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Fractional oscillations and mittag-leffler functions (1996).
| Content Provider | CiteSeerX |
|---|---|
| Author | Gorenflo, Rudolf Mainardi, Francesco |
| Abstract | 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 ff with 1 ! ff ! 2 : Using the method of 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 we also discuss both the cases 0 ! ff ! 1 (fractional relaxation) and 2 ! ff 3 (growing oscillations), showing the key role of the Mittag-Leffler functions. 1991 Mathematics Subject Classification: 26A33, 33E20, 33E30, 45E10, 45J05, 70J99. Key Words: Fractional differential equations, fractional calculus, Mittag-Leffler functions. 1. Introduction For real ff ? 0 (later only for 0 ! ff 3) we consider the fractional differential equation D ff / u(t) \Gamma m\Gamma1 X k=0 t k k! u (k) (0 + ) ! = \Gammau(t) + q(t) ; t ? 0 ; (1:1) where q(t... |
| File Format | |
| Publisher Date | 1996-01-01 |
| Access Restriction | Open |
| Subject Keyword | Mittag-leffler Function Fractional Oscillation Fractional Oscillation Equation Gamma Gamma1 Real Ff Damped Oscillation Second-order Time Finite Number Fractional Relaxation Linear Oscillation Fractional Differential Equation Ff Fractional Differential Equation Key Role Laplace Transform Mathematics Subject Classification Fundamental Solution Order Ff Algebraic Decay Fractional Derivative Fractional Calculus Classical Equation |
| Content Type | Text |
