THERMAL SCIENCE

International Scientific Journal

Thermal Science - Online First

External Links

online first only

Analysis of fractional nonliear diffuision behaviors based on Adomian polynomials

ABSTRACT
A time-fractional nonlinear diffusion equation of two orders is considered to investigate strong nonlinearity through porous media. An equivalent integral equation is established and Adomian polynomials are adopted to linearize nonlinear terms. With the Taylor expansion of fractional order, recurrence formulae are proposed and novel numerical solutions are obtained to depict the diffusion behaviors more accurately. The result shows that the method is suitable for numerical simulation of the fractional diffusion equations of multi-orders.
KEYWORDS
PAPER SUBMITTED: 2016-04-16
PAPER REVISED: 2016-05-21
PAPER ACCEPTED: 2016-06-15
PUBLISHED ONLINE: 2016-12-03
DOI REFERENCE: https://doi.org/10.2298/TSCI160416301W
REFERENCES
  1. Yu, B.M., Analysis of flow in fractal porous media, Applied Mechanics Reviews, 61 (2008), 5 050801
  2. Chen, W., et al., Anomalous diffusion modeling by fractal and fractional derivatives, Computers & Mathematics with Applications, 59 (2010), 5, pp. 1754-1758
  3. Ma, Q., Chen, Z.Q., Lattice Boltzmann simulation of multicomponent noncontinuum diffusion in fractal porous structures, Physical Review E, 92 (2015), 1, 013025
  4. Gmachowski, L., Fractal model of anomalous diffusion, European Biophysics Journal, 44 (2015), 8, pp. 613-621
  5. Xiao, B.Q., et al., Fractal analysis of gas diffusion in porous nanofibers, Fractals, 23(2015), 1, 1540011
  6. Zhuan, X., Sun, Z.Z., A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions, Journal of Computational Physics, 230 (2011), 15, pp. 6061-6074
  7. Sun, H.G., et al., Finite difference schemes for variable-order time fractional diffusion equation, International Journal of Bifurcation and Chaos, 22 (2012), 4, 1250085
  8. Chen, S., et al., Finite difference approximations for the fractional Fokker-Planck equation, Applied Mathematical Modelling, 33 (2009), 1, pp. 256-273
  9. Zhuan, X., Sun, Z.Z., A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions, Journal of Computational Physics, 230 (2011), 15, pp. 6061-6074
  10. Wu, G.C., et al., Lattice fractional diffusion equation in terms of a Riesz-Caputo difference, Physica A, 438 (2015), pp. 335-339
  11. Hristov J., Approximate solutions to time-fractional models by integral balance approach, Chapter 5, In: Fractional Dynamics, C. Cattani, H.M. Srivastava, X.J. Yang, (eds), De Gruyter Open, 2015 , pp.78-109.
  12. Hristov, J., Transient heat diffusion with a non-singular fading memory from the cattaneo constitutive equation with Jeffrey's kernel to the Caputo-Fabrizio time-fractional derivative, Thermal Science, 20 (2016), 2, pp. 757-762
  13. Li, C., Zeng, F., Numerical methods for fractional calculus, Chapman and Hall/CRC, Boca Raton, USA, 2015
  14. Podlubny, I., Fractional differential equations, Academic Press, San Diego, 1999
  15. Yang, Q., et al., Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Applied Mathematical Modelling, 34 (2010), 1, pp. 200-218
  16. Hristov J., Diffusion models with weakly singular kernels in the fading memories: how the integral-balance method can be applied?, Thermal Science, 19 (2015), 3, pp. 947-957
  17. Duan, J.S., Recurrence triangle for Adomian polynomials, Applied Mathematics and Computation, 216 (2010), 4, pp. 1235-1241
  18. Duan, J.S., An efficient algorithm for the multivariable Adomian polynomials, Applied Mathematics and Computation, 217 (2010), 6, pp. 2456-2467
  19. Duan, J.S., Convenient analytic recurrence algorithms for the Adomian polynomials, Applied Mathematics and Computation, 217 (2011), 13, pp. 6337-6348