Fractional calculus extends differentiation and integration to non‑integer orders. Instead of just first or second derivatives, it allows half, 0.3 or even complex order derivatives. This generalization captures memory and hereditary effects in materials and processes that ordinary calculus cannot describe well.

Mathematically, fractional derivatives can be defined in several ways. Common approaches include the Riemann–Liouville and Caputo definitions, which use integrals with power law kernels. These kernels weight past values of a function, so the current rate of change depends on the entire history, not only the present state. This property makes fractional models powerful for complex systems.

In physics and engineering, fractional calculus is used to model anomalous diffusion, viscoelastic behavior, and systems with long‑range interactions. For instance, diffusion in disordered media often deviates from the classical square‑root‑of‑time law, and fractional diffusion equations fit experimental data more accurately. In viscoelasticity, fractional constitutive laws describe materials that behave between pure solids and pure fluids.

In control theory, fractional order controllers may outperform classical PID controllers for processes with significant memory or complexity. In signal processing, fractional derivatives help in edge detection and texture analysis. In biology and medicine, they model phenomena like cardiac tissue dynamics or drug transport where nonlocal effects matter.

Current research focuses on rigorous foundations, efficient numerical methods, and stable algorithms for solving fractional differential equations. Researchers also explore variable‑order derivatives, where the order changes in time or space, to better capture evolving complexity and heterogeneity in real systems.