nonintegerorder

Noninteger order refers to mathematical operations whose order is not an integer, most commonly in fractional calculus. The order α can be any real number (and sometimes complex), generalizing differentiation and integration beyond the integer case. For α>0 a fractional derivative D^α extends the notion of ordinary differentiation; for α<0 a fractional integral I^α extends integration. When α is an integer, the fractional operator reduces to the standard derivative or integral.

Several definitions are used in fractional calculus. The Riemann-Liouville definition expresses D^α f as d^n/dt^n applied to I^{n−α} f, where n−1<α<n. The Caputo definition rearranges the order to apply I^{n−α} after taking n-th order derivatives, which makes initial conditions resemble those in integer-order problems. The Grünwald-Letnikov approach defines D^α f via a limit of finite-difference quotients with binomial coefficients, providing a more direct link to numerical schemes.

A key feature of noninteger order operators is nonlocality: the value of D^α f at a point depends on the function’s history, imparting memory effects that are absent in integer-order calculus. This property helps model real-world phenomena such as materials with fading memory and anomalous diffusion.

Applications span science and engineering. Fractional models appear in viscoelasticity, diffusion processes, electrical circuits, control systems, and signal processing. A common example is D^α t^β = Γ(β+1)/Γ(β+1−α) t^{β−α} for suitable β, illustrating how fractional orders modify scaling and dynamics.

Numerical methods for fractional differential equations (FDEs) include Grünwald-Letnikov discretizations and predictor-corrector schemes, alongside specialized techniques to handle nonlocal operators and initial conditions.