fractionalorder

Fractional order or fractional calculus studies derivatives and integrals of non-integer order. The order α is a real (or complex) number, commonly α>0. Fractional derivatives generalize classical integer-order derivatives and provide models with memory and hereditary properties, useful in describing processes whose future evolution depends on their history.

Several definitions exist for fractional derivatives. The Riemann-Liouville derivative D^α f is defined for n−1<α≤n by D^α f(t) = d^n/dt^n I^{n−α} f(t), where I^{β} is the fractional integral I^{β} f(t) = 1/Γ(β) ∫_0^t (t−τ)^{β−1} f(τ) dτ. The Caputo derivative D^α_C f(t) = I^{n−α} f^{(n)}(t) uses integer-order derivatives of f and is convenient for incorporating standard initial conditions. The Grunwald-Letnikov definition D^α f(t) = lim_{h→0} h^{−α} ∑_{k=0}^{⌊t/h⌋} (−1)^k binomial(α, k) f(t − k h) is widely used for numerical approximations and agrees with RL and Caputo under suitable conditions.

For smooth functions, fractional derivatives of power functions follow D^α t^p = Γ(p+1)/Γ(p+1−α) t^{p−α} (p > −1). Fractional integration satisfies I^α t^p = Γ(p+1)/Γ(p+1+α) t^{p+α}.

Applications span viscoelasticity, anomalous diffusion, control theory, signal processing, and other fields where memory effects and nonlocal behavior are important. Numerical methods include Grunwald-Letnikov discretizations and various approximation techniques, with Mittag-Leffler functions often appearing in analytical solutions. The development of fractional calculus traces to early ideas by Liouville and Riemann, with Caputo and others refining formulations for physical modeling.