Anomalous diffusion in a variable area whose boundary moves with a constant speed

ABSTRACT

In this paper, we study a space-fractional anomalous diffusion in a variable area. The moving boundary is assumed moving with constant speed. The numerical scheme was present by changing the moving boundary to a fixed one. The steady-state approximation was also given to show the properties of the diffusion process.

In this paper, we study a space-fractional anomalous diffusion in a variable area. The moving boundary is assumed moving with constant speed. The numerical scheme was present by changing the moving boundary to a fixed one. The steady-state approximation was also given to show the properties of the diffusion process.

Cite this paper

nullLi, X. (2012) Anomalous diffusion in a variable area whose boundary moves with a constant speed.*Open Journal of Applied Sciences*, **2**, 183-186. doi: 10.4236/ojapps.2012.24B042.

nullLi, X. (2012) Anomalous diffusion in a variable area whose boundary moves with a constant speed.

References

[1] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam: Elsevier, 2006.

[2] I. Podlubny, Fractional Differential Equations, New York: Academic Press, 1999.

[3] A.S. Chaves, A fractional diffusion equation to describe Lévy flights, Phys. Lett. A, vol. 239, pp. 13-16, 1998.

[4] P. Paradisi, R. Cesari, F. Mainardi, A. Maurizi and F. Tampieri, A generalized Fick's law to describe non-local transport effects, Physics and Chemistry of the Earth, vol. 26, pp. 275-279, 2001.

[5] K. Dierhelm, N.J. Ford, A.D. Freed and Y. Luchko, Algorithms for the fractional calculus: A selection of numerical methods, Comput. Methods Appl. Mech. Engrg, vol. 194, pp. 743-773, 2005.

[6] K. Diethelm, N.J. Ford, A.D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., vol. 29, pp. 3-22, 2002.

[7] K. Diethelm, N.J. Ford, A.D. Freed, Detailed error analysis for a fractional Adams method. Numer. Algorithms, vol. 36, pp. 31-52, 2004.

[8] N.J. Ford and A.C. Simpson, The numerical solution of fractional differential equations: Speed versus accuracy, Numerical Algorithms, vol. 26, pp. 333-346, 2001.

[9] W.H. Deng, Short memory principle and a predictor-corrector approach for fractional differential equations, J. Comput. Appl. Math. Vol. 206, pp. 174-188, 2006.

[10] X.X. Li, Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method, Commun Nonlinear Sci Numer Simulat, vol. 17, pp. 3934–3946, 2012.

[11] J.Y. Liu and M.Y. Xu, Some exact solutions to Stefan problems with fractional differential equations, J. Math, Anal. Appl., vol. 351, pp. 536-542, 2009.

[12] X.C. Li, M.Y. Xu, S.W. Wang, Analytical solutions to the moving boundary problems with space-time-fractional derivatives in drug release devices, Journal of Physics A: Mathematical and Theoretical, vol. 40, pp. 12131-12141, 2007.

[13] X.C. Li, M.Y. Xu, S.W. Wang, Scale-invariant solutions to partial differential equations of fractional order with a moving boundary condition, Journal of Physics A: Mathematical and Theoretical, vol. 41, pp. 155202, 2008.

[14] K. Diethelm, N.J. Ford, A.D. Freed and YU. Luchko, Algorithms for the fractional calculus: A selection of numerical methods, Comput. Methods Appl. Mech. Engrg, vol. 194, pp. 743-773, 2005.

[1] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam: Elsevier, 2006.

[2] I. Podlubny, Fractional Differential Equations, New York: Academic Press, 1999.

[3] A.S. Chaves, A fractional diffusion equation to describe Lévy flights, Phys. Lett. A, vol. 239, pp. 13-16, 1998.

[4] P. Paradisi, R. Cesari, F. Mainardi, A. Maurizi and F. Tampieri, A generalized Fick's law to describe non-local transport effects, Physics and Chemistry of the Earth, vol. 26, pp. 275-279, 2001.

[5] K. Dierhelm, N.J. Ford, A.D. Freed and Y. Luchko, Algorithms for the fractional calculus: A selection of numerical methods, Comput. Methods Appl. Mech. Engrg, vol. 194, pp. 743-773, 2005.

[6] K. Diethelm, N.J. Ford, A.D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., vol. 29, pp. 3-22, 2002.

[7] K. Diethelm, N.J. Ford, A.D. Freed, Detailed error analysis for a fractional Adams method. Numer. Algorithms, vol. 36, pp. 31-52, 2004.

[8] N.J. Ford and A.C. Simpson, The numerical solution of fractional differential equations: Speed versus accuracy, Numerical Algorithms, vol. 26, pp. 333-346, 2001.

[9] W.H. Deng, Short memory principle and a predictor-corrector approach for fractional differential equations, J. Comput. Appl. Math. Vol. 206, pp. 174-188, 2006.

[10] X.X. Li, Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method, Commun Nonlinear Sci Numer Simulat, vol. 17, pp. 3934–3946, 2012.

[11] J.Y. Liu and M.Y. Xu, Some exact solutions to Stefan problems with fractional differential equations, J. Math, Anal. Appl., vol. 351, pp. 536-542, 2009.

[12] X.C. Li, M.Y. Xu, S.W. Wang, Analytical solutions to the moving boundary problems with space-time-fractional derivatives in drug release devices, Journal of Physics A: Mathematical and Theoretical, vol. 40, pp. 12131-12141, 2007.

[13] X.C. Li, M.Y. Xu, S.W. Wang, Scale-invariant solutions to partial differential equations of fractional order with a moving boundary condition, Journal of Physics A: Mathematical and Theoretical, vol. 41, pp. 155202, 2008.

[14] K. Diethelm, N.J. Ford, A.D. Freed and YU. Luchko, Algorithms for the fractional calculus: A selection of numerical methods, Comput. Methods Appl. Mech. Engrg, vol. 194, pp. 743-773, 2005.