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

Distributed-order PDEs are tractable mathematical models for complex multiscaling anomalous transport, where derivative orders are distributed over a range of values. We develop a fast and stable Petrov-Galerkin spectral method for such models by employing Jacobi \textit{poly-fractonomial}s and Legendre polynomials as temporal and spatial basis/test functions, respectively. By defining the proper underlying \textit{ distributed Sobolev} spaces and their equivalent norms, we prove the well-posedness of the weak formulation, and thereby carry out the corresponding stability and error analysis. We finally provide several numerical simulations to study the performance and convergence of proposed scheme.