A Quintic Spline Collocation Method for the Fractional Sub-Diffusion Equation with Variable Coefficients

A Quintic Spline Collocation Method for the Fractional Sub-Diffusion Equation with Variable Coefficients

Gao Li 

Experimental center of Liberal Arts, Neijiang Normal University, Neijing, China

Corresponding Author Email: 
15 March 2017
15 April 2017
30 March 2017
| Citation



For the time fractional sub-diffusion equation with variable coefficients, a quintic spline method is presented, along the time direction, the recursion formula obtained from the Lagrange interpolation functions is used, along the space direction, the quintic spline interpolation functions, which have high order accuracy when being used to approximate smooth functions and their 1,2,3 order derivatives, are used as the basis functions. Theoretical analyses and numerical examples show that 4 order accuracy in space can be achieved for this scheme.


Sub-diffusion equation, quintic spline collocation method, fractional

1. Introduction
2. Preparation
3. Quintic Spline Collocation Method
4. Numerical Examples
5. Conclusion

1. R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.

2. C. Ingo, R. Magin, T. Parrish, New insights into the fractional order diffusion equation using entropy and kurtosis, Entropy 16 (2014) 5838–5852.

3. I. Podlubny, Fractional Differential Equations, Academic Press, California, USA, 1999.

4. S. Das, R. Kumar, Fractional diffusion equations in the presence of reaction terms, J. Comput. Complex. Appl. 1 (1) (2015) 15-21.

5.  S.B. Yuste, L. Acedo, K. Lindenberg, Reaction front in an   reaction-subdiffusion process, Phys. Rev. E, 69 (2004) 036126:1-10.

6. G.H. Gao, Z.Z. Sun, H.W. Zhang. A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys., 259(2)(2014), 33-50.

7. G.-H. Gao, Z.-Z. Sun, A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys., 230 (2011), 586-595.

8.  C.-C. Ji, Z.-Z. Sun, A high-order compact finite difference scheme for the fractional sub-diffusion equation, J. Sci. Comput., 64 (3) (2015), 959-985.

9. B. Jin, R. Lazarov, Z. Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal., 51 (1) (2013), 445-466.

10. F. Zeng, C. Li, F. Liu, I. Turner, The use of finite difference/element approaches for solving the time-fractional subdiffusion equation,  SIAM J. Sci. Comput., 35 (6) (2013), A2976-A3000.

11. F. Zeng, C. Li,  F. Liu, I. Turner, Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy, SIAM J. Sci. Comput., 37 (2015), A55-A78.

12. F. Chen, Q. Xu, J.S. Hesthaven, A multi-domain spectral method for time-fractional differential equations, J. Comput. Phys., 293 (2013), 157-172.

13. M. Zheng, F.Liu, I.Turner, V. Anh, A novel high order space-time spectral method for the time fractional fokker--planck equation, SIAM J. Sci. Comput., 37 (2015), A701-A724.

14. F. Liu, P. Zhuang, I. Turner, K. Burrage, V. Anh, A new fractional finite volume method for solving the fractional diffusion equation, Appl. Math. Model., 38 (2014), 3871-3878.

15. I.J. Schoenberg, Contribution to the problem of approximation of equidistant data by analytic functions, Quart. Appl. Math. 4 (1946), 112–141.

16. C. De Boor, Practical Guide to Splines, Springer-Verlag, Berlin, New York, 1978.

17. M.I. Ellina, E.N. Houstis, An O(h6) quintic spline collocation method for fourth order two-point boundary value problems, BIT 28 (1988), 288–301.