Fractional time evolution. (English) Zbl 0994.34050

Hilfer, R.

Hilfer, R. (ed.), Applications of fractional calculus in physics. Singapore: World Scientific. 87-130 (2000).

The author discusses the terms semigroup, continuity, homogeneity, casuality and coarse graining in order to define time evolution. The main interest of this article lies in fractional evolution equations and their emergence from coarse graining. Explicit solutions to generalized fractional relaxation equations are obtained in terms of Mittag-Leffler functions by the application of Laplace transform. Similarly, generalized fractional relaxation equations are solved in terms of Fox’s \(H\)-function by the application of Fourier-Laplace transforms. At the end of the article, some basic properties of Fox’s \(H\)-function are given in the Appendix.
For the entire collection see [Zbl 0998.26002].

Reviewer: Ram Kishore Saxena (Jodhpur)

MSC:

34G25	Evolution inclusions
60H35	Computational methods for stochastic equations (aspects of stochastic analysis)

Keywords:

evolution inclusions; coarse graining; stable averages; fractional relaxation; fractional diffusion and \(H\)-function; Mittag-Leffler functions