  The Laplace transform method for solving of a wide class of initial value
problems for fractional differential equations is introduced. The method is
based on the Laplace transform of the Mittag-Leffler function in two
parameters. To extend the proposed method for the case of so-called
"sequential" fractional differential equations, the Laplace transform for the
''sequential'' fractional derivative is also obtained. Besides that, tools
necessary for testing candidate solutions by direct substitution in
corresponding equations are introduced: fractional derivatives of the
Mittag-Leffler function and the rule for the fractional differentiation of
integrals depending on a parameter. Definition of the fractional Green's
function is given and some of its properties, necessary for constructing
solutions of initial-value problems for fractional linear differential
equations, are presented. Explicit expressions for the fractional Green's
function for the special cases of one-, two-, three- and four-term equations
are given, as well as the explicit expression for an arbitrary n-term
fractional linear ordinary differential equation with constant coefficients.
Several examples of solution of various types of fractional differential
equations, including fractional diffusion-wave equation. The bibliography (54
items) covers many field of possible application of fractional derivatives.
