×

A Petrov-Galerkin spectral method of linear complexity for fractional multiterm ODEs on the half line. (English) Zbl 1367.65108

Summary: 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 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.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34A30 Linear ordinary differential equations and systems
34A08 Fractional ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] T. M. Atanackovic, M. Budincevic, and S. Pilipovic, {\it On a fractional distributed-order oscillator}, J. Phys. A, 38 (2005), pp. 6703-6713, . · Zbl 1074.74030
[2] D. Baleanu, A. Bhrawy, and T. Taha, {\it A modified generalized Laguerre spectral method for fractional differential equations on the half line}, Abstract Appl. Anal., (2013), 413529, . · Zbl 1291.65239
[3] A. Bhrawy, D. Baleanu, and L. Assas, {\it Efficient generalized Laguerre spectral methods for solving multi-term fractional differential equations on the half line}, J. Vibration Control, 20 (2013), pp. 973-985, . · Zbl 1348.65060
[4] H. Brunner, {\it Collocation Methods for Volterra Integral and Related Functional Differential Equations}, vol. 15, Cambridge University Press, Cambridge, UK, 2004. · Zbl 1059.65122
[5] M. Caputo, {\it Distributed order differential equations modelling dielectric induction and diffusion}, Fract. Calc. Appl. Anal., 4 (2001), pp. 421-442. · Zbl 1042.34028
[6] M. Caputo, {\it Diffusion with space memory modelled with distributed order space fractional differential equations}, Ann. Geophys., 46 (2003), pp. 223-234, .
[7] C. Celik and M. Duman, {\it Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative}, J. Comput. Phys., 231 (2012), pp. 1743-1750, . · Zbl 1242.65157
[8] M. Chen and W. Deng, {\it Fourth order accurate scheme for the space fractional diffusion equations}, SIAM J. Numer. Anal., 52 (2014), pp. 1418-1438, . · Zbl 1318.65048
[9] L. Delves and J. Mohamed, {\it Computational Methods for Integral Equations}, Cambridge University Press, Cambridge, UK, 1985. · Zbl 0592.65093
[10] W. Deng, {\it Finite element method for the space and time fractional Fokker-Planck equation}, SIAM J. Numer. Anal., 47 (2008), pp. 204-226, . · Zbl 1416.65344
[11] K. Diethelm and N. J. Ford, {\it Numerical analysis for distributed-order differential equations}, J. Comput. Appl. Math., 225 (2009), . · Zbl 1159.65103
[12] H. Ding, C. Li, and Y. Chen, {\it High-order algorithms for Riesz derivative and their applications}, Abstract Appl. Anal., 293 (2013), pp. 218-237, . · Zbl 1349.65284
[13] J. Edwards, N. J. Ford, and A. C. Simpson, {\it The numerical solution of linear multi-term fractional differential equations: Systems of equations}, J. Comput. Appl. Math., 148 (2002), pp. 401-418, . · Zbl 1019.65048
[14] V. Ervin and J. Roop, {\it Variational solution of fractional advection dispersion equations on bounded domains in \(\mathbb{R}^d\)}, Numer. Methods Partial Differential Equations, 23 (2007), pp. 256-281, . · Zbl 1117.65169
[15] D. Gottlieb and S. A. Orszag, {\it Numerical Analysis of Spectral Methods: Theory and Applications}, CBMS-NSF Regional Conf. Ser. in Appl. Math. 26, SIAM, Philadelphia, 1977. · Zbl 0412.65058
[16] J. Hesthaven, S. Gottlieb, and D. Gottlieb, {\it Spectral Methods for Time-Dependent Problems}, Cambridge University Press, Cambridge, UK, 2007. · Zbl 1111.65093
[17] E. Kharazmi, M. Zayernouri, and G. E. Karniadakis, {\it Petrov-Galerkin and Spectral Collocation Methods for Distributed Order Differential Equations}, preprint, , 2016. · Zbl 1367.65113
[18] H. Khosravian-Arab, M. Dehghan, and M. Eslahchi, {\it Fractional Sturm-Liouville boundary value problems in unbounded domains: Theory and applications}, J. Comput. Phys., 229 (2015), pp. 526-560, . · Zbl 1352.65202
[19] X. Li and C. Xu, {\it A space-time spectral method for the time fractional diffusion equation}, SIAM J. Numer. Anal., 47 (2009), pp. 2108-2131, . · Zbl 1193.35243
[20] F. Liu, M. Meerschaert, R. McGough, P. Zhuang, and Q. Liu, {\it Numerical methods for solving the multi-term time fractional wave equations}, Fract. Calc. Appl. Anal., 16 (2013), pp. 9-25, . · Zbl 1312.65138
[21] P. Martinsson, V. Rokhlin, and M. Tygert, {\it A fast algorithm for the inversion of general Toeplitz matrices}, Comput. Math. Appl., 50 (2005), pp. 741-752, . · Zbl 1087.65025
[22] C. Ming, F. Liu, L. Zheng, I. Turner, and V. Anh, {\it 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, . · Zbl 1398.35277
[23] M. Naghibolhosseini, {\it 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.
[24] I. Podlubny, {\it Fractional Differential Equations}, Academic Press, San Diego, CA, 1999. · Zbl 0924.34008
[25] S. Samko, A. Kilbas, and O. Marichev, {\it Fractional Integrals and Derivatives}, Gordon and Breach Science Publishers, Yverdon, 1993. · Zbl 0818.26003
[26] J. Shen, T. Tang, and L.-L. Wang, {\it Spectral Methods: Algorithms, Analysis and Applications}, Springer, 8 2011. · Zbl 1227.65117
[27] I. Sokolov, A. Chechkin, and J. Klafter, {\it Distributed order fractional kinetics}, Acta Phys. Polonica B, 35 (2004), pp. 1323-1341.
[28] H. Ye, F. Liu, I. Turner, V. Anh, and K. Burrage, {\it 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, .
[29] M. Zayernouri, M. Ainsworth, and G. E. Karniadakis, {\it Tempered fractional Sturm-Liouville eigenproblems}, SIAM J. Sci. Comput., 37 (2015), pp. A1777-A1800, . · Zbl 1323.34012
[30] M. Zayernouri, M. Ainsworth, and G. E. Karniadakis, {\it A unified Petrov-Galerkin spectral method for fractional PDEs}, Comput. Methods Appl. Mech. Engrg., 283 (2015), pp. 1545-1569, . · Zbl 1425.65127
[31] M. Zayernouri and G. E. Karniadakis, {\it Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation}, J. Comput. Phys., 252 (2013), pp. 495-517, . · Zbl 1349.34095
[32] M. Zayernouri and G. E. Karniadakis, {\it Discontinuous spectral element methods for time- and space-fractional advection equations}, SIAM J. Sci. Comput., 36 (2014), pp. B684-B707, . · Zbl 1304.35757
[33] M. Zayernouri and G. E. Karniadakis, {\it Exponentially accurate spectral and spectral element methods for fractional ODEs}, J. Comput. Phys., 257 (2014), pp. 460-480, . · Zbl 1349.65257
[34] M. Zayernouri and G. E. Karniadakis, {\it Fractional spectral collocation method}, SIAM J. Sci. Comput., 36 (2014), pp. A40-A62, . · Zbl 1294.65097
[35] Z. Zhang, F. Zeng, and G. E. Karniadakis, {\it 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, . · Zbl 1326.65100
[36] Y. Zhao, Y. Zhang, F. Liu, I. Turner, Y. Tang, and V. Anh, {\it Convergence and superconvergence of a fully-discrete scheme for multi-term time fractional diffusion equations}, Appl. Math. Model., 73 (2016), pp. 1087-1099, . · Zbl 1412.65159
[37] M. Zheng, F. Liu, V. Anh, and I. Turner, {\it A high order spectral method for the multi-term time-fractional diffusion equation}, Appl. Math. Model., 40 (2016), pp. 4970-4985, . · Zbl 1459.65205
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.