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The role of the Fox-Wright functions in fractional sub-diffusion of distributed order. (English) Zbl 1120.35002
Summary: 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.

MSC:
35A08Fundamental solutions of PDE
35A22Transform methods (PDE)
26A33Fractional derivatives and integrals (real functions)
33E12Mittag-Leffler functions and generalizations
33C45Orthogonal polynomials and functions of hypergeometric type
33C60Hypergeometric integrals and functions defined by them
44A10Laplace transform
45K05Integro-partial differential equations