Solving Time-Fractional Diffusion Equation: A Finite-Element Approach

  • Hussein J. Zekri Dept. of Mechanical Engineering, University of Zakho, Kurdistan Region, Iraq
Keywords: Time-Fractional Diffusion Equation, Finite Element Method

Abstract

The numerical solution for a time-fractional diffusion equation supplemented with initial and boundary conditions is considered. The scheme is based on the Galerkin finite element method. The uniform space discretization is applied to study the stability of the solution of the problem within our approach. An analytically solvable example is presented to make a comparison between the exact solution and our numerical solution. By presenting the absolute error with different step-sizes and different values for time-fractional derivative, reliability and efficiency of our proposed numerical method is manifested.

Author Biography

Hussein J. Zekri, Dept. of Mechanical Engineering, University of Zakho, Kurdistan Region, Iraq

Dept. of Mechanical Engineering, University of Zakho, Kurdistan Region, Iraq – (Hussein.zekri@uoz.edu.krd)

References

Atangana A. (2015). Numerical analysis of time fractional three dimensional di_usion equation. Thermal Science, 19(suppl. 1):7-12.
Bellman R. (1997). Introduction to matrix analysis. Siam.
Esen, A., Uucar, Y., Yagmurlu, M., & Tasbozan, O. (2014). Solving fractional diffusion and fractional diffusion-wave equations by Petrov-Galerkin finite element method. TWMS Journal of Applied and Engineering Mathematics, 4(2), 155-168.
Esen, A., Ucar, Y., Yagmurlu, N. and Tasbozan, O. (2013). A Galerkin finite element method to solve fractional diffusion and fractional diffusion-wave equations. Mathematical Modelling and Analysis, 18(2), pp.260-273.
http://dx.doi.org/10.3846/13926292.2013.783884
Esen, A., Tasbozan, O., Ucar, Y., & Yagmurlu, N. M. (2015). A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations. Tbilisi Mathematical Journal, 8(2), 181-193.
Tasbozan, O., Esen, A., Yagmurlu, N. M., & Ucar, Y. (2013). A numerical solution to fractional diffusion equation for force-free case. In Abstract and applied analysis, 2013. Hindawi. http://dx.doi.org/10.1155/2013/187383
Ferziger J. H. and Peric M. (2012). Computational methods for uid dynamics. SpringerScience & Business Media.
Hanert E. and Piret C. (2012). Numerical solution of the space-time fractional diffusion equation:Alternatives to finite differences. In 5th IFAC Symposium on Fractional Differentiation and Its Applications-FDA2012.
Hanert E. (2010). A comparison of three eulerian numerical methods for fractional-order transport models. Environmental uid mechanics, 10(1-2):7-20.
Hilfer R. (2000). Applications of fractional calculus in physics. World Scientific.
Horn R. A. and Johnson C. R. (1991). Topics in matrix analysis. Cambridge University Presss, Cambridge, 37:39, 1991.
Kilbas A. A. et al (2006). Theory and applications of fractional differential equations, 204 (north-holland mathematics studies).
Lin, Z., Wang, D., Qi, D., & Deng, L. (2020). A Petrov–Galerkin finite element-meshfree formulation for multi-dimensional fractional diffusion equations. Computational Mechanics, 66, 323-350.
Ling L. and Yamamoto M. (2013). Numerical simulations for space{time fractional diffusion equations. International Journal of Computational Methods, 10(02):1341001.
Mainardi F. (1996). The fundamental solutions for the fractional diffusion-wave equation. Applied Mathematics Letters, 9(6):23-28.
Francesco Mainardi and Gianni Pagnini. The wright functions as solutions of the timefractional diffusion equation. Applied Mathematics and Computation, 141(1):51-62,2003.
Podlubny I. et al. (2009). Matrix approach to discrete fractional calculus II : Partial fractional differential equations. Journal of Computational Physics, 228(8):3137-3153. ISSN 0021-9991. doi: 10.1016/j.jcp.2009.01.014. URL http://dx.doi.org/10.1016/j.jcp. 2009.01.014.
Samko S. G. et al. (1993). Fractional integrals and derivatives. Theory and Applications, Gordon and Breach, Yverdon.
Sun Z. and Wu. X. (2006). A fully discrete difference scheme for a diffusion-wavesystem. Applied Numerical Mathematics, 56(2):193-209.
Tikhonov A. et al. (2013). Numerical methods for the solution of ill-posed problems. Springer Science & Business Media.
Published
2021-03-30
How to Cite
Zekri, H. (2021). Solving Time-Fractional Diffusion Equation: A Finite-Element Approach. Science Journal of University of Zakho, 9(1), 38-42. https://doi.org/10.25271/sjuoz.2021.9.1.791
Section
Science Journal of University of Zakho