Caputo

Caputo's fractional derivative is a mathematical concept used to describe processes that exhibit memory and hereditary properties, unlike traditional derivatives that consider only local, instant rates of change. It extends the idea of differentiation to non-integer (fractional) orders, allowing for more accurate modeling of complex systems such as viscoelastic materials, anomalous diffusion, and signal processing. The Caputo derivative is particularly useful because it incorporates initial conditions in a way that's compatible with standard physical and engineering problems, making it both flexible and practical for real-world applications involving non-local dynamics.