Abstract
The eigenfunctions of fractional Sturm–Liouville problems (FSLPs) serve as basis functions for the construction of efficient spectral methods for fractional differential equations. In this work, we provide a rigorous error analysis of spectral approximations using eigenfunctions of FSLPs in the uniform norm. Specifically, we first establish bounds on the coefficients in spectral expansions in terms of eigenfunctions of regular and singular FSLPs and then apply these bounds to derive error estimates for spectral approximations in the uniform norm. We also clarify the effect of the parameters in these eigenfunctions on the decay rate of the expansion coefficients. Numerical examples are provided to confirm our analysis.
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References
Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (2000)
Chen, S., Shen, J., Wang, L.-L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85, 1603–1638 (2016)
Elliott, D.: The evaluation and estimation of the coefficients in the Chebyshev series expansion of a function. Math. Comput. 18(86), 274–284 (1964)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Academic Press, Cambridge (2007)
Liu, W.-J., Wang, L.-L., Li, H.-Y.: Optimal error estimates for Chebyshev approximation of functions with limited regularity in fractional Sobolev-type spaces. Math. Comput. 88(320), 2857–2895 (2019)
Mao, Z.-P., Karniadakis, G.E.: A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative. SIAM J. Numer. Anal. 56(1), 24–49 (2018)
Nevai, P., Erdélyi, T., Magnus, A.P.: Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials. SIAM J. Math. Anal. 25, 602–614 (1994)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)
Protter, M.H., Morrey, C.B.: A First Course in Real Analysis. Springer, Berlin (1991)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego, CA (1999)
Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications. Springer Series in Computational Mathematics, vol. 41. Springer, Berlin (2011)
Szegő, G.: Orthogonal Polynomials, vol. 23. American Mathematical Society, Providence (1939)
Trefethen, L.N.: Approximation Theory and Approximation Practice. SIAM, New Delhi (2013)
Wang, H.-Y., Xiang, S.-H.: On the convergence rates of Legendre approximation. Math. Comput. 81(278), 861–877 (2012)
Wang, H.-Y.: On the optimal estimates and comparison of Gegenbauer expansion coefficients. SIAM J. Numer. Anal. 54(3), 1557–1581 (2016)
Wang, H.-Y.: A new and sharper bound for Legendre expansion of differentiable functions. Appl. Math. Lett. 85, 95–102 (2018)
Xiang, S.-H.: On error bounds for orthogonal polynomial expansions and Gauss-type quadrature. SIAM J. Numer. Anal. 50(3), 1240–1263 (2012)
Xiang, S.-H., Liu, G.-D.: Optimal asymptotic orders on orthogonal polynomial expansions for functions of limited regularities (2018) (submitted)
Zayernouri, M., Karniadakis, G.E.: Fractional Sturm–Liouville eigen-problems: theory and numerical approximation. J. Comput. Phys. 252, 495–517 (2013)
Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36, A40–A62 (2014)
Zayernouri, M., Karniadakis, G.E.: Exponentially accurate spectral and spectral element methods for fractional ODEs. J. Comput. Phys. 257, 460–480 (2014)
Zayernouri, M., Ainsworth, M., Karniadakis, G.E.: A unified Petrov–Galerkin spectral method for fractional PDEs. Comput. Methods Appl. Mech. Eng. 283, 1545–1569 (2015)
Zhang, Z.-Q., Zeng, F.-H., Karniadakis, G.E.: Optimal error estimates of spectral Petrov–Galerkin and collocation methods for initial value problems of fractional differential equations. SIAM J. Numer. Anal. 53(4), 2074–2096 (2015)
Zhao, X.-D., Wang, L.-L., Xie, Z.-Q.: Sharp error bounds for Jacobi expansions and Gegenbauer–Gauss quadrature of analytic functions. SIAM J. Numer. Anal. 51(3), 1443–1469 (2013)
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant 11671160. The author wishes to thank anonymous reviewers for their helpful comments.
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Wang, H. Analysis of Spectral Approximations Using Eigenfunctions of Fractional Sturm–Liouville Problems. J Sci Comput 81, 1655–1677 (2019). https://doi.org/10.1007/s10915-019-01056-4
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DOI: https://doi.org/10.1007/s10915-019-01056-4