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Distributed-order PDEs are tractable mathematical models for complexmultiscaling anomalous transport, where derivative orders are distributed overa range of values. We develop a fast and stable Petrov-Galerkin spectral methodfor such models by employing Jacobi \textit{poly-fractonomial}s and Legendrepolynomials as temporal and spatial basis/test functions, respectively. Bydefining the proper underlying \textit{ distributed Sobolev} spaces and theirequivalent norms, we prove the well-posedness of the weak formulation, andthereby carry out the corresponding stability and error analysis. We finallyprovide several numerical simulations to study the performance and convergenceof proposed scheme.

Petrov-Galerkin Method for Fully Distributed-Order Fractional Partial Differential Equations