Methods and Algorithms for Scientific Computing

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


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.


  1. spectral accuracy
  2. linear complexity
  3. singular solutions
  4. tunable accuracy
  5. distributed order

MSC codes

  1. 34L10
  2. 58C40
  3. 34K28
  4. 65M70
  5. 65M60

Get full access to this article

View all available purchase options and get full access to this article.


T. M. Atanackovic, M. Budincevic, and S. Pilipovic, On a fractional distributed-order oscillator, J. Phys. A, 38 (2005), pp. 6703--6713,
D. Baleanu, A. Bhrawy, and T. Taha, A modified generalized Laguerre spectral method for fractional differential equations on the half line, Abstract Appl. Anal., (2013), 413529,
A. Bhrawy, D. Baleanu, and L. Assas, Efficient generalized Laguerre spectral methods for solving multi-term fractional differential equations on the half line, J. Vibration Control, 20 (2013), pp. 973--985,
H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, vol. 15, Cambridge University Press, Cambridge, UK, 2004.
M. Caputo, Distributed order differential equations modelling dielectric induction and diffusion, Fract. Calc. Appl. Anal., 4 (2001), pp. 421--442.
M. Caputo, Diffusion with space memory modelled with distributed order space fractional differential equations, Ann. Geophys., 46 (2003), pp. 223--234,
C. Celik and M. Duman, Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phys., 231 (2012), pp. 1743--1750,
M. Chen and W. Deng, Fourth order accurate scheme for the space fractional diffusion equations, SIAM J. Numer. Anal., 52 (2014), pp. 1418--1438,
L. Delves and J. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, UK, 1985.
W. Deng, Finite element method for the space and time fractional Fokker--Planck equation, SIAM J. Numer. Anal., 47 (2008), pp. 204--226,
K. Diethelm and N. J. Ford, Numerical analysis for distributed-order differential equations, J. Comput. Appl. Math., 225 (2009),
H. Ding, C. Li, and Y. Chen, High-order algorithms for Riesz derivative and their applications, Abstract Appl. Anal., 293 (2013), pp. 218--237,
J. Edwards, N. J. Ford, and A. C. Simpson, The numerical solution of linear multi-term fractional differential equations: Systems of equations, J. Comput. Appl. Math., 148 (2002), pp. 401--418,
V. Ervin and J. Roop, Variational solution of fractional advection dispersion equations on bounded domains in $\mathbb{R}^d$, Numer. Methods Partial Differential Equations, 23 (2007), pp. 256--281,
D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conf. Ser. in Appl. Math. 26, SIAM, Philadelphia, 1977.
J. Hesthaven, S. Gottlieb, and D. Gottlieb, Spectral Methods for Time-Dependent Problems, Cambridge University Press, Cambridge, UK, 2007.
E. Kharazmi, M. Zayernouri, and G. E. Karniadakis, Petrov-Galerkin and Spectral Collocation Methods for Distributed Order Differential Equations, preprint,, 2016.
H. Khosravian-Arab, M. Dehghan, and M. Eslahchi, Fractional Sturm-Liouville boundary value problems in unbounded domains: Theory and applications, J. Comput. Phys., 229 (2015), pp. 526--560,
X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), pp. 2108--2131,
F. Liu, M. Meerschaert, R. McGough, P. Zhuang, and Q. Liu, Numerical methods for solving the multi-term time fractional wave equations, Fract. Calc. Appl. Anal., 16 (2013), pp. 9--25,
P. Martinsson, V. Rokhlin, and M. Tygert, A fast algorithm for the inversion of general Toeplitz matrices, Comput. Math. Appl., 50 (2005), pp. 741--752,
C. Ming, F. Liu, L. Zheng, I. Turner, and V. Anh, Analytical solutions of multi-term time fractional differential equations and application to unsteady flows of generalized viscoelastic fluid, Comput. Math. Appl., 72 (2016), pp. 2084--2097,
M. Naghibolhosseini, Estimation of Outer-Middle Ear Transmission Using DPOAEs and Fractional-Order Modeling of Human Middle Ear, Ph.D. thesis, Department of Speech-Language-Hearing Services, City University of New York, NY, 2015.
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999.
S. Samko, A. Kilbas, and O. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993.
J. Shen, T. Tang, and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, 8 2011.
I. Sokolov, A. Chechkin, and J. Klafter, Distributed order fractional kinetics, Acta Phys. Polonica B, 35 (2004), pp. 1323--1341.
H. Ye, F. Liu, I. Turner, V. Anh, and K. Burrage, Series expansion solutions for the multi-term time and space fractional partial differential equations in two and three dimensions, Eur. Phys. J. Special Topics, 222 (2013), pp. 1901--1914,
M. Zayernouri, M. Ainsworth, and G. E. Karniadakis, Tempered fractional Sturm--Liouville eigenproblems, SIAM J. Sci. Comput., 37 (2015), pp. A1777--A1800,
M. Zayernouri, M. Ainsworth, and G. E. Karniadakis, A unified Petrov-Galerkin spectral method for fractional PDEs, Comput. Methods Appl. Mech. Engrg., 283 (2015), pp. 1545--1569,
M. Zayernouri and G. E. Karniadakis, Fractional Sturm--Liouville eigen-problems: Theory and numerical approximation, J. Comput. Phys., 252 (2013), pp. 495--517,
M. Zayernouri and G. E. Karniadakis, Discontinuous spectral element methods for time- and space-fractional advection equations, SIAM J. Sci. Comput., 36 (2014), pp. B684--B707,
M. Zayernouri and G. E. Karniadakis, Exponentially accurate spectral and spectral element methods for fractional ODEs, J. Comput. Phys., 257 (2014), pp. 460--480,
M. Zayernouri and G. E. Karniadakis, Fractional spectral collocation method, SIAM J. Sci. Comput., 36 (2014), pp. A40--A62,
Z. Zhang, F. Zeng, and G. E. Karniadakis, Optimal error estimates of spectral Petrov--Galerkin and collocation methods for initial value problems of fractional differential equations, SIAM J. Numer. Anal., 53 (2015), pp. 2074--2096,
Y. Zhao, Y. Zhang, F. Liu, I. Turner, Y. Tang, and V. Anh, Convergence and superconvergence of a fully-discrete scheme for multi-term time fractional diffusion equations, Appl. Math. Model., 73 (2016), pp. 1087--1099,
M. Zheng, F. Liu, V. Anh, and I. Turner, A high order spectral method for the multi-term time-fractional diffusion equation, Appl. Math. Model., 40 (2016), pp. 4970--4985,

Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A922 - A946
ISSN (online): 1095-7197


Submitted: 23 January 2017
Accepted: 30 January 2017
Published online: 25 May 2017


  1. spectral accuracy
  2. linear complexity
  3. singular solutions
  4. tunable accuracy
  5. distributed order

MSC codes

  1. 34L10
  2. 58C40
  3. 34K28
  4. 65M70
  5. 65M60



Funding Information

Office of the Secretary of Defense : W911NF-15-1-0562

Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account