A Petrov-Galerkin Spectral Method of Linear Complexity for Fractional Multiterm ODEs on the Half Line

Abstract. We present a new tunably-accurate Laguerre Petrov-Galerkin spectral method for solving linear multi-term fractional initial value problems with derivative orders at most one and constant coe cients on the half line. Our method results in a matrix equation of special structure which can be solved in O(N logN) operations. We also take advantage of recurrence relations for the generalized associated Laguerre functions (GALFs) in order to derive explicit expressions for the entries of the sti↵ness and mass matrices, which can be factored into the product of a diagonal matrix and a lower-triangular Toeplitz matrix. The resulting spectral method is e cient for solving multi-term fractional di↵erential equations with arbitrarily many terms, which we demonstrate by solving a fifty-term example. We apply this method to a distributed order di↵erential equation, which is approximated by linear multi-term equations through the Gauss-Legendre quadrature rule. We provide numerical examples demonstrating the spectral convergence and linear complexity of the method.