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

The Laplace transform of fractional order α is determined as

lim M α 0 M E α (-s α x α )f(x)(M-x) α-1 dx

with the Mittag-Leffler function

E α (u)= k=0 u k /Γ(αk+1)·

Replacing derivatives by fractional derivatives, the usual properties of the Laplace transform and also its inversion formula are transferred to this generalized transform.

MSC:
44A10Laplace transform
44A20Integral transforms of special functions
26A33Fractional derivatives and integrals (real functions)
33E12Mittag-Leffler functions and generalizations
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