541 research outputs found
Computable solutions of fractional partial differential equations related to reaction-diffusion systems
The object of this paper is to present a computable solution of a fractional
partial differential equation associated with a Riemann-Liouville derivative of
fractional order as the time-derivative and Riesz-Feller fractional derivative
as the space derivative. The method followed in deriving the solution is that
of joint Laplace and Fourier transforms. The solution is derived in a closed
and computable form in terms of the H-function. It provides an elegant
extension of the results given earlier by Debnath, Chen et al., Haubold et al.,
Mainardi et al., Saxena et al., and Pagnini et al. The results obtained are
presented in the form of four theorems. Some results associated with fractional
Schroeodinger equation and fractional diffusion-wave equation are also derived
as special cases of the findings.Comment: 12 pages, Plain Te
Space-time fractional reaction-diffusion equations associated with a generalized Riemann-Liouville fractional 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 Riemann-Liouville fractional derivative defined in Hilfer et
al. , and the space derivative of second order by the Riesz-Feller fractional
derivative, and adding a function . 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 earlier by Mainardi et al., and the
result very recently given by Tomovski et al.. At the end, extensions of the
derived results, associated with a finite number of Riesz-Feller space
fractional derivatives, are also investigated.Comment: 15 pages, LaTe
Boltzmann-Gibbs Entropy Versus Tsallis Entropy: Recent Contributions to Resolving the Argument of Einstein Concerning "Neither Herr Boltzmann nor Herr Planck has given a definition of W"?
Classical statistical mechanics of macroscopic systems in equilibrium is
based on Boltzmann's principle. Tsallis has proposed a generalization of
Boltzmann-Gibbs statistics. Its relation to dynamics and nonextensivity of
statistical systems are matters of intense investigation and debate. This essay
review has been prepared at the occasion of awarding the 'Mexico Prize for
Science and Technology 2003'to Professor Constantino Tsallis from the Brazilian
Center for Research in Physics.Comment: 5 pages, LaTe
Unified Fractional Kinetic Equation and a Fractional Diffusion Equation
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
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