Laplace transform Mittag-Leffler function

The Mittag-Leffler function is defined through the inverse Laplace transform... 

Note the difference in the 5-dependence of the two coefficients of the right-hand side of Eq. (132). The inverse Laplace transform of the first term yields the Mittag-Leffler function as found in the homogeneous case above. The inverse Laplace transform of the second term is the convolution of the random force and a stationary kernel. The kernel is given by the series... 

It is convenient to use Fox functions (generalized Mellin-Barnes integras) when solving equations with fractional derivatives because Laplace and Fourier transformations for Fox functions may be expressed via Fox functions with given parameters. The connection of Mittag-Leffler functions with Fox functions is as follows [216,217] ... 

Now, on noting that the Laplace transform of the Mittag-Leffler function Ea[-(t/x)°] is [31]... 

For the solution of the fractional-order differential equations, most effective and easy analytic methods were developed based on the formula of the Laplace transform method of the Mittag-Leffler function in two parameters for further details see Refs. [19,38]. 

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