Theoretical and Numerical Results for Fractional Difference and Differential Equations

The fractional calculus is relatively an old topic dating back to more than 300 years. In 1695, L’Hopital wrote to Leibniz asking him about the notation of the nth derivative (Dnf/dxn) which was mentioned for the first time in his publications. L’Hopital asked Leibniz about the result if n = 1/2. Leibniz’s response was as follows: An apparent paradox, from which one day useful consequences will be drawn. This communication was what created the so-called fractional calculus. However, the gate to the fractional calculus subject was opened in 1832 by Liouville in a series of papers from 1832 to 1837. Later, the intensive investigations on the results of Liouville by Riemann led to the construction of the RiemannLiouville fractional integral operator, given by Jαy (x) = 1 Γ (α) ∫ x