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The Numerical Simulation of Space-Time Variable Fractional Order Diffusion Equation

Published online by Cambridge University Press:  28 May 2015

Hongmei Zhang  and
Shujun Shen
Hongmei Zhang*
Affiliation:
School of Mathematical and Computer Sciences, Fuzhou University, Fuzhou, China
Shujun Shen*
Affiliation:
School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian, China
*
Corresponding author.Email address:zhm-fzu@163.com
Corresponding author.Email address:shensj12@sina.com
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Abstract

Many physical processes appear to exhibit fractional order behavior that may vary with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. Numerical methods and analysis of stability and convergence of numerical scheme for the variable fractional order partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the space-time variable fractional order diffusion equation on a finite domain. It is worth mentioning that here we use the Coimbra-definition variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation is proposed and then the stability and convergence of the numerical scheme are investigated. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

Keywords

26A3334K2865M1260J70Variable fractional derivativediffusion equationimplicit Euler schemestabilityconvergence
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 4 , November 2013 , pp. 571 - 585
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Metzler, R. and Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), pp. 1–77.CrossRefGoogle Scholar
[2]Podlubny, I., Fractional Differential Equations, Academic Press, 1999.Google Scholar
[3]Samko, S.G., A.Kilbas, A. and O.Marichev, I., Fractional Integrals and Derivatives: Theory and Applications (Translation from the Russian), Amsterdam: Gordon and Breach, 1993.Google Scholar
[4]Coimbra, C.F.M., Mechanica with variable-order differential operators, Annalen der Physik, No. 11-12, 12 (2003), pp. 692–703.CrossRefGoogle Scholar
[5]Cooper, G.R.J. and Cowan, D.R., Filtering using variable order vertical derivatives, Computers and Geosciences, 30 (2004), pp. 455–459.CrossRefGoogle Scholar
[6]Tseng, C.C., Design of variable and adaptive fractional order FIR differentiators, Signal Processing, 86 (2006), pp. 2554–2566.CrossRefGoogle Scholar
[7]Sun, H., Chen, W. and Chen, Y., Variable-order fractional differential operators in anomalous diffusion modeling, Physica A, 388 (2009), pp. 4586–4592.CrossRefGoogle Scholar
[8]Samko, S.G. and Ross, B., Intergation and differentiation to a variable fractional order, Integral Transforms and Special Functions, 1(4) (1993), pp. 277–300.CrossRefGoogle Scholar
[9]Ross, B. and Samko, S. G., Fractional integration operator of variable order in the HSlder spaces Hλ(x), Internat. J. Math. and Math. Sci., 18(4) (1995),pp.777-788.CrossRefGoogle Scholar
[10]Samko, S.G., Fractional integration and differentiation of variable order, Analysis Mathematica, 21 (1995), pp. 213–236.CrossRefGoogle Scholar
[11]Kikuchi, K. and Negoro, A., On Markov process generated by pseudodifferential operator of variable order, Osaka J. Math., 34 (1997), pp. 319–335.Google Scholar
[12]Lorenzo, C.F. and Hartley, T.T., Initialization, conceptualization, and application in the generalized fractional calculus, NASA Technical Publication 98-208415, NASA, Lewis Reseach Center, 1998.Google Scholar
[13]Lorenzo, C.F. and Hartley, T.T., Variable order and distributed order fractional operators, Nonlinear Dynamics, 29 (2002), pp. 57–98.CrossRefGoogle Scholar
[14]Ingman, D. and Suzdalnitsky, J., Control of damping oscilations by fractional differential operator with time-dependent order, Comput. Methods Appl. Mech. Eng., 193 (2004), pp. 5585–5595.CrossRefGoogle Scholar
[15]Pedro, H.T.C., Kobayashi, M.H., Pereira, J.M.C. and Coimbra, C.F.M., Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere, Journal of Vibration and Control, 14 (2008), pp. 1659–1672.CrossRefGoogle Scholar
[16]Ramirez, L.E.S. and Coimbra, C.F.M., On the selection and meaning of variable order operators for dynamic modeling, International Journal of Differential Equations, 2010, doi:10.1155/2010,846107.CrossRefGoogle Scholar
[17]Lin, R., Liu, F., Anh, V. and Turner, I., Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation, Appl. Math. Comput., 212 (2009), pp. 435–445.Google Scholar
[18]Zhuang, P., Liu, F., Anh, V. and Turner, I., Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal., 47(3) (2009), pp. 1760-1781.CrossRefGoogle Scholar
[19]Chen, C.-M., Liu, F., Anh, V. and Turner, I., Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation, SIAM J. Sci. Comput., 32(4) (2010), pp. 1740-1760.CrossRefGoogle Scholar
[20]Liu, F., Anh, V. and Turner, I., Numerical solution of space fractional Fokker-Planck equation, J. Comp. and Appl. Math., 166 (2004), pp. 209–219.CrossRefGoogle Scholar
[21]Meerschaert, M.M. and Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, Journal of Computational and Applied Mathematics, 172 (2004), pp. 65–77.CrossRefGoogle Scholar
[22]Lin, Y.M. and C.Xu, J., Finite diffrence/spectral approximation for the time fractional diffusion equations, J. Comput. Phys., 2(3) (2007), pp. 1533-1552.Google Scholar
[23]Smith, G. D., Numerical Solution of Partial Differential Equations: Finite Difference Methods, Clarendon Press, Oxford, 1985.Google Scholar
[24]Yu, D. and Tan, H., Numerical Methods of Differential Equations, Science Publisher, Beijing, 2003.Google Scholar