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Fox function representation of non-Debye relaxation processes. (English) Zbl 0945.82559
Summary: Applying the Liouville-Riemann fractional calculus, we derive and solve a fractional operator relaxation equation. We demonstrate how the exponent $\beta$ of the asymptotic power law decay $\sim t^{-\beta}$ relates to the order $\nu$ of the fractional operator $d^{\nu} / dt^{\nu}$ $(0 < \nu < 1)$. Continuous-time random walk (CTRW) models offer a physical interpretation of fractional order equations, and thus we point out a connection between a special type of CTRW and our fractional relaxation model. Exact analytical solutions of the fractional relaxation equation are obtained in terms of Fox functions by using Laplace and Mellin transforms. Apart from fractional relaxation, Fox functions are further used to calculate Fourier integrals of Kohlrausch-Williams-Watts type relaxation functions. Because of its close connection to integral transforms, the rich class of Fox functions forms a suitable framework for discussing slow relaxation phenomena.

82C41Dynamics of random walks, random surfaces, lattice animals, etc.
33C60Hypergeometric integrals and functions defined by them
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