Towards a Unified theory of Fractional and Nonlocal Vector Calculus

Nonlocal and fractional-order models capture effects that classical partial differential equations cannot describe; for this reason, they are suitable a broad class of engineering scientific applications feature multiscale or anomalous behavior. This has driven desire vector calculus includes nonlocal fractional gradient, divergence Laplacian type operators, as well tools such Green’s identities, to model subsurface transport, turbulence, conservation laws. In the literature, several independent definitions theories have been put forward. Some studied rigorously in depth, while others introduced ad-hoc specific applications. The goal work is provide foundations unified by (1) consolidating special case calculus, (2) relating unweighted weighted operators introducing an equivalence kernel, (3) proving form identity unify corresponding variational frameworks resulting volume-constrained problems. proposed framework goes beyond analysis supporting new discovery, establishing theory interpretation providing useful analogues standard from calculus.