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

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*Article:*

“**A Petrov-Galerkin Spectral Method of Linear Complexity for Fractional Multi-Term ODEs on the Half Line**”

Anna Lischke, Mohsen Zayernouri, George Em Karniadakis

*SIAM Journal of Scientific Computing, Volume 39, Issue 3, Pages A922-A946, 25 May 2017*

Link: __http://epubs.siam.org/doi/abs/10.1137/17M1113060__

**Abstract**

We present a new tunably accurate Laguerre Petrov-Galerkin spectral method for solving linear multiterm fractional initial value problems with derivative orders at most one and constant coefficients on the half line. Our method results in a matrix equation of special structure which can be solved in $\mathcal{O}(N \log N)$ 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 stiffness 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 efficient for solving multiterm fractional differential equations with arbitrarily many terms, which we demonstrate by solving a fifty-term example. We apply this method to a distributed order differential equation, which is approximated by linear multiterm equations through the Gauss-Legendre quadrature rule. We provide numerical examples demonstrating the spectral convergence and linear complexity of the method.