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A High-Order Difference Scheme for the Generalized Cattaneo Equation

Published online by Cambridge University Press:  28 May 2015

Seak-Weng Vong ,
Hong-Kui Pang  and
Xiao-Qing Jin
Seak-Weng Vong*
Affiliation:
Department of Mathematics, University of Macau, Macao, P.R. China
Hong-Kui Pang*
Affiliation:
School of Mathematical Sciences, Jiangsu Normal University, Xuzhou, P.R. China
Xiao-Qing Jin*
Affiliation:
Department of Mathematics, University of Macau, Macao, P.R. China
*
Corresponding author. Email: swvong@umac.mo
Corresponding author. Email: panghongkui@163.com
Corresponding author. Email: xqjin@umac.mo
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Abstract

A high-order finite difference scheme for the fractional Cattaneo equation is investigated. The L1 approximation is invoked for the time fractional part, and a compact difference scheme is applied to approximate the second-order space derivative. The stability and convergence rate are discussed in the maximum norm by the energy method. Numerical examples are provided to verify the effectiveness and accuracy of the proposed difference scheme.

Keywords

65M0665M1265M1535Q51Fractional Cattaneo equationL1 approximationcompact finite differencestabilityconvergence
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 2 , Issue 2 , May 2012 , pp. 170 - 184
Copyright
Copyright © Global-Science Press 2012

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References

[1]Compte, A., Metzler, R., The generalized Cattaneo equation for the description of anomalous transport processes, J. Phys. A: Math. Gen., 30 (1997), pp. 7277–7289.CrossRefGoogle Scholar
[2]Cui, M., Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228 (2009), pp. 7792–7804.Google Scholar
[3]Du, R., Cao, W., Sun, Z. Z., A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model., 34 (2010), pp. 2998–3009.Google Scholar
[4]Gao, G., Sun, Z. Z., A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys., 230 (2011), pp. 586–595.Google Scholar
[5]Ghazizadeh, H. R., Maerefat, M., Azimi, A., Explicit and implicit finite difference schemes for fractional Cattaneo equation, J. Comput. Phys., 229 (2010), pp. 7042–7057.CrossRefGoogle Scholar
[6]Godoy, S., Garcia-Colin, L. S., From the quantum random walk to classical mesoscopic diffusion in crystalline solids, Phys. Rev. E, 53 (1996), pp. 5779–5785.Google Scholar
[7]Gomez, H., Colominas, I., Navarrina, F., Casteleiro, M., A mathematical model and a numerical model for hyperbolic mass transport in compressible flows, Heat Mass Transfer, 45 (2008), pp. 219–226CrossRefGoogle Scholar
[8]Jou, D., Casas-Vazquez, J., Lebon, G., Extended Irreversible Thermodynamics, Springer-Verlag, Berlin Heidelberg, 2001.Google Scholar
[9]Kilbas, A., Srivastava, H., Trujillo, J., Theory and Applications of Fractional Differential Equations, Elsevier Science and Technology, 2006.Google Scholar
[10]Kosztołowicz, T., Lewandowska, K. D., Hyperbolic subdiffusive impedance, J. Phys. A: Math. Theor., 42 (2009) 055004.CrossRefGoogle Scholar
[11]Langlands, T., Henry, B., The accuracy and stability of an implicit solution method for the fractional diffusion equaition, J. Comput. Phys., 205 (2005), pp. 719–736.Google Scholar
[12]Lopez-Marcos, J. C., A difference scheme for a nonlinear partial integro-differential equation, SIAM. J. Numer. Anal., 27 (1990), pp. 20–31.Google Scholar
[13]Meerschaert, M. M., Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), pp. 65–77.CrossRefGoogle Scholar
[14]Meerschaert, M. M., Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56 (2006), pp. 80–90.Google Scholar
[15]Odibat, Z. M., Shawagfeh, N. T., Generalized Taylor's formula, Appl. Math. Comput., 186 (2007), pp. 286–293.Google Scholar
[16]Podlubny, I., Fractional Differential Equations, Academic Press, New York, 1999.Google Scholar
[17]Povstenko, Y. Z., Fractional Cattaneo-type equations and generalized thermoelasticity, J. Therm. Stresses, 34 (2011), pp. 97–114.Google Scholar
[18]Qi, H., Jiang, X., Solutions of the space-time fractional Cattaneo diffusion equation, Physica A, 390 (2011), pp. 1876–1883.Google Scholar
[19]Samarskii, A. A., Andreev, B. B., Finite Difference Methods for Elliptic Equation, Nauka, Moscow, 1976 (in Russian).Google Scholar
[20]Sun, Z. Z., On the compact difference schemes for heat equation with Neumann boundary conditions, Numer. Method Partial Differen. Equat., 25 (2009), pp. 1320–1341.CrossRefGoogle Scholar
[21]Sun, Z. Z., Wu, X., A fully discrete scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), pp. 193–209.Google Scholar
[22]Tang, T., Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, Numer. Math., 61 (1992), pp. 373–382.CrossRefGoogle Scholar
[23]Tang, T., A finite difference scheme for partial integro-differential equations with weakly singular kernel, Appl. Numer. Math., 11 (1993), pp. 309–319.Google Scholar
[24]Tang, T., A note on collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Anal., 13 (1993), pp. 93–99.Google Scholar
[25]Tzou, D. Y., A unified field approach for heat conduction from macro- to micro-scales, ASME J. Heat Transfer, 117 (1995), pp. 8–16.Google Scholar
[26]Yuste, S. B., Acedo, L., An explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42 (2005), pp. 1862–1874.Google Scholar
[27]Zakari, M., Jou, D., Equations of state and transport equations in viscous cosmological models, Phys. Rev. D, 48 (1993), pp. 1597–1601.Google Scholar
[28]Zhuang, P., Liu, F., Anh, V., Turner, I., New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. Numer. Anal., 46 (2008), pp. 1079–1095.Google Scholar