differentioituvia

Differentioituvia is a theoretical framework in mathematics that generalizes the notions of differentiation and integration through a one-parameter family of linear operators {D_α} defined for α in [0,1]. For suitably smooth functions on an interval, D_1 f is the standard derivative f', and D_0 f is an accumulation operator that behaves like an integral. The intermediate operators D_α interpolate between local rates of change and cumulative effects, providing a continuum of transforms intended to model processes that exhibit both instantaneous and memory-based dynamics.

Symbolically, the operators are linear and can be realized through kernel representations or via a functional calculus with a distinguished operator L, such that D_α = φ_α(L) for a family of functions φ_α. They are designed to satisfy regularity conditions and a tractable composition rule, allowing the combination of consecutive transforms to yield another member of the family. This unifies certain classes of differential and integral equations under one framework.

Origin and nomenclature: the idea emerged in the late 2010s within the field of applied analysis, and the name incorporates roots meaning "to differentiate" and "to accumulate."

Applications and reception: potential uses include modeling systems with memory (for example, viscoelastic materials or anomalous diffusion), formulating boundary value problems that interpolate between differential and integral formulations, and guiding numerical schemes that adapt the α parameter to data. Some critics question redundancy with established approaches such as fractional calculus, while others see it as a conceptual umbrella for memory-enabled processes. The framework remains exploratory and under active development.