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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:
35A08 Fundamental solutions to PDEs
35A22 Transform methods (e.g., integral transforms) applied to PDEs
26A33 Fractional derivatives and integrals
33E12 Mittag-Leffler functions and generalizations
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
44A10 Laplace transform
45K05 Integro-partial differential equations
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