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

  • Wenjie Liu
  • Li-Lian WangEmail author
  • Shuhuang Xiang
Original Paper
  • 1 Downloads

Abstract

In this paper, we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches, the nonstandard singular basis functions are defined from some generalised Birkhoff 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.

Keywords

Fractional differential equations Generalised Birkhoff interpolation Nonstandard singular basis functions 

Mathematics Subject Classification

41A05 41A10 41A25 65M70 65N35 

References

  1. 1.
    Babuška, I., Banerjee, U.: Stable generalized finite element method (SGFEM). Comput. Methods Appl. Mech. Eng. 201, 91–111 (2012)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Borwein, P., Erdélyi, T., Zhang, J.: Müntz systems and orthogonal Müntz-Legendre polynomial. Comput. Math. Appl. 342, 523–542 (1994)zbMATHGoogle Scholar
  3. 3.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)zbMATHGoogle Scholar
  4. 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)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chen, S., Shen, J.: Enriched spectral methods and applications to problems with weakly singular solutions. J. Sci. Comput. 77, 1468–1489 (2018)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chen, S., Shen, J., Wang, L.L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85(300), 1603–1638 (2016)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Costabile, F., Napoli, A.: A class of Birkhoff-Lagrange-collocation methods for high order boundary value problems. Appl. Numer. Comput. 116, 129–140 (2017)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Diethelm, K.: The analysis of fractional differential equations. Lecture Notes in Math, vol. 2004. Springer, Berlin (2010)Google Scholar
  9. 9.
    Ervin, V.J., Heuer, N., Roop, J.P.: Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation. SIAM J. Numer. Anal. 45(2), 572–591 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Ervin, V.J., Roop, J.P.: Variational solution of fractional advection dispersion equations on bounded domains in \(\mathbb{R}^d\). Numer. Methods Partial Differ. Equ. 23(2), 256–281 (2007)zbMATHGoogle Scholar
  11. 11.
    Esmaeili, S., Garrappa, R.: Exponential quadrature rules for linear fractional differential equations. Mediterr. J. Math. 12, 219–244 (2015)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Garrappa, R., Moret, I., Popolizio, M.: Solving the time-fractional Schrödinger equation by Krylov projection methods. J. Comput. Phys. 293, 115–134 (2015)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gottlieb, D., Orszag, S.: Numerical Analysis of Spectral Methods: Theory and Applications. Pa, SIAM, Philadelphia (1977)Google Scholar
  14. 14.
    Guo, B.Y., Shen, J., Wang, L.L.: Generalized Jacobi polynomials/functions and their applications. Appl. Numer. Math. 59(5), 1011–1028 (2009)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Guo, B.Y., Shen, J., Wang, L.L.: Optimal spectral-Galerkin methods using generalized Jacobi polynomials. J. Sci. Comput. 27, 305–322 (2006)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Hong, Y., Jung, C.Y.: Enriched spectral method for stiff convection-dominated equations. J. Sci. Comput. 74(3), 1325–1346 (2018)MathSciNetzbMATHGoogle Scholar
  17. 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)MathSciNetzbMATHGoogle Scholar
  18. 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)MathSciNetzbMATHGoogle Scholar
  19. 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)MathSciNetzbMATHGoogle Scholar
  20. 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. 21.
    Jiao, Y.J., Wang, L.L., Huang, C.: Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis. J. Comput. Phys. 305, 1–28 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Jin, B.T., Lazarov, R., Zhou, Z.: Error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM J. Numer. Anal. 51(1), 445–466 (2013)MathSciNetzbMATHGoogle Scholar
  23. 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)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Kharazmi, E., Zayernouri, M., Karniadakis, G.E.: Petrov-Galerkin and spectral collocation methods for distributed order differential equations. SIAM J. Sci. Comput. 39(3), A1003–A1037 (2017)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Li, X.J., Xu, C.J.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3), 2108–2131 (2009)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Li, X.J., Xu, C.J.: Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 8(5), 1016–1051 (2010)MathSciNetzbMATHGoogle Scholar
  27. 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)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 166, 209–219 (2004)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Lorentz, G.G., Jetter, K., Riemenschneider, S.D.: Birkhoff Interpolation, volume 19 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Co., Reading (1983)Google Scholar
  30. 30.
    Mao, Z.P., Chen, S., Shen, J.: Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations. Appl. Numer. Math. 106, 165–181 (2016)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Mao, Z.P., Shen, J.: Efficient spectral-Galerkin methods for fractional partial differential equations with variable coefficients. J. Comput. Phys. 307, 243–261 (2016)MathSciNetzbMATHGoogle Scholar
  32. 32.
    McCoid, C., and Trummer, M: Preconditioning of spectral methods via Birkhoff interpolation. Numer. Algor.,  https://doi.org/10.1007/s11075-017-0450-6,online since (Dec. 12, 2017)
  33. 33.
    Meerschaert, M.M., Scheffler, H.P., Tadjeran, C.: Finite difference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 211(1), 249–261 (2006)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172(1), 65–77 (2004)MathSciNetzbMATHGoogle Scholar
  35. 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)Google Scholar
  36. 36.
    Mokhtarya, P., Ghoreishi, F., Srivastava, H.M.: The Müntz-Legendre Tau method for fractional differential equations. Appl. Math. Model. 40, 671–684 (2016)MathSciNetGoogle Scholar
  37. 37.
    Podlubny, I.: Fractional Differential Equations, volume 198 of Mathematics in Science and Engineering. Academic Press Inc., San Diego, CA. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (1999)Google Scholar
  38. 38.
    Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426–447 (2011)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Shen, J.: Efficient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials. SIAM J. Sci. Comput. 15, 1489–1505 (1994)MathSciNetzbMATHGoogle Scholar
  40. 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)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Shen, J., Tang, T., Wang, L.L.: Spectral methods: algorithms, analysis and applications. Series in computational mathematics, vol. 41. Springer, Berlin (2011)Google Scholar
  42. 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)MathSciNetzbMATHGoogle Scholar
  43. 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)zbMATHGoogle Scholar
  44. 44.
    Shi, Y.G.: Theory of Birkhoff Interpolation. Nova Science Pub Incorporated, New York (2003)zbMATHGoogle Scholar
  45. 45.
    Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56(2), 193–209 (2006)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Szegö, G.: Orthogonal polynomials, 4th edn. Amer. Math. Soc, Providence, RI (1975)Google Scholar
  47. 47.
    Tadjeran, C., Meerschaert, M.M.: A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220(2), 813–823 (2007)MathSciNetzbMATHGoogle Scholar
  48. 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)MathSciNetzbMATHGoogle Scholar
  49. 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)MathSciNetzbMATHGoogle Scholar
  50. 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)MathSciNetzbMATHGoogle Scholar
  51. 51.
    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)MathSciNetzbMATHGoogle Scholar
  52. 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)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation methods for linear and nonlinear variable order FPDEs. J. Comput. Phys. 293, 312–338 (2015)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Zayernouri, M., Karniadakis, G.E.: Fractional Sturm-Liouville eigen-problems: theory and numerical approximation. J. Comput. Phys. 252, 495–517 (2013)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Zeng, F.H., Mao, Z.P., Karniadakis, G.E.: A generalized spectral collocation method with tunable accuracy for fractional differential equations with end-point singularities. SIAM J. Sci. Comput. 39(1), A360–A383 (2017)MathSciNetzbMATHGoogle Scholar
  56. 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)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Zhang, C., Wang, L.L., Gu, D.Q., Liu, W.J.: On approximate inverse of Hermite and Laguerre collocation differentiation matrices and new collocation schemes in unbounded domains. J. Comput. Appl. Math. 344, 553–571 (2018)MathSciNetzbMATHGoogle Scholar
  58. 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 differential equations. SIAM J. Numer. Anal. 53(4), 2074–2096 (2015)MathSciNetzbMATHGoogle Scholar

Copyright information

© Shanghai University 2019

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinChina
  2. 2.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore
  3. 3.Mathematics and StatisticsCentral South UniversityChangshaChina

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