Numerical Algorithms

, Volume 80, Issue 3, pp 849–877 | Cite as

A high-order numerical method for solving the 2D fourth-order reaction-diffusion equation

  • Haixiang Zhang
  • Xuehua YangEmail author
  • Da XuEmail author
Original Paper


In the present work, orthogonal spline collocation (OSC) method with convergence order O(τ3−α + hr+ 1) is proposed for the two-dimensional (2D) fourth-order fractional reaction-diffusion equation, where τ, h, r, and α are the time-step size, space size, polynomial degree of space, and the order of the time-fractional derivative (0 < α < 1), respectively. The method is based on applying a high-order finite difference method (FDM) to approximate the time Caputo fractional derivative and employing OSC method to approximate the spatial fourth-order derivative. Using the argument developed recently by Lv and Xu (SIAM J. Sci. Comput. 38, A2699–A2724, 2016) and mathematical induction method, the optimal error estimates of proposed fully discrete OSC method are proved in detail. Then, the theoretical analysis is validated by a number of numerical experiments. To the best of our knowledge, this is the first proof on the error estimates of high-order numerical method with convergence order O(τ3−α + hr+ 1) for the 2D fourth-order fractional equation.


Fourth-order fractional equation Orthogonal spline collocation Finite difference method Error estimate 

Mathematics Subject Classification (2010)

65M12 65M06 65M70 35S10 


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The authors thank the anonymous reviewers for their constructive comments and suggestions and Professor Graeme Fairweather for stimulating discussions and for his constant encouragement and support.

Funding information

The work is supported by the National Natural Science Foundation of China (11701168, 11601144, 11626096), Hunan Provincial Natural Science Foundation of China (2018JJ3108, 2018JJ3109, 2018JJ4062), Scientific Research Fund of Hunan Provincial Education Department (16K026,YB2016B033), China Postdoctoral Science Foundation (2016M600964), Science Challenge Project (TZ2016002).


  1. 1.
    Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Sun, Z.Z., Wu, X.N.: A fully difference scheme for a diffusion-wave system. Appl. Numer. Math. 2, 193–209 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gao, G., Sun, Z., Zhang, H.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 33–50 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Li, C.P., Wu, R.F., Ding, H.F.: High-order approximation to Caputo derivative and Caputo-type advection-diffusion equations. Commun. Appl. Ind. Math 6(2), e-536 (2014)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cao, J., Li, C., Chen, Y.Q.: High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (ii). Fract. Calc. Appl. Anal. 18, 735–761 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Li, H., Cao, J., Li, C.: High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (III). J. Comput. Appl. Math. 299, 159–175 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lv, C., Xu, C.: Error analysis of a high order method for time-fractional diffusion equations. SIAM J. Sci. Comput. 38, A2699–A2724 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Li, Z.Q., Liang, Z.Q., Yan, Y.B.: High-order numerical methods for solving time fractional partial differential equations. J. Sci. Comput. 71, 785–803 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Li, Z.Q., Yan, Y.B., Ford, N.J.: Error estimates of a high order numerical method for solving linear fractional differential equations. Appl. Numer. Math. 114, 201–220 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Yan, Y.B., Pal, K., Ford, N.J.: Higher order numerical methods for solving fractional differential equations. BIT Numer. Math. 54, 555–584 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dehghan, M., Abbaszadeh, M., Mohebbib, A.: Error estimate for the numerical solution of fractional reaction-subdiffusion process based on a meshless method. J. Comput. Appl. Math. 280, 14–36 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dehghan, M., Fakhar-Izadi, F.: The spectral collocation method with three different bases for solving a nonlinear partial differential equation arising in modeling of nonlinear waves. Math. Comput. Model. 53, 1865–1877 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dehghan, M., Abbaszadeh, M.: Variational multiscale element free Galerkin (VMEFG) and local discontinuous Galerkin (LDG) methods for solving two-dimensional Brusselator reaction-diffusion system with and without cross-diffusion. Comput. Methods Appl. Mech. Eng. 300, 770–797 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Dehghan, M., Abbaszadeh, M.: Two meshless procedures: moving Kriging interpolation and element-free Galerkin for fractional PDEs. Appl. Anal. 96, 936–969 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dehghan, M., Abbaszadeh, M.: Element free galerkin approach based on the reproducing kernel particle method for solving 2d fractional tricomi-type equation with robin boundary condition. Comput. Math. Appl. 73, 1270–1285 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Oldhan, K.B., Spainer, J.: The Fractional Calculus. Academic Press, New York (1974)Google Scholar
  18. 18.
    Karpman, V.I.: Stabilization of soliton instabilities by higher-order dispersion: fourth order nonlinear Schrödinger-type equations. Phys. Rev. E 53, 1336–1339 (1996)CrossRefGoogle Scholar
  19. 19.
    Ji, C.C., Sun, Z.Z., Hao, Z.P.: Numerical algorithms with high spatial accuracy for the fourth-order fractional sub-diffusion equations with the first Dirichlet boundary conditions. J. Sci. Comput. 66, 1148–1174 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hu, X.L., Zhang, L.M.: On finite difference methods for fourth-order fractional diffusion-wave and subdiffusion systems. Appl. Math. Comput. 218, 5019–5034 (2012)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Hu, X.L., Zhang, L.M.: A compact finite difference scheme for the fourth-order fractional diffusion-wave system. Comput. Phys. Commun. 230, 1645–1650 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Guo, J., Li, C.P., Ding, H.F.: Finite difference methods for time subdiffusion equation with space fourth order. Commun. Appl. Math. Comput. 28, 96–108 (2014). in ChineseMathSciNetzbMATHGoogle Scholar
  23. 23.
    Vong, S., Wang, Z.: Compact finite difference scheme for the fourth-order fractional subdiffusion system. Adv. Appl. Math. Mech. 6, 419–435 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zhang, P., Pu, H.: A second-order compact difference scheme for the fourth-order fractional sub-diffusion equation. Numer. Algor. (2017)
  25. 25.
    Wei, L.L., He, Y.N.: Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems. Appl. Math. Model. 38, 1511–1522 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Liu, Y., Fang, Z. C., Li, H., He, S.: A mixed finite element method for a time-fractional fourth-order partial differential equation. Appl. Math. Comput. 243, 703–717 (2014)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Liu, Y., Du, Y.W., Li, H., He, S., Gao, W.: Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction diffusion problem. Comput. Math. Appl. 70, 573–591 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Liu, Y., Du, Y.W., Li, H., Li, J.C., He, S.: A two-grid mixed finite element method for a nonlinear fourth-order reaction diffusion problem with time-fractional derivative. Comput. Math. Appl. 70, 2474–2492 (2015)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Siddiqi, S.S., Arshed, S.: Numerical solution of time-fractional fourth-order partial differential equations. Int. J. Comput. Math. 92, 1496–1518 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Cao, J., Xu, C., Wang, Z.: A high order finite difference/spectral approximations to the time fractional diffusion equations. Adv. Mater. Res. 875, 781–785 (2014)CrossRefGoogle Scholar
  31. 31.
    Li, B., Fairweather, G., Bialecki, B.: Discrete-time orthogonal spline collocation methods for Schrödinger equations in two space variables. SIAM J. Numer. Anal. 35, 453–477 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Fairweather, G., Gladwell, I.: Algorithms for almost block diagonal linear systems. SIAM Rev. 46, 49–58 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Bialecki, B.: Convergence analysis of orthogonal spline collocation for elliptic boundary value problems. SIAM J. Numer. Anal. 35, 617–631 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Percell, P., Wheeler, M.F.: A C 1 finite element collocation method for elliptic equations. SIAM J. Numer. Anal. 17, 605–622 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Greenwell-Yanik, C.E., Fairweather, G.: Analyses of spline collocation methods for parabolic and hyperbolic problems in two space variables. SIAM J. Numer. Anal. 23, 282–296 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Jiang, Y.J., Ma, J.T.: High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 235, 3285–3290 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Zhao, Y.M., Chen, P., Bu, W.P., Liu, X.T., Tang, Y.F.: Two mixed finite element methods for time-fractional diffusion equations. J. Sci. Comput. 70, 407–428 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Huang, J.F., Tang, Y.F., Vázquez, L., Yang, J.Y.: Two finite difference schemes for time fractional diffusion-wave equation. Numer. Algor. 64, 707–720 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Manickam, A.V., Moudgalya, K.M., Pani, A.K.: Second order splitting and orthogonal cubic spline collocation methods for Kuramoto-Sivashinsky equation. Comput. Math. Appl. 35, 5–25 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Yan, Y., Fairweather, G.: Orthogonal spline collocation methods for some partial integro-differential equations. SIAM J. Numer. Anal. 29, 755–768 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Ren, J.C., Sun, Z.Z., Zhao, X.: Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 232, 456–467 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.School of ScienceHunan University of TechnologyZhuzhouChina
  2. 2.National Key Laboratory of Science and Technology on Computational PhysicsInstitute of Applied Physics and Computational MathematicsBeijingChina
  3. 3.Department of MathematicsHunan Normal UniversityChangshaChina

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