THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

External Links

Wei Jiang

,

Beibei Guo

A NEW NUMERICAL METHOD FOR SOLVING TWO-DIMENSIONAL VARIABLE-ORDER ANOMALOUS SUB-DIFFUSION EQUATION

ABSTRACT
The novelty and innovativeness of this paper are the combination of reproducing kernel theory and spline, this leads to a new simple but effective numerical method for solving variable-order anomalous sub-diffusion equation successfully. This combination overcomes the weaknesses of piecewise polynomials that can not be used to solve differential equations directly because of lack of the smoothness. Moreover, new bases of reproducing kernel spaces are constructed. On the other hand, the existence of any ε-approximate solution is proved and an effective method for obtaining the ε-approximate solution is established. A numerical example is given to show the accuracy and effectiveness of theoretical results.
KEYWORDS
variable-order anomalous sub-diffusion equation, spline, reproducing kernel theory, approximate solution
PAPER SUBMITTED: 2015-11-15
PAPER REVISED: 2016-01-25
PAPER ACCEPTED: 2016-02-18
PUBLISHED ONLINE: 2016-09-24
DOI REFERENCE: https://doi.org/10.2298/TSCI16S3701J
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2016, VOLUME 20, ISSUE Supplement 3, PAGES [S701 - S710]
REFERENCES
  1. Zhao, D., et al., Some Fractal Heat-Transfer Problems with Local Fractional Calculus, Thermal Science, 19 (2015), 5, pp. 1867-1871
  2. Yang, X. J., et al., Nonlinear Dynamics for Local Fractional Burger's Equation Arising in Fractal Flow, Nonlinear Dynamics, 84 (2015), 1, pp. 1-5
  3. Jiang, W., et al., Numerical Solution of Nonlinear Volterra Integro-Differential Equations of Fractional- Order by the Reproducing Kernel Method, Applied Mathematical Modelling, 39 (2015), 16, pp. 4871- 4876
  4. Guo, B. B., et al., Numerical Application for Volterra's Population Growth Model with Fractional Order by the Modified Reproducing Kernel Method, Journal of Computational Complexity and Applications, 1 (2015), 1, pp. 1-9
  5. Ji, J., Discrete Fractional Diffusion Equation with a Source Term, Journal of Computational Complexity and Applications, 1 (2015), 1, pp. 10-14
  6. Wu, F., et al., Discrete Fractional Creep Model of Salt Rock, Journal of Computational Complexity and Applications, 2 (2016), 1, pp. 1-6
  7. Zhou, X. J., et al., Numerical Method for Differential-Algebraic Equations of Fractional Order, Journal of Computational Complexity and Applications, 1 (2015), 2, pp. 54-63
  8. Geng, F. Z., et al., A Numerical Method for Solving Fractional Singularly Perturbed Initial Value Problems Based on the Reproducing Kernel Method, Journal of Computational Complexity and Applications, 1 (2015), 2, pp. 89-94
  9. Sun, H. G., et al., On Mean Square Displacement Behaviors of Anomalous Diffusions with Variable and Random Orders, Physics Letters A, 374 (2010) 7, pp. 906-910
  10. Sun, H. G., et al., Variable-Order Fractional Differential Operators in Anomalous Diffusion Modeling, Physical A, 388 (2009), 21, pp. 4586-4592
  11. Chen, C. M., et al., Numerical Methods for Solving a Two-Dimensional Variable-Order Anomalous Subdiffusion Equation, Mathematics of Computation, 81 (2012), 81, pp. 345-366
  12. Chen, C., et al., Numerical Schemes with High Spatial Accuracy for a Variable-Order Anomalous Sub- Diffusion Equations, SIAM Journal on Scientific Computing, 32 (2010), 4, pp. 1740-1760
  13. Lin, R., et al., Stability and Convergence of a New Explicit Finite-Difference Approximation for the Variable-Order Nonlinear Fractional Diffusion Equation, Applied Mathematics and Computation, 212 (2009), 2, pp. 435-445
  14. Chen, C., et al., Numerical Schemes with High Spatial Accuracy for a Variable-Order Anomalous Subdiffusion Equations, SIAM Journal on Scientific Computing, 32 (2010), 4, pp. 1740-1760
  15. Wu, B.Y., et al., Applied Reproducing Kernel Theory, Science Publisher, New York, USA, 2012
  16. Wang, Y. L., et al., Using Reproducing Kernel for Solving a Class of Partial Differential Equation with Variable-Coeffcients, Applied Mathematics and Mechanics, 29 (2008), 1, pp. 129-137
  17. Yan, Q., Numerical Analysis, Beihang University Press, Beijing, China, 2011
PDF VERSION [DOWNLOAD]
A new numerical method for solving two-dimensional variable-order anomalous sub-diffusion equation

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, 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