Distributed order reaction-diffusion systems associated with Caputo derivatives

This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation of distributed order associated with the Caputo derivatives as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the joint Laplace and Fourier transforms in compact and closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by other authors, notably by Mainardi et al. [“The fundamental solution of the space-time fractional diffusion equation,” Fractional Calculus Appl. Anal. 4, 153–202 (2001); Mainardi et al. “Fox H-functions in fractional diffusion,” J. Comput. Appl. Math. 178, 321–331 (2005)] for the fundamental solution of the space-time fractional equation, including Haubold et al. [“Solutions of reaction-diffusion equations in terms of the H-function,” Bull. Astron. Soc. India 35, 681–689 (2007)] and Saxena et al. [“Fractional reaction-diffusio...

[1]  A. Wiman.  Über den Fundamentalsatz in der Teorie der FunktionenEa(x) , 1905 .

[2]  A. Wiman,et al.  Über die Nullstellen der FunktionenEa(x) , 1905 .

[3]  William Feller,et al.  An Introduction to Probability Theory and Its Applications, Vol. 2 , 1967 .

[4]  P. A. P. Moran,et al.  An introduction to probability theory , 1968 .

[5]  T. R. Prabhakar.  A SINGULAR INTEGRAL EQUATION WITH A GENERALIZED MITTAG LEFFLER FUNCTION IN THE KERNEL , 1971 .

[6]  H. Srivastava,et al.  A Note on the Convergence of KAMP DE FRIET's Double Hypergeometric Series , 1972 .

[7]  I. Prigogine,et al.  Formative Processes. (Book Reviews: Self-Organization in Nonequilibrium Systems. From Dissipative Structures to Order through Fluctuations) , 1977 .

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

[9]  W. Schneider,et al.  Fractional diffusion and wave equations , 1989 .

[10]  M. Cross,et al.  Pattern formation outside of equilibrium , 1993 .

[11]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[12]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[13]  Igor M. Sokolov,et al.  ANOMALOUS TRANSPORT IN EXTERNAL FIELDS : CONTINUOUS TIME RANDOM WALKS AND FRACTIONAL DIFFUSION EQUATIONS EXTENDED , 1998 .

[14]  I. Podlubny.  Fractional differential equations , 1998 .

[15]  Francesco Mainardi,et al.  Approximation of Levy-Feller Diffusion by Random Walk , 1999 .

[16]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

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

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

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

[20]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

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

[22]  M. Naber.  DISTRIBUTED ORDER FRACTIONAL SUB-DIFFUSION , 2003, math-ph/0311047.

[23]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[24]  L. Beghin,et al.  Time-fractional telegraph equations and telegraph processes with brownian time , 2004 .

[25]  Fawang Liu,et al.  The time fractional diffusion equation and the advection-dispersion equation , 2005, The ANZIAM Journal.

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

[27]  F. Mainardi,et al.  Fox H functions in fractional diffusion , 2005 .

[28]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[29]  Xiaoyi Guo,et al.  Some physical applications of fractional Schrödinger equation , 2006 .

[30]  A. M. Mathai,et al.  Solutions of fractional reaction-diffusion equations in terms of Mittag-Leffler functions , 2006 .

[31]  A. M. Mathai,et al.  Solution of Generalized Fractional Reaction-Diffusion Equations , 2006 .

[32]  V. V. Gafiychuk,et al.  Mathematical modeling of pattern formation in sub- and supperdiffusive reaction-diffusion systems , 2006, nlin/0611005.

[33]  A. M. Mathai,et al.  Reaction-Diffusion Systems and Nonlinear Waves , 2006 .

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

[35]  F. Mainardi,et al.  The fundamental solution of the space-time fractional diffusion equation , 2007, cond-mat/0702419.

[36]  V. V. Gafiychuk,et al.  Nonlinear oscillations and stability domains in fractional reaction-diffusion systems , 2007 .

[37]  Fawang Liu,et al.  THE FUNDAMENTAL AND NUMERICAL SOLUTIONS OF THE RIESZ SPACE-FRACTIONAL REACTION–DISPERSION EQUATION , 2008, The ANZIAM Journal.

[38]  A. M. Mathai,et al.  The H-Function: Theory and Applications , 2009 .

[39]  Hans Engler,et al.  On the Speed of Spread for Fractional Reaction-Diffusion Equations , 2009, 0908.0024.

[40]  Francesco Mainardi,et al.  Evolution equations for the probabilistic generalization of the Voigt profile function , 2007, J. Comput. Appl. Math..

[41]  L. Boyadjiev,et al.  INTEGRAL TRANSFORMS METHOD TO SOLVE A TIME-SPACE FRACTIONAL DIFFUSION EQUATION , 2010 .

[42]  Arak M. Mathai,et al.  The H-Function , 2010 .

[43]  R. K. Saxena,et al.  Computational solution of a fractional generalization of the Schrödinger equation occurring in quantum mechanics , 2010, Appl. Math. Comput..

[44]  Arak M. Mathai,et al.  Further solutions of fractional reaction-diffusion equations in terms of the H-function , 2007, J. Comput. Appl. Math..

[45]  A. M. Mathai,et al.  Analysis of Solar Neutrino Data from SuperKamiokande I and II: Back to the Solar Neutrino Problem , 2012 .

[46]  M. T. Cicero.  FRACTIONAL CALCULUS AND WAVES IN LINEAR VISCOELASTICITY , 2012 .

[47]  Arak M. Mathai,et al.  Analysis of Solar Neutrino Data from Super-Kamiokande I and II , 2014, Entropy.

[48]  William Feller,et al.  On a Generalization of Marcel Riesz’ Potentials and the Semi-Groups generated by them , 2015 .