Unified Fractional Kinetic Equation and a Fractional Diffusion Equation

Abstract

In earlier papers Saxena et al. (2002, 2003) derived the solutions of a number of fractional kinetic equations in terms of generalized Mittag-Leffler functions which extended the work of Haubold and Mathai (2000). The object of the present paper is to investigate the solution of a unified form of fractional kinetic equation in which the free term contains any integrable function f(t), which provides the unification and extension of the results given earlier recently by Saxena et al. (2002, 2003). The solution has been developed in terms of the Wright function in a closed form by the method of Laplace transform. Further we derive a closed-form solution of a fractional diffusion equation. The asymptotic expansion of the derived solution with respect to the space variable is also discussed. The results obtained are in a form suitable for numerical computation.Comment: 13 pages, LaTe

Subordination Principle for a Class of Fractional Order Differential Equations

“…The asymptotic behavior of the three-parameter Mittag-Leffler function (10) can be obtained from the identity [28]:…”

“…The reason for this is that then the operator S(t), defined by (28), will satisfy (16) and (17). Indeed, by (28) and (29) and the identity for the Laplace transform of a C 0 -semigroup…”

Analysis of Solar Neutrino Data from Super-Kamiokande I and II

“…Two distinct processes, the pp-chain and the sub-dominant CNO-cycle, are producing solar neutrinos with different energy spectra and fluxes (see Figure 1). To date, only fluxes from the pp-chain have been measured: 7 Be, 8 B and, indirectly, pp.…”

“…This paper summarizes briefly a research programme, comprised of five elements: (i) standard deviation analysis and diffusion entropy analysis of solar neutrino data [1,2]; (ii) Mathai's entropic pathway model [3,4]; (iii) fractional reaction and extended thermonuclear functions [5,6]; (iv) fractional reaction and diffusion [7,8]; and (v) fractional reaction-diffusion [9][10][11][12]. Boltzmann translated Clausius' second law of thermodynamics "The entropy of the Universe tends to a maximum" into a crucial quantity that links equilibrium and non-equilibrium (time dependent) properties of physical systems and related entropy to probability, S = k log W, which, later, Einstein called Boltzmann's principle [13].…”

Velocity and displacement correlation functions for fractional generalized Langevin equations

“…[41] and [52]), fractional Fokker-Planck equation [34,35,37], generalized Chapman-Kolmogorov equation [33], fractional generalized Langevin equation (FGLE) [12,28,15]. To analyze these equations the properties of different Mittag-Leffler (M-L) type functions [40,48,44,22,38,7,51,19] are of great importance. Thus, Mainardi and Pironi [30] introduced a fractional Langevin equation as a particular case of a GLE, and for the first time represented the velocity and displacement correlation functions in terms of the M-L functions (see also Ref.…”