On the solutions of time-fractional reaction–diffusion equations

Abstract In this paper, a new application of generalized differential transform method (GDTM) has been used for solving time-fractional reaction–diffusion equations. To illustrate the reliability of the method, some examples are provided.

[1]  L. Acedo,et al.  Some exact results for the trapping of subdiffusive particles in one dimension , 2004 .

[2]  I M Sokolov,et al.  Reaction-subdiffusion equations. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  S. Yoshizawa,et al.  An Active Pulse Transmission Line Simulating Nerve Axon , 1962, Proceedings of the IRE.

[4]  Markus Bär,et al.  Spiral waves in a surface reaction: Model calculations , 1994 .

[5]  J. Brandts [Review of: W. Hundsdorfer, J.G. Verwer (2003) Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations] , 2006 .

[6]  Niels F. Otani,et al.  A primary mechanism for spiral wave meandering. , 2002, Chaos.

[7]  Kazuhiko Seki,et al.  Fractional reaction-diffusion equation , 2003 .

[8]  Necdet Bildik,et al.  Solution of different type of the partial differential equation by differential transform method and Adomian's decomposition method , 2006, Appl. Math. Comput..

[9]  José Canosa,et al.  Diffusion in Nonlinear Multiplicative Media , 1969 .

[10]  J. Keener A geometrical theory for spiral waves in excitable media , 1986 .

[11]  A. M. Mathai,et al.  Fractional Reaction-Diffusion Equations , 2006, math/0604473.

[12]  V. M. Kenkre,et al.  Applicability of the Fisher equation to bacterial population dynamics. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Matjaz Perc,et al.  Spatial coherence resonance in excitable biochemical media induced by internal noise. , 2007, Biophysical chemistry.

[14]  Abba B. Gumel,et al.  Nonstandard discretizations of the generalized Nagumo reaction‐diffusion equation , 2003 .

[15]  O. Onyejekwe Green element procedures for transport accompanied by nonlinear reaction , 2003 .

[16]  E. Lazzaro,et al.  Reaction-Diffusion Problems in the Physics of Hot Plasmas , 2000 .

[17]  Henry C. Tuckwell,et al.  Introduction to theoretical neurobiology , 1988 .

[18]  W. Malfliet Solitary wave solutions of nonlinear wave equations , 1992 .

[19]  S L Wearne,et al.  Turing pattern formation in fractional activator-inhibitor systems. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Shaher Momani,et al.  A generalized differential transform method for linear partial differential equations of fractional order , 2008, Appl. Math. Lett..

[21]  V. Krinsky,et al.  Models of defibrillation of cardiac tissue. , 1998, Chaos.

[22]  J. Tyson What Everyone Should Know About the Belousov-Zhabotinsky Reaction , 1994 .

[23]  Ronald E. Mickens,et al.  A best finite‐difference scheme for the fisher equation , 1994 .

[24]  J. Bouchaud,et al.  Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications , 1990 .

[25]  Juan I. Ramos,et al.  On diffusive methods and exponentially fitted techniques , 1999, Appl. Math. Comput..

[26]  I. H. Abdel-Halim Hassan,et al.  Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems , 2008 .

[27]  V. V. Gafiychuk,et al.  Pattern formation in a fractional reaction diffusion system , 2006 .

[28]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

[29]  M. Perc Stochastic resonance on excitable small-world networks via a pacemaker. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  Ibrahim Özkol,et al.  Solution of boundary value problems for integro-differential equations by using differential transform method , 2005, Appl. Math. Comput..

[31]  F. Mahomed,et al.  Approximate conditional symmetries and approximate solutions of the perturbed Fitzhugh-Nagumo equation , 2005 .

[32]  Maury Bramson,et al.  Maximal displacement of branching brownian motion , 1978 .

[33]  Department of Physics,et al.  Some Applications of Fractional Equations , 2003 .

[34]  Juan I. Ramos,et al.  Numerical methods for one-dimensional reaction-diffusion equations arising in combustion theory , 1987 .

[35]  S. Wearne,et al.  Fractional Reaction-Diffusion , 2000 .

[36]  D. Barkley A model for fast computer simulation of waves in excitable media , 1991 .

[37]  T. C. Chawla Annual review of numerical fluid mechanics and heat transfer. Volume 1 , 1986 .

[38]  Shaher Momani,et al.  Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation , 2007 .

[39]  M. A. Aziz-Alaoui,et al.  A multi-step differential transform method and application to non-chaotic or chaotic systems , 2010, Comput. Math. Appl..

[40]  Matjaž Perc,et al.  Effects of small-world connectivity on noise-induced temporal and spatial order in neural media , 2007 .

[41]  M. Caputo Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[42]  Ralf Metzler,et al.  Diffusion on random-site percolation clusters: theory and NMR microscopy experiments with model objects. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  R. Barrio,et al.  Travelling Turing patterns with anomalous diffusion , 2004 .

[44]  S L Wearne,et al.  Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[45]  Shaher Momani,et al.  Generalized differential transform method: Application to differential equations of fractional order , 2008, Appl. Math. Comput..

[46]  Fatma Ayaz,et al.  Solutions of the system of differential equations by differential transform method , 2004, Appl. Math. Comput..

[47]  M. Perc Spatial coherence resonance in excitable media. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[48]  R. FitzHugh Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[49]  S. Levin Lectu re Notes in Biomathematics , 1983 .

[50]  J. Verwer,et al.  Numerical solution of time-dependent advection-diffusion-reaction equations , 2003 .

[51]  D. Frank-Kamenetskii,et al.  Diffusion and heat exchange in chemical kinetics , 1955 .

[52]  Karma Meandering transition in two-dimensional excitable media. , 1990, Physical review letters.

[53]  R. Fisher THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

[54]  F. Fenton,et al.  Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity. , 2002, Chaos.

[55]  Rizwan Uddin Comparison of the Nodal Integral Method and Nonstandard Finite-Difference Schemes for the Fisher Equation , 2001, SIAM J. Sci. Comput..

[56]  T. J. Chung,et al.  Numerical modeling in combustion , 1993 .

[57]  V. Gafiychuk,et al.  Stability analysis and oscillatory structures in time-fractional reaction-diffusion systems. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[58]  BROWNIAN MOTION IN SHORT RANGE RANDOM POTENTIALS , 1998 .

[59]  J. Ross,et al.  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. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[60]  Zaid M. Odibat,et al.  Generalized Taylor's formula , 2007, Appl. Math. Comput..

[61]  S. Wearne,et al.  Existence of Turing Instabilities in a Two-Species Fractional Reaction-Diffusion System , 2002, SIAM J. Appl. Math..