  We introduce the linear operators of fractional integration and fractional
differentiation in the framework of the Riemann-Liouville fractional calculus.
Particular attention is devoted to the technique of Laplace transforms for
treating these operators in a way accessible to applied scientists, avoiding
unproductive generalities and excessive mathematical rigor. By applying this
technique we shall derive the analytical solutions of the most simple linear
integral and differential equations of fractional order. We show the
fundamental role of the Mittag-Leffler function, whose properties are reported
in an ad hoc Appendix. The topics discussed here will be: (a) essentials of
Riemann-Liouville fractional calculus with basic formulas of Laplace
transforms, (b) Abel type integral equations of first and second kind, (c)
relaxation and oscillation type differential equations of fractional order.
