Towards a Unified Theory of Fractional and Nonlocal Vector Calculus

Understand

Nonlocal and fractional-order models capture effects that classical partial differential equations cannot describe; for this reason, they are suitable for a broad class of engineering and scientific applications that feature multiscale or anomalous behavior.

• This has driven a desire for a vector calculus that includes nonlocal and fractional gradient, divergence and Laplacian type operators, as well as tools such as Green's identities, to model subsurface transport, turbulence, and conservation laws.
• In the literature, several independent definitions and theories of nonlocal and fractional vector calculus have been put forward.
• Some have been studied rigorously and in depth, while others have been introduced ad-hoc for specific applications.