The role of the Fox-Wright functions in fractional sub-diffusion of distributed order

Authors:
Francesco Mainardi;Gianni Pagnini
Affiliations:
Department of Physics, University of Bologna, and INFN, Via Irnerio 46, I-40126 Bologna, Italy;ENEA: National Agency for New Technologies, Energy and the Environment, Centre "E. Clementel", Via Martiri di Monte Sole 4, I-40129 Bologna, Italy
Venue:
Journal of Computational and Applied Mathematics
Year:
2007

Citing 8
Cited 2

Integrals and series of special functions

Integrals and series of special functions
Wright functions as scale-invariant solutions of the diffusion-wave equation

Journal of Computational and Applied Mathematics - Special issue on higher transcendental functions and their applications
On Mittag-Leffler-type functions in fractional evolution processes

Journal of Computational and Applied Mathematics - Special issue on higher transcendental functions and their applications
Salvatore Pincherle: the pioneer of the Mellin-Barnes integrals

Journal of Computational and Applied Mathematics - Proceedings of the sixth international symposium on orthogonal polynomials, special functions and their applications
The Wright functions as solutions of the time-fractional diffusion equation

Applied Mathematics and Computation - Special issue: Advanced special functions and related topics in differential equations, third Melfi workshop, proceedings of the Melfi school on advanced topics in mathematics and physics
Fox H functions in fractional diffusion

Journal of Computational and Applied Mathematics - Special issue: Proceedings of the seventh international symposium on Orthogonal polynomials, special functions and applications
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,

Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies)

Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies)

Anomalous diffusion modeling by fractal and fractional derivatives

Computers & Mathematics with Applications
On distributed order integrator/differentiator

Signal Processing

Quantified Score

Hi-index	7.29

Visualization

Abstract

The fundamental solution of the fractional diffusion equation of distributed order in time (usually adopted for modelling sub-diffusion processes) is obtained based on its Mellin-Barnes integral representation. Such solution is proved to be related via a Laplace-type integral to the Fox-Wright functions. A series expansion is also provided in order to point out the distribution of time-scales related to the distribution of the fractional orders. The results of the time fractional diffusion equation of a single order are also recalled and then re-obtained from the general theory.