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We present a new tunably-accurate Laguerre Petrov-Galerkin spectral methodfor solving linear multi-term fractional initial value problems with derivativeorders at most one and constant coefficients on the half line. Our methodresults in a matrix equation of special structure which can be solved in$\mathcal{O}(N \log N)$ operations. We also take advantage of recurrencerelations for the generalized associated Laguerre functions (GALFs) in order toderive explicit expressions for the entries of the stiffness and mass matrices,which can be factored into the product of a diagonal matrix and alower-triangular Toeplitz matrix. The resulting spectral method is efficientfor solving multi-term fractional differential equations with arbitrarily manyterms. We apply this method to a distributed order differential equation, whichis approximated by linear multi-term equations through the Gauss-Legendrequadrature rule. We provide numerical examples demonstrating the spectralconvergence and linear complexity of the method.

A Tunably-Accurate Laguerre Petrov-Galerkin Spectral Method for Multi-Term Fractional Differential Equations on the Half Line