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Fractional oscillations and Mittag-Leffler functions. (English) Zbl 0916.34011
Hamoui, Adnan (ed.) et al., International workshop on the recent advances in applied mathematics, RAAM ’96, State of Kuwait, Kuwait, May 4–7, 1996. Kuwait: Kuwait Univ., Department of Mathematics and Computer Science, 193-208 (1996).
Summary: The fractional oscillation equation is obtained from the classical equation for linear oscillations by replacing the second-order time derivative by a fractional derivative of order $$\alpha$$ with $$1<\alpha<2$$. Using the Laplace transform, it is shown that the fundamental solutions can be expressed in terms of Mittag-Leffler functions, and exhibit a finite number of damped oscillations with an algebraic decay. For completeness the authors discuss both the cases $$0<\alpha<1$$ (fractional relaxation) and $$2<\alpha\leq 3$$ (growing oscillations), showing the key role of the Mittag-Leffler functions.
For the entire collection see [Zbl 0879.00037].

##### MSC:
 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. 26A33 Fractional derivatives and integrals 33E20 Other functions defined by series and integrals 33E30 Other functions coming from differential, difference and integral equations 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 45J05 Integro-ordinary differential equations 70J35 Forced motions in linear vibration theory
##### Keywords:
linear oscillations; Mittag-Leffler functions