THERMAL SCIENCE

International Scientific Journal

SOLVING THE TIME-FRACTIONAL DIFFUSION EQUATION USING A LIE GROUP INTEGRATOR

ABSTRACT
In this paper, we propose a numerical method to approximate the solutions of time fractional diffusion equation which is in the class of Lie group integrators. Our utilized method, namely fictitious time integration method transforms the unknown dependent variable to a new variable with one dimension more. Then the group preserving scheme is used to integrate the new fractional partial differential equations in the augmented space R3+1. Effectiveness and validity of method demonstrated using two examples.
KEYWORDS
PAPER SUBMITTED: 2014-11-15
PAPER REVISED: 2015-02-02
PAPER ACCEPTED: 2015-03-04
PUBLISHED ONLINE: 2015-08-02
DOI REFERENCE: https://doi.org/10.2298/TSCI15S1S77H
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2015, VOLUME 19, ISSUE Supplement 1, PAGES [S77 - S83]
REFERENCES
  1. Kilbas, A., et al., Theory and Applications of Fractional Differential Equations, Elsevier, 2006
  2. Podlubny, I., Fractional Differential Equations, Academic Press, New York, USA, 1999
  3. Hilfer, R., Applications of Fractional Calculus in Physics, Academic press, Orlando, Fla, USA, 1999
  4. Diethelm, K., The Analysis of Fractional Differential Equations, An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer-Verlag, Berlin-Heidelberg, Germany, 2010
  5. Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, Inc., New York, USA, 1993
  6. Chen, W., et al., Fractional Diffusion Equations by the Kansa Method, Computers and Mathematics with Applications, 59 (2010), 5, pp. 1614-1620
  7. Liu, Q., et al., An Implicit RBF Meshless Approach for Time Fractional Diffusion Equations, Computational Mechanics, 48 (2011), 1, pp. 1-12
  8. Fu, Z. J., et al., Boundary Particle Method for Laplace Transformed Time Fractional Diffusion Equations, Journal of Computational Physics, 235 (2013), Feb., pp. 52-66
  9. Gu, Y. T., et al., An Advanced Meshless Method for Time Fractional Diffusion Equation, International Journal of Computational Methods, 8 (2011), 4, pp. 653-665
  10. Pirkhedri, A., Javadi, H. H. S., Solving the Time-Fractional Diffusion Equation via Sinc-Haar Collocation Method, Applied Mathematics and Computation, 257 (2015), Apr., pp. 317-326
  11. Cui, M., Convergence Analysis of High-Order Compact Alternating Direction Implicit Schemes for the Two-Dimensional Time Fractional Diffusion Equation, Numerical Algorithms, 62 (2013), 3, pp. 383-409
  12. Alikhanov, A. A., A New Difference Scheme for the Time Fractional Diffusion Equation, Journal of Computational Physics, 280 (2015), Feb., pp. 424-438
  13. Zhang, Y. N., et al., Finite Difference Methods for the Time Fractional Diffusion Equation on Non- Uniform Meshes, Journal of Computational Physics, 265 (2014), May, pp. 195-210
  14. Liu, C.-S., Solving an Inverse Sturm-Liouville Problem by a Lie-Group Method, Boundary Value Problems, 2008 (2008), Jan., 749865
  15. Liu, C.-S., Cone of Non-Linear Dynamical System and Group Preserving Schemes, International Journal of Non-Linear Mechanics, 36 (2001), 7, pp. 1047-1068
  16. Abbasbandy, S., Hashemi, M., Group Preserving Scheme for the Cauchy Problem of the Laplace Equation, Engineering Analalysis with Boundary Elements, 35 (2011), 8, pp. 1003-1009
  17. Abbasbandy, S., et al., The Lie-Group Shooting Method for Solving the Bratu Equation, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 11, pp. 4238-4249
  18. Hashemi, M., Constructing a New Geometric Numerical Integration Method to the Nonlinear Heat Transfer Equations, Commun. in Nonlin. Sci. and Num. Simulat., 22 (2015), 1-3, pp. 990-1001

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