
45. Fisher R.A. The wave of advance of advantageous genes. Ann Eugene 7 (1937) 335-369
46. Malflict W. Solitary wave solutions of nonlinear wave equations. Am J Phys 60 (1992) 650-654
47. Frank D.A. Diffusion and heat exchange in chemical kinetics (1955), Princeton University Press, Princeton (NJ)
48. Aronson D.J., and Weinberg H.F. Nonlinear diffusion in population genetics combustion and never pulse propagation (1988), Springer, New York
49. Canosa J. Diffusion in nonlinear multiplication media. J Math Phys (1969) 1862-1869
50. Tuckwell H.C. Introduction to theoretical neurobiology (1988), Cambridge University Press, Cambridge
51. Bramson M.D. Maximal displacement of branching Brownian motion. Commun Pure Appl Math 31 (1978) 531-581
52. Fitzhugh R. Impulse and physiological states in models of nerve membrane. Biophys J 1 (1961) 445-466
53. Nagumo J.S., Arimoto S., and Yoshizawa S. An active pulse transmission line simulating nerve axon. Proc IRE 50 (1962) 2061-2070
54. Shih M., Momoniat E., and Mahomed F.M. Approximate conditional symmetries and approximate solutions of the perturbed Fitzhugh-Nagumo equation. J Math Phys 46 (2005) 023503
55. Odibat Z., Momani S., and Erturk V.S. Generalized differential transform method: for solving a space- and time-fractional diffusion-wave equation. Phys Lett A 370 (2007) 379-387
56. Momani S., Odibat Z., and Erturk V.S. Generalized differential transform method: application to differential equations of fractional order. Appl Math Comput 19 (2008) 467-477
57. Odibat Z., and Momani S. A generalized differential transform method for linear partial differential equations of fractional order. Appl Maths Lett 21 (2008) 194-199
58. Ayaz F. Solutions of the system of differential equations by differential transform method. Appl Math Comput 147 (2004) 547-567
59. Arikoglu A., and Ozkol I. Solution of boundary value problems for integro-differential equations by using differential transform method. Appl Math Comput 168 (2005) 1145-1158
60. Bildik N., Konuralp A., Bek F., and Kucukarslan S. Solution of different type of the partial differential equation by differential transform method and Adomian's decomposition method. Appl Math Comput 172 (2006) 551-567
61. Hassan I.H. Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems. Chaos Soliton Fract 36 (2008) 53-65
62. Odibat Z., Bertelle C., Aziz-Alaoui M.A., and Duchamp G.H.E. A multi-step differential transform method and application to non-chaotic and chaotic systems. Comput Math Appl 59 (2010) 1462-1472
63. Odibat Z., and Shawagfeh N. Generalized Taylor's formula. Appl Math Comput 186 (2007) 286
64. Caputo M. Linear models of dissipation whose Q is almost frequency independent. Part II. J Roy Astral Soc 13 (1967) 529-539