RiemannLiouville

Riemann-Liouville calculus refers to a pair of fractional operators that generalize the ordinary integral and derivative to non-integer orders. Developed in the 19th century by Liouville and Riemann, these operators are nonlocal: the value of the operator at a given point depends on the function over an interval from a starting point to that point. This nonlocality makes them useful for modeling memory effects and anomalous processes in various fields.

The Riemann-Liouville fractional integral of order α > 0, with lower limit a, is defined by

I^α_a f(t) = (1 / Γ(α)) ∫_a^t (t − τ)^(α−1) f(τ) dτ,

for t > a, where Γ is the gamma function. The kernel (t − τ)^(α−1) imparts a power-law weighting of the history of f.

The Riemann-Liouville fractional derivative of order α > 0 is defined as

D^α_a f(t) = d^n/dt^n [ I^(n−α)_a f(t) ],

where n = ⌈α⌉ is the smallest integer not less than α. For integer α = m, D^m_a f(t) reduces to the ordinary m-th derivative provided f is sufficiently smooth and appropriate initial data are specified. In general, RL derivatives require initial terms expressed through fractional integrals of f.

Key properties include linearity, and the semigroup relation I^α_a I^β_a = I^(α+β)_a. In the Laplace domain, L{I^α_a f} = s^(−α) F(s), and L{D^α_a f} involves an algebraic term dependent on initial values at a.

Applications span fractional differential equations, viscoelasticity, anomalous diffusion, control theory, and signal processing. The Riemann-Liouville framework is often compared with the Caputo derivative, which uses more conventional initial conditions.