A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Diferential Equations

doi:10.1007/s42967-019-00012-1

Communications on Applied Mathematics and Computation ›› 2019, Vol. 1 ›› Issue (2): 207-230.doi: 10.1007/s42967-019-00012-1

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A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Diferential Equations

Wenjie Liu¹, Li-Lian Wang², Shuhuang Xiang³

1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China;
2 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore;
3 Mathematics and Statistics, Central South University, Changsha 410083, China

收稿日期:2018-08-28 修回日期:2018-09-10 出版日期:2019-06-20 发布日期:2019-06-20
通讯作者: Li-Lian Wang E-mail:LiLian@ntu.edu.sg
基金资助:
W. Liu:The research of this author is partially supported by the China Postdoctoral Science Foundation Funded Project (No. 2017M620113), the National Natural Science Foundation of China (Nos. 11801120, 71773024 and 11771107), the Fundamental Research Funds for the Central Universities (Grant No.HIT.NSRIF.2019058) and the Natural Science Foundation of Heilongjiang Province of China (No. G2018006). L. Wang:The research of this author is partially supported by Singapore MOE AcRF Tier 2 Grants (MOE2017-T2-2-014 and MOE2018-T2-1-059). S. Xiang:This work of this author is partially supported by National Science Foundation of China (No. 11371376) and the Innovation-Driven Project and Mathematics.

A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Diferential Equations

Wenjie Liu¹, Li-Lian Wang², Shuhuang Xiang³

1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China;
2 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore;
3 Mathematics and Statistics, Central South University, Changsha 410083, China

Received:2018-08-28 Revised:2018-09-10 Online:2019-06-20 Published:2019-06-20
Contact: Li-Lian Wang E-mail:LiLian@ntu.edu.sg
Supported by:
W. Liu:The research of this author is partially supported by the China Postdoctoral Science Foundation Funded Project (No. 2017M620113), the National Natural Science Foundation of China (Nos. 11801120, 71773024 and 11771107), the Fundamental Research Funds for the Central Universities (Grant No.HIT.NSRIF.2019058) and the Natural Science Foundation of Heilongjiang Province of China (No. G2018006). L. Wang:The research of this author is partially supported by Singapore MOE AcRF Tier 2 Grants (MOE2017-T2-2-014 and MOE2018-T2-1-059). S. Xiang:This work of this author is partially supported by National Science Foundation of China (No. 11371376) and the Innovation-Driven Project and Mathematics.

摘要/Abstract

摘要： In this paper, we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial diferential equations (PDEs). Diferent from many other approaches, the nonstandard singular basis functions are defned from some generalised Birkhof interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such, the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs, leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems.

关键词: Fractional diferential equations, Generalised Birkhof interpolation, Nonstandard singular basis functions

Abstract: In this paper, we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial diferential equations (PDEs). Diferent from many other approaches, the nonstandard singular basis functions are defned from some generalised Birkhof interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such, the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs, leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems.

Key words: Fractional diferential equations, Generalised Birkhof interpolation, Nonstandard singular basis functions

中图分类号:

Wenjie Liu, Li-Lian Wang, Shuhuang Xiang. A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Diferential Equations[J]. Communications on Applied Mathematics and Computation, 2019, 1(2): 207-230.

参考文献

1. Babuška, I., Banerjee, U.:Stable generalized fnite element method (SGFEM). Comput. Methods Appl. Mech. Eng. 201, 91-111(2012)
2. Borwein, P., Erdélyi, T., Zhang, J.:Müntz systems and orthogonal Müntz-Legendre polynomial. Comput. Math. Appl. 342, 523-542(1994)
3. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.:Spectral Methods:Fundamentals in Single Domains. Springer, Berlin (2006)
4. Cao, Y., Herdman, T., Xu, Y.:A hybrid collocation method for Volterra integral equations with weakly singular kernels. SIAM J. Numer. Anal. 41, 364-381(2003)
5. Chen, S., Shen, J.:Enriched spectral methods and applications to problems with weakly singular solutions. J. Sci. Comput. 77, 1468-1489(2018)
6. Chen, S., Shen, J., Wang, L.L.:Generalized Jacobi functions and their applications to fractional diferential equations. Math. Comput. 85(300), 1603-1638(2016)
7. Costabile, F., Napoli, A.:A class of Birkhof-Lagrange-collocation methods for high order boundary value problems. Appl. Numer. Comput. 116, 129-140(2017)
8. Diethelm, K.:The analysis of fractional diferential equations. Lecture Notes in Math, vol. 2004. Springer, Berlin (2010)
9. Ervin, V.J., Heuer, N., Roop, J.P.:Numerical approximation of a time dependent, nonlinear, spacefractional difusion equation. SIAM J. Numer. Anal. 45(2), 572-591(2007)
10. Ervin, V.J., Roop, J.P.:Variational solution of fractional advection dispersion equations on bounded domains in ℝ^d. Numer. Methods Partial Difer. Equ. 23(2), 256-281(2007)
11. Esmaeili, S., Garrappa, R.:Exponential quadrature rules for linear fractional diferential equations. Mediterr. J. Math. 12, 219-244(2015)
12. Garrappa, R., Moret, I., Popolizio, M.:Solving the time-fractional Schrödinger equation by Krylov projection methods. J. Comput. Phys. 293, 115-134(2015)
13. Gottlieb, D., Orszag, S.:Numerical Analysis of Spectral Methods:Theory and Applications. Pa, SIAM, Philadelphia (1977)
14. Guo, B.Y., Shen, J., Wang, L.L.:Generalized Jacobi polynomials/functions and their applications. Appl. Numer. Math. 59(5), 1011-1028(2009)
15. Guo, B.Y., Shen, J., Wang, L.L.:Optimal spectral-Galerkin methods using generalized Jacobi polynomials. J. Sci. Comput. 27, 305-322(2006)
16. Hong, Y., Jung, C.Y.:Enriched spectral method for stif convection-dominated equations. J. Sci. Comput. 74(3), 1325-1346(2018)
17. Hou, D.M., Xu, C.J.:A fractional spectral method with applications to some singular problems. Adv. Comput. Math. 43(5), 911-944(2017)
18. Huang, C., Stynes, M.:A spectral collocation method for a weakly singular Volterra integral equation of the second kind. Adv. Comput. Math. 42(5), 1015-1030(2016)
19. Huang, C., Jiao, Y.J., Wang, L.L., Zhang, Z.M.:Optimal fractional integration preconditioning and error analysis of fractional collocation method using nodal generalized Jacobi functions. SIAM J. Numer. Anal. 54(6), 3357-3387(2016)
20. Huang, C., Wang, L.L.:An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media. Adv. Comput. Math. (2018). https://doi.org/10.1007/s10444-018-9636-2
21. Jiao, Y.J., Wang, L.L., Huang, C.:Well-conditioned fractional collocation methods using fractional Birkhof interpolation basis. J. Comput. Phys. 305, 1-28(2016)
22. Jin, B.T., Lazarov, R., Zhou, Z.:Error estimates for a semidiscrete fnite element method for fractional order parabolic equations. SIAM J. Numer. Anal. 51(1), 445-466(2013)
23. Kharazmi, E., Zayernouri, M., Karniadakis, G.E.:A Petrov-Galerkin spectral element method for fractional elliptic problems. Comput. Methods Appl. Mech. Eng. 324, 512-536(2017)
24. Kharazmi, E., Zayernouri, M., Karniadakis, G.E.:Petrov-Galerkin and spectral collocation methods for distributed order diferential equations. SIAM J. Sci. Comput. 39(3), A1003-A1037(2017)
25. Li, X.J., Xu, C.J.:A space-time spectral method for the time fractional difusion equation. SIAM J. Numer. Anal. 47(3), 2108-2131(2009)
26. Li, X.J., Xu, C.J.:Existence and uniqueness of the weak solution of the space-time fractional difusion equation and a spectral method approximation. Commun. Comput. Phys. 8(5), 1016-1051(2010)
27. Li, C.P., Zeng, F.H., Liu, F.W.:Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15, 383-406(2012)
28. Liu, F., Anh, V., Turner, I.:Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 166, 209-219(2004)
29. Lorentz, G.G., Jetter, K., Riemenschneider, S.D.:Birkhof Interpolation, volume 19 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Co., Reading (1983)
30. Mao, Z.P., Chen, S., Shen, J.:Efcient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional diferential equations. Appl. Numer. Math. 106, 165-181(2016)
31. Mao, Z.P., Shen, J.:Efcient spectral-Galerkin methods for fractional partial diferential equations with variable coefcients. J. Comput. Phys. 307, 243-261(2016)
32. McCoid, C., and Trummer, M:Preconditioning of spectral methods via Birkhof interpolation. Numer. Algor., https://doi.org/10.1007/s11075-017-0450-6,onlinesince (Dec. 12, 2017)
33. Meerschaert, M.M., Schefer, H.P., Tadjeran, C.:Finite diference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 211(1), 249-261(2006)
34. Meerschaert, M.M., Tadjeran, C.:Finite diference approximations for fractional advection-dispersion fow equations. J. Comput. Appl. Math. 172(1), 65-77(2004)
35. Milovanović, G.V.:Müntz orthogonal polynomials and their numerical evaluation. In W. Gautschi, G. Opfer, and G.H. Golub, editors, Applications and Computation of Orthogonal Polynomials, pages 179-194. Birkhäuser Basel, (1999)
36. Mokhtarya, P., Ghoreishi, F., Srivastava, H.M.:The Müntz-Legendre Tau method for fractional diferential equations. Appl. Math. Model. 40, 671-684(2016)
37. Podlubny, I.:Fractional Diferential Equations, volume 198 of Mathematics in Science and Engineering. Academic Press Inc., San Diego, CA. An introduction to fractional derivatives, fractional diferential equations, to methods of their solution and some of their applications (1999)
38. Sakamoto, K., Yamamoto, M.:Initial value/boundary value problems for fractional difusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426-447(2011)
39. Shen, J.:Efcient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials. SIAM J. Sci. Comput. 15, 1489-1505(1994)
40. Shen, J., Sheng, C.T., Wang, Z.Q.:Generalized Jacobi spectral-Galerkin method for nonlinear Volterra integral equations with weakly singular kernels. J. Math. Study 48(4), 315-329(2015)
41. Shen, J., Tang, T., Wang, L.L.:Spectral methods:algorithms, analysis and applications. Series in computational mathematics, vol. 41. Springer, Berlin (2011)
42. Shen, J., Wang, L.L.:Fourierization of the Legendre-Galerkin method and a new space-time spectral method. Appl. Numer. Math. 57, 710-720(2007)
43. Shen, J., Wang, Y.W.:Müntz-Galerkin methods and applications to mixed Dirichlet-Neumann boundary value problems. SIAM J. Sci. Comput. 38(4), A2357-A2381(2016)
44. Shi, Y.G.:Theory of Birkhof Interpolation. Nova Science Pub Incorporated, New York (2003)
45. Sun, Z.Z., Wu, X.N.:A fully discrete diference scheme for a difusion-wave system. Appl. Numer. Math. 56(2), 193-209(2006)
46. Szegö, G.:Orthogonal polynomials, 4th edn. Amer. Math. Soc, Providence, RI (1975)
47. Tadjeran, C., Meerschaert, M.M.:A second-order accurate numerical method for the two-dimensional fractional difusion equation. J. Comput. Phys. 220(2), 813-823(2007)
48. Wang, L.L., Samson, M., Zhao, X.D.:A well-conditioned collocation method using pseudospectral integration matrix. SIAM J. Sci. Comput. 36, A907-A929(2014)
49. Wang, L.L., Zhang, J., Zhang, Z.:On hp-convergence of prolate spheroidal wave functions and a new well-conditioned prolate-collocation scheme. J. Comput. Phys. 268, 377-398(2014)
50. Xiang, S.H.:On interpolation approximation:convergence rates for polynomial interpolation for functions of limited regularity. SIAM J. Numer. Anal. 54(4), 2081-2113(2016)
51. Zayernouri, M., Ainsworth, M., Karniadakis, G.E.:A unifed Petrov-Galerkin spectral method for fractional PDEs. Comput. Methods Appl. Mech. Eng. 283, 1545-1569(2015)
52. Zayernouri, M., Karniadakis, G.E.:Exponentially accurate spectral and spectral element methods for fractional ODEs. J. Comput. Phys. 257(part A), 460-480(2014)
53. Zayernouri, M., Karniadakis, G.E.:Fractional spectral collocation methods for linear and nonlinear variable order FPDEs. J. Comput. Phys. 293, 312-338(2015)
54. Zayernouri, M., Karniadakis, G.E.:Fractional Sturm-Liouville eigen-problems:theory and numerical approximation. J. Comput. Phys. 252, 495-517(2013)
55. Zeng, F.H., Mao, Z.P., Karniadakis, G.E.:A generalized spectral collocation method with tunable accuracy for fractional diferential equations with end-point singularities. SIAM J. Sci. Comput. 39(1), A360-A383(2017)
56. Zhang, C., Liu, W.J., Wang, L.L.:A new collocation scheme using non-polynomial basis functions. J. Sci. Comput. 70(2), 793-818(2017)
57. Zhang, C., Wang, L.L., Gu, D.Q., Liu, W.J.:On approximate inverse of Hermite and Laguerre collocation diferentiation matrices and new collocation schemes in unbounded domains. J. Comput. Appl. Math. 344, 553-571(2018)
58. 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 diferential equations. SIAM J. Numer. Anal. 53(4), 2074-2096(2015)

A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Diferential Equations

A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Diferential Equations

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