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Jumarie, Guy

Laplace’s transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative. (English) Zbl 1181.44001

Appl. Math. Lett. 22, No. 11, 1659-1664 (2009).
The Laplace transform of fractional order \(\alpha\) is determined as \[ \lim_{M\to\infty} \alpha\int^M_0 E_\alpha(-s^\alpha x^\alpha) f(x)(M- x)^{\alpha-1} dx \] with the Mittag-Leffler function \[ E_\alpha(u)= \sum^\infty_{k=0} u^k/\Gamma(\alpha k+ 1). \] Replacing derivatives by fractional derivatives, the usual properties of the Laplace transform and also its inversion formula are transferred to this generalized transform.
Reviewer: Lothar Berg (Rostock)

Cited in 56 Documents

MSC:

44A10 Laplace transform
44A20 Integral transforms of special functions
26A33 Fractional derivatives and integrals
33E12 Mittag-Leffler functions and generalizations

Keywords:

fractional Laplace transform; Mittag-Leffler function; fractional derivatives; inversion formula
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\textit{G. Jumarie}, Appl. Math. Lett. 22, No. 11, 1659--1664 (2009; Zbl 1181.44001)
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References:

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