Advertisement

The Laguerre-Hermite spectral methods for the time-fractional sub-diffusion equations on unbounded domains

  • Hao Yu
  • Boying Wu
  • Dazhi Zhang
Original Paper
  • 36 Downloads

Abstract

This study uses Laguerre-Hermite spectral Galerkin and spectral collocation methods for solving time-fractional sub-diffusion equations on unbounded domains. In the time domain, the generalized associated Laguerre functions of the first kind are employed as basis functions. In the Galerkin method, the fractional derivative of the basis functions can be obtained using the Laplace transform and its inverse. In the collocation method, the solution is expanded in terms of suitable global basis functions, and collocation conditions are imposed on the Gauss points. The corresponding errors are estimated in the time-space domain, and numerical examples verify the results and provide additional insight into the convergence behavior of the proposed method.

Keywords

Spectral Galerkin method Spectral collocation method The generalized associated Laguerre functions Hermite function Time-fractional sub-diffusion equation Unbounded domain 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Funding information

This work is partially supported by the National Natural Science Foundation of China (Grant No. U1637208), National Key Research and Development Program of China (2017YFB1401801), Natural Scientific Foundation of Heilongjiang Province in China (A2016003), and Research Project of China Scholarship Council (No. 201706120212).

References

  1. 1.
    Abbaszadeh, M., Dehghan, M.: An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate. Numer. Algorithms 75(1), 173–211 (2017)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aboelenen, T., Bakr, S.A., El-Hawary, H.M.: Fractional Laguerre spectral methods and their applications to fractional differential equations on unbounded domain. Int. J. Comput. Math. 94(3), 570–596 (2017)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Andrews, G.E., Askey, R., Roy, R.: Special functions, volume 71 of encyclopedia of mathematics and its applications (1999)Google Scholar
  4. 4.
    Arara, A., Benchohra, M., Hamidi, N., Nieto, J.: Fractional order differential equations on an unbounded domain. Nonlinear Anal. Theory Methods Appl. 72(2), 580–586 (2010)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bateman, H.: Tables of integral transforms. McGraw-Hill, New York (1954)Google Scholar
  6. 6.
    Bhrawy, A.: A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations. Numer. Algorithms 73(1), 91–113 (2016)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bhrawy, A., Abdelkawy, M., Alzahrani, A., Baleanu, D., Alzahrani, E.: A Chebyshev-Laguerre-Gauss-Radau collocation scheme for solving a time fractional sub-diffusion equation on a semi-infinite domain (2015)Google Scholar
  8. 8.
    Chandru, M., Das, P., Ramos, H.: Numerical treatment of two-parameter singularly perturbed parabolic convection diffusion problems with non-smooth data. Math. Methods Appl. Sci. 41(14), 5359–5387 (2018)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chandru, M., Prabha, T., Das, P., Shanthi, V.: A numerical method for solving boundary and interior layers dominated parabolic problems with discontinuous convection coefficient and source terms. Differential Equations and Dynamical Systems.  https://doi.org/10.1007/s12591-017-0385-3 (2017)
  10. 10.
    Chen, H., Lü, S., Chen, W.: Spectral methods for the time fractional diffusion–wave equation in a semi-infinite channel. Comput. Math. Appl. 71(9), 1818–1830 (2016)MathSciNetGoogle Scholar
  11. 11.
    Das, P.: Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problems. J. Comput. Appl. Math. 290, 16–25 (2015)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Das, P.: An a posteriori based convergence analysis for a nonlinear singularly perturbed system of delay differential equations on an adaptive mesh. Numerical Algorithms.  https://doi.org/10.1007/s11075-018-0557-4 (2018)
  13. 13.
    Das, P.: A higher order difference method for singularly perturbed parabolic partial differential equations. J. Differ. Equ. Appl. 24(3), 452–477 (2018)MathSciNetGoogle Scholar
  14. 14.
    Das, P., Mehrmann, V.: Numerical solution of singularly perturbed convection-diffusion-reaction problems with two small parameters. BIT Numer. Math. 56(1), 51–76 (2016)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Das, P., Natesan, S.: Optimal error estimate using mesh equidistribution technique for singularly perturbed system of reaction–diffusion boundary-value problems. Appl. Math. Comput. 249, 265–277 (2014)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Das, P., Vigo-Aguiar, J.: Parameter uniform optimal order numerical approximation of a class of singularly perturbed system of reaction diffusion problems involving a small perturbation parameter. Journal of Computational and Applied Mathematics (2017)Google Scholar
  17. 17.
    Dehghan, M., Abbaszadeh, M.: Spectral element technique for nonlinear fractional evolution equation, stability and convergence analysis. Appl. Numer. Math. 119, 51–66 (2017)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Dehghan, M., Abbaszadeh, M., Mohebbi, A.: Analysis of a meshless method for the time fractional diffusion-wave equation. Numer. Algorithms 73(2), 445–476 (2016)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Dehghan, M., Abbaszadeh, M., Mohebbi, A.: Legendre spectral element method for solving time fractional modified anomalous sub-diffusion equation. Appl. Math. Model. 40(5-6), 3635–3654 (2016)MathSciNetGoogle Scholar
  20. 20.
    Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Methods Partial Differ. Equ.: Int. J. 26(2), 448–479 (2010)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Gao, G., Alikhanov, A.A., Sun, Z.: The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations. J. Sci. Comput. 73(1), 93–121 (2017)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gao, G., Sun, Z., Zhang, Y.: A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions. J. Comput. Phys. 231(7), 2865–2879 (2012)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Gómez-Aguilar, J.: Space–time fractional diffusion equation using a derivative with nonsingular and regular kernel. Physica A: Stat. Mech. Appl. 465, 562–572 (2017)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products. Academic Press, New York (2014)zbMATHGoogle Scholar
  25. 25.
    Guo, B., Wang, L., Wang, Z.: Generalized Laguerre interpolation and pseudospectral method for unbounded domains. SIAM J. Numer. Anal. 43(6), 2567–2589 (2006)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Huang, C., Zhang, Z., Song, Q.: Spectral methods for substantial fractional differential equations. J. Sci. Comput. 74(3), 1554–1574 (2018)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Huang, J., Yang, D.: A unified difference-spectral method for time–space fractional diffusion equations. Int. J. Comput. Math. 94(6), 1172–1184 (2017)MathSciNetzbMATHGoogle Scholar
  28. 28.
    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)MathSciNetGoogle Scholar
  29. 29.
    Jiang, W., Li, H.: A space–time spectral collocation method for the two-dimensional variable-order fractional percolation equations. Comput. Math. Appl. 75(10), 3508–3520 (2018)MathSciNetGoogle Scholar
  30. 30.
    Khosravian-Arab, H., Dehghan, M., Eslahchi, M.: Fractional Sturm-Liouville boundary value problems in unbounded domains: theory and applications. J. Comput. Phys. 299, 526–560 (2015)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Khosravian-Arab, H., Dehghan, M., Eslahchi, M.: Fractional spectral and pseudo-spectral methods in unbounded domains: Theory and applications. J. Comput. Phys. 338, 527–566 (2017)MathSciNetGoogle Scholar
  32. 32.
    Lenka, B.K., Banerjee, S.: Sufficient conditions for asymptotic stability and stabilization of autonomous fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 56, 365–379 (2018)MathSciNetGoogle Scholar
  33. 33.
    Li, H., Jiang, W.: A space-time spectral collocation method for the 2-dimensional nonlinear Riesz space fractional diffusion equations. Mathematical Methods in the Applied SciencesGoogle Scholar
  34. 34.
    Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3), 2108–2131 (2009)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Li, X., Xu, C.: 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 (2010)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Mao, Z., Shen, J.: Hermite spectral methods for fractional PDEs in unbounded domains. SIAM J. Sci. Comput. 39(5), A1928–A1950 (2017)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Olver, F.W.: NIST Handbook of mathematical functions hardback and CD-ROM. Cambridge University Press, Cambridge (2010)Google Scholar
  38. 38.
    Parand, K., Shahini, M., Dehghan, M.: Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane–Emden type. J. Comput. Phys. 228(23), 8830–8840 (2009)MathSciNetzbMATHGoogle Scholar
  39. 39.
    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. Elsevier, New York (1998)Google Scholar
  40. 40.
    Povstenko, Y.: Fundamental solutions to time-fractional heat conduction equations in two joint half-lines. Open Phys. 11(10), 1284–1294 (2013)Google Scholar
  41. 41.
    Ren, J., Mao, S., Zhang, J.: Fast evaluation and high accuracy finite element approximation for the time fractional subdiffusion equation. Numer. Methods Partial Differ. Equ. 34(2), 705–730 (2018)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Ren, L., Wang, Y.: A fourth-order extrapolated compact difference method for time-fractional convection-reaction-diffusion equations with spatially variable coefficients. Appl. Math. Comput. 312, 1–22 (2017)MathSciNetGoogle Scholar
  43. 43.
    Salehi, R.: A meshless point collocation method for 2-D multi-term time fractional diffusion-wave equation. Numer. Algorithms 74(4), 1145–1168 (2017)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Shan, Y., Liu, W., Wu, B.: Space–time Legendre–Gauss–Lobatto collocation method for two-dimensional generalized Sine-Gordon equation. Appl. Numer. Math. 122, 92–107 (2017)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Shen, J., Tang, T., Wang, L.: Spectral methods: algorithms, analysis and applications, vol. 41. Springer Science & Business Media, Berlin (2011)Google Scholar
  46. 46.
    Tang, T., Yuan, H., Zhou, T.: Hermite spectral collocation methods for fractional PDEs in unbounded domains. arXiv:1801.09073.2018 (2018)
  47. 47.
    Wang, T., Jiao, Y.: A fully discrete pseudospectral method for Fisher’s equation on the whole line. Appl. Numer. Math. 120, 243–256 (2017)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Wei, L.: Analysis of a new finite difference/local discontinuous Galerkin method for the fractional Cattaneo equation. Numer. Algorithms 77(3), 675–690 (2018)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Wei, S., Chen, W., Zhang, J.: Time-fractional derivative model for chloride ions sub-diffusion in reinforced concrete. Eur. J. Environ. Civ. Eng. 21(3), 319–331 (2017)Google Scholar
  50. 50.
    Yu, H., Wu, B., Zhang, D.: A generalized laguerre spectral Petrov–Galerkin method for the time-fractional subdiffusion equation on the semi-infinite domain. Appl. Math. Comput. 331, 96–111 (2018)MathSciNetGoogle Scholar
  51. 51.
    Zeng, F., Li, C.: A new Crank–Nicolson finite element method for the time-fractional subdiffusion equation. Appl. Numer. Math. 121, 82–95 (2017)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Zhang, C., Liu, W., Wang, L.: A new collocation scheme using non-polynomial basis functions. J. Sci. Comput. 70(2), 793–818 (2017)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Zhang, Q., Zhang, J., Jiang, S., Zhang, Z.: Numerical solution to a linearized time fractional KDV equation on unbounded domains. Math. Comput. 87 (310), 693–719 (2018)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Zhang, S.: Monotone method for initial value problem for fractional diffusion equation. Sci. China Ser. A: Math. 49(9), 1223–1230 (2006)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Zhao, X., Ge, W.: Unbounded solutions for a fractional boundary value problems on the infinite interval. Acta Appl. Math. 109(2), 495–505 (2010)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Zhao, Z.: Bäcklund transformations, rational solutions and soliton-cnoidal wave solutions of the modified Kadomtsev–Petviashvili equation. Appl. Math. Lett. 89, 103–110 (2019)MathSciNetGoogle Scholar
  57. 57.
    Zhokh, A., Trypolskyi, A., Strizhak, P.: An investigation of anomalous time-fractional diffusion of isopropyl alcohol in mesoporous silica. Int. J. Heat Mass Transf. 104, 493–502 (2017)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinPeople’s Republic of China

Personalised recommendations