Fractional PDE models generalize the standard (integer-order) calculus and PDEs to any differential form of fractional orders. Fractional PDEs open up new possibilities for robust mathematical modeling of physical processes that exhibit anomalous (sub- or super-) diffusion, nonlocal interactions, self-similar structures, long memory dependence, and power-law effects. Fractional PDEs are emerging as the right tool for exploring fractal operators and for modeling sharp interfaces in multi-phase problems, wave propagation in disordered media, and multi-scale materials. Such phenomena occur in many applications, including non-Gaussian (Levy) processes in turbulent flows, non-Newtonian fluids and rheology, non-Brownian transport phenomena in porous and disordered materials, and non-Markovian processes in multi-scale complex fluids.

In such applications, fractional PDEs naturally appear as the right governing equations leading to multi-fidelity modeling and predictive simulations, which otherwise cannot be achieved by employing the standard PDEs. However, the extension of existing numerical methods to fractional PDEs is not trivial because of their non-local and history-dependent nature. To this end, we first present a new theory on Fractional Sturm-Liouville Problems, which serves as a fundamental spectral theory providing explicit (non-polynomial) eigenfunctions, namely as "Jacobi Poly-fractonomials." These eigenfunctions extend the well-known family of Jacobi polynomials to their fractional counterparts. Based upon this base fractional spectral theory, we develop a series of high-order numerical methods that efficiently treat fixed-order, variable-order, and distributed-order fractional PDEs in low-to-high dimensions. Finally, introduced is the wealth of fractional-order modeling in the context of fractional conservation laws, chaotic quasi-geostrophic flows, zonal flows & incomplete mixing, in addition to the fractional turbulence modeling.