THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

External Links

NUMERICAL ANALYSIS OF TIME FRACTIONAL THREE DIMENSIONAL DIFFUSION EQUATION

ABSTRACT
The three dimensional diffusion equations were extended to the scope of fractional order derivative. The fractional operator used here is in Caputo sense. The resulting equation was solved using two numerical approaches: The forward in time and central in space method and the Crank-Nicholson method. The stability analysis of both methods was studied, and the study showed that the Crank-Nicholson method is unconditionally stable while the forward method is stable if some conditions are satisfied.
KEYWORDS
PAPER SUBMITTED: 2014-10-10
PAPER REVISED: 2015-03-15
PAPER ACCEPTED: 2015-03-18
PUBLISHED ONLINE: 2015-08-02
DOI REFERENCE: https://doi.org/10.2298/TSCI15S10S7A
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2015, VOLUME 19, ISSUE Supplement 1, PAGES [S7 - S12]
REFERENCES
  1. Li, B. Q., Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer, Springer-Verlag London, 2006
  2. Versteeg, H., Malalasekera, W., An Introduction to Computational Fluid Dynamics, The Finite Volume Methods, 2nd ed., Pearson Education, London, 2000, pp. 262-263
  3. Smoluchowski, M. von, About Brownian Motion under the Action of External Forces and the Relationship with the Generalized Diffusion Equation (in German), Ann. Phys., 353 (1915), pp. 1103-1112
  4. Granville Sewell, E., The Numerical Solution of Ordinary and Partial Differential Equations, John Wiley & Sons, Inc., New York, USA, 1988 Atangana, A.: Numerical Analysis of Time Fractional Three Dimensional … S12 THERMAL SCIENCE, Year 2015, Vol. 19, Suppl. 1, pp. S7-S12
  5. Doungmo Goufo, E. F., et al., Some Properties of the Kermack-McKendrick Epidemic Model with Fractional Derivative and Nonlinear Incidence, Advances in Difference Equations, 278 (2014), Oct., pp. 1-9
  6. Doungmo Goufo, E. F., A Mathematical Analysis of Fractional Fragmentation Dynamics with Growth, Journal of Function Spaces, 2014 (2014), ID 201520
  7. Doungmo Goufo, E. F., Stella. M., Mathematical Solvability of a Caputo Fractional Polymer Degradation Model Using Further Generalized Functions, Mathematical Problems in Engineering, 2014 (2014), ID 392792
  8. Atangana, A., Kilicman, A., Analytical Solutions of the Space-Time Fractional Derivative of Advection Dispersion Equation, Mathematical Problems in Engineering, 2013 (2013), ID 853127
  9. Zhang, Y., Finite Difference Method for Fractional Partial Differential Equation, Applied Mathematics and Computation, 215 (2009), 2, pp. 524-529
  10. Atangana, A., Alabaraoye, E., Solving System of Fractional Partial Differential Equations Arisen in the Model of HIV Infection of CD4+ Cells and Attractor One-Dimensional Keller-Segel Equation, Advances in Difference Equations, 2013 (2013), 1, pp.1-12
  11. Podlubny, I., El-Sayed, A. M. A., On Two Definitions of Fractional Calculus, Slovak Academy of Science, Institute of Experimental Physics, Kosice, Slovakia, 1996
  12. Kilbas, A. A., et al., Theory and Applications of Fractional Differential Equations, vol. 204 of North- Holland Mathematics Studies, Elsevier Science B. V., Amsterdam, The Netherlands, 2006
  13. Yuste, S. B., Acedo, L., On Explicit Finite Difference Method and a New von Neumann-Type Stability Analysis for Fractional Diffusion Equations, SIAM Journal on Numerical Analysis, 42 (2005), 5, pp. 1862-1874
  14. Crank, J., Nicolson P., A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat Conduction Type, Proceedings, The Cambridge Philosophical Society, 43 (1947), 1, pp. 50-67

© 2019 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence