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Distributed order fractional operators offer a rigorous tool for mathematicalmodelling of multi-physics phenomena, where the differential orders aredistributed over a range of values rather than being just a fixedinteger/fraction as it is in standard/fractional ODEs/PDEs. We develop twospectrally-accurate schemes, namely a Petrov-Galerkin spectral method and aspectral collocation method for distributed order fractional differentialequations. These schemes are developed based on the fractional Sturm-Liouvilleeigen-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 employanother space of test functions as the span of poly-fractonomial eigenfunctionsof the FSLP of second kind. We define the underlying \textit{distributedSobolev space} and the associated norms, where we carry out the correspondingdiscrete stability and error analyses of the proposed scheme. In thecollocation scheme, we employ fractional (non-polynomial) Lagrange interpolantssatisfying the Kronecker delta property at the collocation points.Subsequently, we obtain the corresponding distributed differentiation matricesto be employed in the discretization of the strong problem. We performsystematic numerical tests to demonstrate the efficiency and conditioning ofeach method.

Petrov-Galerkin and Spectral Collocation Methods for distributed Order Differential Equations