Computational solutions of unified fractional reaction-diffusion equations with composite fractional time derivative

This paper deals with the investigation of the computational solutions of an unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized fractional time-derivative defined by Hilfer (2000), the space derivative of second order by the Riesz-Feller fractional derivative and adding the function phi(x,t) which is a nonlinear function overning reaction. The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of the H-function. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained earlier by Mainardi et al. (2001, 2005) and a result very recently given by Tomovski et al. (2011). Computational representation of the fundamental solution is also obtained explicitly. Fractional order moments of the distribution are deduced. At the end, mild extensions of the derived results associated with a finite number of Riesz-Feller space fractional derivatives are also discussed.

[1]  A. M. Mathai,et al.  On generalized fractional kinetic equations , 2004 .

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

[3]  R. Gorenflo,et al.  Fractional Calculus: Integral and Differential Equations of Fractional Order , 2008, 0805.3823.

[4]  A. M. Mathai,et al.  The fractional kinetic equation and thermonuclear functions , 2000 .

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

[6]  Sune Jespersen,et al.  LEVY FLIGHTS IN EXTERNAL FORCE FIELDS : LANGEVIN AND FRACTIONAL FOKKER-PLANCK EQUATIONS AND THEIR SOLUTIONS , 1999 .

[7]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

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

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

[10]  H. Haken.  Synergetics: Introduction and Advanced Topics , 2004 .

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

[12]  Trifce Sandev,et al.  Generalized space–time fractional diffusion equation with composite fractional time derivative , 2012 .

[13]  ON FRACTIONAL RELAXATION , 2003 .

[14]  Bruce Ian Henry,et al.  Existence of Turing Instabilities in a Two-Species Fractional Reaction-Diffusion System , 2002, SIAM J. Appl. Math..

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

[16]  A. M. Mathai,et al.  A heuristic remark on the periodic variation in the number of solar neutrinos detected on Earth , 1995 .

[17]  W. R. Schneider,et al.  Stable distributions: Fox function representation and generalization , 1986 .

[18]  Hari M. Srivastava,et al.  Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel , 2009, Appl. Math. Comput..

[19]  A. M. Mathai,et al.  On fractional kinetic equations , 2002 .

[20]  Shyam L. Kalla,et al.  Solution of Space-Time Fractional Schrödinger Equation Occurring in Quantum Mechanics , 2010 .

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

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

[23]  Arak M. Mathai,et al.  The H-Function with Applications in Statistics and Other Disciplines. , 1981 .

[24]  J. Gates.  Introduction to Probability and its Applications , 1992 .

[25]  R. Hilfer.  FRACTIONAL TIME EVOLUTION , 2000 .

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

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

[28]  V E Lynch,et al.  Front dynamics in reaction-diffusion systems with Levy flights: a fractional diffusion approach. , 2002, Physical review letters.

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

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

[31]  G. Mittag-Leffler.  Sur la représentation analytique , 1891 .

[32]  A. M. Mathai,et al.  Solutions of certain fractional kinetic equations and a fractional diffusion equation , 2007, 0704.1916.

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

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

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

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

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

[38]  M. M. Djrbashian,et al.  Harmonic analysis and boundary value problems in the complex domain , 1993 .

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

[40]  R. Hilfer.  Threefold Introduction to Fractional Derivatives , 2008 .

[41]  Astrophysical thermonuclear functions , 1993, math/9308209.

[42]  Diego del-Castillo-Negrete,et al.  Front propagation and segregation in a reaction–diffusion model with cross-diffusion , 2002 .

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

[44]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[45]  V. Zolotarev,et al.  Chance and Stability, Stable Distributions and Their Applications , 1999 .

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

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

[48]  Unified Fractional Kinetic Equation and a Fractional Diffusion Equation , 2004, math-ph/0406047.

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

[50]  J. Trujillo,et al.  On the solution of fractional evolution equations , 2004 .