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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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