This paper deals with the investigation of the computational solutions of a unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized Riemann–Liouville fractional derivative defined by others and the space derivative of second order by the Riesz–Feller fractional derivative and adding a function ɸ(x, t). The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of Mittag–Leffler functions. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained by others and the result very recently given by others. At the end, extensions of the derived results, associated with a finite number of Riesz–Feller space fractional derivatives, are also investigated.

We are going back to the roots of the original solar neutrino problem: the analysis of data from solar neutrino experiments. The application of standard deviation analysis (SDA) and diffusion entropy analysis (DEA) to the Super-Kamiokande I and II data reveals that they represent a non-Gaussian signal. The Hurst exponent is different from the scaling exponent of the probability density function, and both the Hurst exponent and scaling exponent of the probability density function of the Super-Kamiokande data deviate considerably from the value of 0.5, which indicates that the statistics of the underlying phenomenon is anomalous. To develop a road to the possible interpretation of this finding, we utilize Mathai’s pathway model and consider fractional reaction and fractional diffusion as possible explanations of the non-Gaussian content of the Super-Kamiokande data.

INCOMPLETE EXTENDED HURWITZ-LERCH ZETA FUNCTIONS AND ASSOCIATED PROPERTIES gamma functions;incomplete gamma functions;Pochhammer symbol;incomplete Pochhammer symbols;incomplete generalized hypergeometric functions;incomplete Fox-Wright function;generalized Hurwitz-Lerch zeta function;incomplete Hurwitz-Lerch zeta function;fractional derivative operator;probability density function; Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [Integral Transforms Spec. Funct. 23 (2012), 659-683] by means of the incomplete Pochhammer symbols $({\lambda};{\kappa})_{\nu}$ and $[{\lambda};{\kappa}]_{\nu}$, we first introduce incomplete Fox-Wright function. We then define the families of incomplete extended Hurwitz-Lerch Zeta function. We then systematically investigate several interesting properties of these incomplete extended Hurwitz-Lerch Zeta function which include various integral representations, summation formula, fractional derivative formula. We also consider an application to probability distributions and some special cases of our main results.

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-diffusion equations,” Astrophys. Space Sci. 305, 289–296 (2006a)] for fractional reaction-diffusion equations. The advantage of using the Riesz-Feller derivative lies in the fact that the solution of the fractional reaction-diffusion equation, containing this derivative, includes the fundamental solution for space-time fractional diffusion, which itself is a generalization of fractional diffusion, space-time fraction diffusion, and time-fractional diffusion, see Schneider and Wyss [“Fractional diffusion and wave equations,” J. Math. Phys. 30, 134–144 (1989)]. These specialized types of diffusion can be interpreted as spatial probability density functions evolving in time and are expressible in terms of the H-function in compact forms. The convergence conditions for the double series occurring in the solutions are investigated. It is interesting to observe that the double series comes out to be a special case of the Srivastava-Daoust hypergeometric function of two variables given in Appendix B of this paper. Fractional reaction-diffusion equations are of specific interest in physics for non-Gaussian, non-Markovian, and non-Fickian phenomena.

In this paper, we consider generalized space-time fractional cable equation in presence of external source. By using the Fourier-Laplace transform we obtain the Green function in terms of infinite series in H-functions. The fractional moments of the fundamental solution are derived and their asymptotic behavior in the short and long time limit is analyzed. Some previously obtained results are compared with those presented in this paper. By using the Bernstein characterization theorem we find the conditions under which the even moments are non-negative.