  Distributed order fractional operators offer a rigorous tool for mathematical
modelling of multi-physics phenomena, where the differential orders are
distributed over a range of values rather than being just a fixed
integer/fraction as it is in standard/fractional ODEs/PDEs. We develop two
spectrally-accurate schemes, namely a Petrov-Galerkin spectral method and a
spectral collocation method for distributed order fractional differential
equations. These schemes are developed based on the fractional Sturm-Liouville
eigen-problems (FSLPs). In the Petrov-Galerkin method, we employ fractional
(non-polynomial) basis functions, called \textit{Jacobi poly-fractonomials},
which are the eigenfunctions of the FSLP of first kind, while, we employ
another space of test functions as the span of poly-fractonomial eigenfunctions
of the FSLP of second kind. We define the underlying \textit{distributed
Sobolev space} and the associated norms, where we carry out the corresponding
discrete stability and error analyses of the proposed scheme. In the
collocation scheme, we employ fractional (non-polynomial) Lagrange interpolants
satisfying the Kronecker delta property at the collocation points.
Subsequently, we obtain the corresponding distributed differentiation matrices
to be employed in the discretization of the strong problem. We perform
systematic numerical tests to demonstrate the efficiency and conditioning of
each method.
