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An Exponentially Convergent Scheme in Time for Time Fractional Diffusion Equations with Non-smooth Initial Data

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Abstract

In this paper we aim at developing highly accurate and stable method in temporal direction for time-fractional diffusion equations with initial data \(v\in L^2(\varOmega )\). To this end we begin with a kind of (time-)fractional ODEs, and a hybrid multi-domain Petrov–Galerkin spectral method is proposed. To match the singularity at the beginning of time, we use geometrically graded mesh together with fractional power Jacobi functions as basis on the first interval. Jacobi polynomials are then chosen to approximate the solution in temporal direction on the intervals hereafter. The algorithm is motivated by the discovery that the solution in temporal direction possesses high regularity in proper weighted Sobolev space on a special mesh in piecewise sense. Combining standard finite element method, we extend it to solve time-fractional diffusion equations with initial data \(v\in L^2(\varOmega )\), in which the mesh in temporal direction is determined by spatial mesh size and finial time T. Numerical tests show that the scheme is stable and converges exponentially in time.

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References

  1. Chen, S., Shen, J., Wang, L.L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85(300), 1603–1638 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, Y., Tang, T.: Spectral methods for weakly singular volterra integral equations with smooth solutions. J. Comput. Appl. Math. 233(4), 938–950 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, Y., Tang, T.: Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Math. Comput. 79(269), 147–167 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Duan, B., Jin, B., Lazarov, R., Pasciak, J., Zhou, Z.: Space-time Petrov–Galerkin fem for fractional diffusion problems. Comput. Methods Appl. Math. 18(1), 1–20 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Duan, B., Zheng, Z., Cao, W.: Spectral approximation methods and error estimates for Caputo fractional derivative with applications to initial-value problems. J. Comput. Phys. 319, 108–128 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hou, D., Hasan, M.T., Xu, C.: Müntz spectral methods for the time-fractional diffusion equation. Comput. Methods Appl. Math. 18(1), 43–62 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hou, D., Xu, C.: A fractional spectral method with applications to some singular problems. Adv. Comput. Math. 43(5), 1–34 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jiang, S., Zhang, J., Zhang, Q., Zhang, Z.: Fast evaluation of the caputo fractional derivative and its applications to fractional diffusion equations. Commun. Comput. Phys. 21(3), 650–678 (2017)

    Article  MathSciNet  Google Scholar 

  9. Jin, B., 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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jin, B., Lazarov, R., Zhou, Z.: An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36(2), 197–221 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Li, X., Tang, T.: Convergence analysis of Jacobi spectral collocation methods for Abel–Volterra integral equations of second kind. Front. Math. China 7(1), 69–84 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225(2), 1533–1552 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704–719 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lubich, C.: Convolution quadrature and discretized operational calculus. I. Numer. Math. 52(2), 129–145 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lubich, C.: Convolution quadrature and discretized operational calculus. II. Numer. Math. 52(4), 413–425 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lubich, C., Sloan, I., Thomée, V.: Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term. Math. Comput. Am. Math. Soc. 65(213), 1–17 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Persson, P.O., Strang, G.: A simple mesh generator in matlab. SIAM Rev. 46(2), 329–345 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Academic Press, Cambridge (1998)

    MATH  Google Scholar 

  19. Pollard, H.: The completely monotonic character of the Mittag–Leffler function \({E}_{\alpha }(-x)\). Bull. Am. Math. Soc. 54(12), 1115–1116 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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(1), 426–447 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications, vol. 41. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  22. Shen, Jy, Sun, Zz, Du, R.: Fast finite difference schemes for time-fractional diffusion equations with a weak singularity at initial time. East Asian J. Appl. Math. 8(4), 834–858 (2018)

    Article  MathSciNet  Google Scholar 

  23. Sheng, C., Shen, J.: A hybrid spectral element method for fractional two-point boundary value problems. Numer. Math. Theory Methods Appl. 10(2), 437–464 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sheng, C., Shen, J.: A space-time Petrov–Galerkin spectral method for time fractional diffusion equation. Numer. Math. Theory Methods Appl. 11, 854–876 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Stynes, M.: Too much regularity may force too much uniqueness. Fract. Calc. Appl. Anal. 19(6), 1554–1562 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55(2), 1057–1079 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sun, T., Liu, Rq, Wang, L.L.: Generalised müntz spectral galerkin methods for singularly perturbed fractional differential equations. East Asian J. Appl. Math. 8(4), 611–633 (2018)

    Article  MathSciNet  Google Scholar 

  28. Sun, Zz, Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56(2), 193–209 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wu, S.L., Zhou, T.: Parareal algorithms with local time-integrators for time fractional differential equations. J. Comput. Phys. 358, 135–149 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yan Yonggui, S.Z.Z., Zhang, J.: Fast evaluation of the caputo fractional derivative and its applications to fractional diffusion equations: a second-order scheme. Commun. Comput. Phys. 22(4), 1028–1048 (2017)

    Article  MathSciNet  Google Scholar 

  31. Yan, Y., Khan, M., Ford, N.J.: An analysis of the modified l1 scheme for time-fractional partial differential equations with nonsmooth data. SIAM J. Numer. Anal. 56(1), 210–227 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. 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)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zayernouri, M., Karniadakis, G.E.: Fractional sturm-liouville eigen-problems: theory and numerical approximation. J. Comput. Phys. 252, 495–517 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1), A40–A62 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, People’s Republic of China

    Beiping Duan & Zhoushun Zheng

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  1. Beiping Duan
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  2. Zhoushun Zheng
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Correspondence to Zhoushun Zheng.

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This work is partially supported by the National Key Research and Development Program of China (Grant No. 2017YFB0701700). Beiping Duan really appreciates the financial support by the Fundamental Research Funds for the Central Universities of Central South University (2016zzts015).

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Duan, B., Zheng, Z. An Exponentially Convergent Scheme in Time for Time Fractional Diffusion Equations with Non-smooth Initial Data. J Sci Comput 80, 717–742 (2019). https://doi.org/10.1007/s10915-019-00953-y

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  • DOI: https://doi.org/10.1007/s10915-019-00953-y

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