GrünwaldLetnikov

The Grünwald–Letnikov derivative is a definition of a fractional derivative of order α, extending the notion of integer-order differentiation. It is based on limits of generalized finite differences and provides a natural link between differentiation and discretization.

Formally, for a function f defined on [0, ∞), the Grünwald–Letnikov derivative of order α is given by

D^α f(t) = lim_{h→0+} h^(-α) ∑_{k=0}^{⌊t/h⌋} (-1)^k binom(α, k) f(t − k h),

where the generalized binomial coefficient is binom(α, k) = Γ(α+1)/(Γ(k+1) Γ(α−k+1)). The limit requires suitable regularity of f near t = 0. The definition recovers the ordinary derivative when α is a nonnegative integer and reduces to a fractional integral for negative α.

Relation to other fractional derivatives: for α ∈ (n−1, n] with n ∈ N, and f sufficiently smooth (n-th derivative exists), the Grünwald–Letnikov derivative coincides with the Riemann–Liouville derivative: D^α f(t) = d^n/dt^n [I^{n−α} f](t). The operator is linear and nonlocal, since the value at t depends on the entire past history of f.

Examples and applications: for f(t) = t^β with β > α−1, D^α f(t) = Γ(β+1)/Γ(β−α+1) t^{β−α}. The definition provides a convenient framework for numerical approximation via discrete convolutions, and it is used in modeling anomalous diffusion, viscoelasticity, and other systems with memory effects.

History: the derivative is named after Grünwald and Letnikov, who introduced it independently in the early 20th century, and it remains a foundational definition in fractional calculus alongside Riemann–Liouville and Caputo derivatives.