Hostname: page-component-7c8c6479df-hgkh8 Total loading time: 0 Render date: 2024-03-28T22:49:19.869Z Has data issue: false hasContentIssue false

Numerical Solution to the Multi-Term Time Fractional Diffusion Equation in a Finite Domain

Published online by Cambridge University Press:  20 July 2016

Gongsheng Li*
Affiliation:
School of Sciences, Shandong University of Technology, Zibo, Shandong 255049, China
Chunlong Sun*
Affiliation:
School of Sciences, Shandong University of Technology, Zibo, Shandong 255049, China
Xianzheng Jia*
Affiliation:
School of Sciences, Shandong University of Technology, Zibo, Shandong 255049, China
Dianhu Du*
Affiliation:
School of Sciences, Shandong University of Technology, Zibo, Shandong 255049, China
*
*Corresponding author. Email addresses:ligs@sdut.edu.cn (Gongsheng Li), sunchunlong527@163.com (Chunlong Sun), kathy1978@126.com (Xianzheng Jia), dudianhu@126.com (Dianhu Du)
*Corresponding author. Email addresses:ligs@sdut.edu.cn (Gongsheng Li), sunchunlong527@163.com (Chunlong Sun), kathy1978@126.com (Xianzheng Jia), dudianhu@126.com (Dianhu Du)
*Corresponding author. Email addresses:ligs@sdut.edu.cn (Gongsheng Li), sunchunlong527@163.com (Chunlong Sun), kathy1978@126.com (Xianzheng Jia), dudianhu@126.com (Dianhu Du)
*Corresponding author. Email addresses:ligs@sdut.edu.cn (Gongsheng Li), sunchunlong527@163.com (Chunlong Sun), kathy1978@126.com (Xianzheng Jia), dudianhu@126.com (Dianhu Du)

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper deals with numerical solution to the multi-term time fractional diffusion equation in a finite domain. An implicit finite difference scheme is established based on Caputo's definition to the fractional derivatives, and the upper and lower bounds to the spectral radius of the coefficient matrix of the difference scheme are estimated, with which the unconditional stability and convergence are proved. The numerical results demonstrate the effectiveness of the theoretical analysis, and the method and technique can also be applied to other kinds of time/space fractional diffusion equations.

Type
Research Article
Copyright
Copyright © Global-Science Press 2016 

References

[1]Adams, E. E., Gelhar, L. W., Field study of dispersion in a heterogeneous aquifer 2: Spatial moments analysis, Water Resources Research, 28 (1992), pp. 32933307.Google Scholar
[2]Benson, D. A., The Fractional Advection-Dispersion Equation: Development and Application, Dissertation of Doctorial Degree, University of Nevada, Reno, USA, 1998.Google Scholar
[3]Berkowitz, B., Scher, H., Silliman, S. E., Anomalous transport in laboratory-scale heterogeneous porus media, Water Resources Research, 36 (2000), pp. 149158.CrossRefGoogle Scholar
[4]Caponetto, R., Dongola, G., Fortuna, L., Petras, I., Fractional Order Systems: Modeling and Control Applications, World Scientific, Singapore, 2010.CrossRefGoogle Scholar
[5]Chen, S., Liu, F., Burrage, K., Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media, Computers and Mathematics with Applications, 67 (2014), pp. 16731681.Google Scholar
[6]Coimbra, C. F. M., Mechanics with variable-order differential operators, Ann. Phys., 12 (2003), pp. 692703.Google Scholar
[7]Daftardar-Gejji, V., Bhalekar, S., Boundary value problems for multi-term fractional differential equationas, J. Math. Anal. Appl., 345 (2008), pp. 754765.Google Scholar
[8]Giona, M., Gerbelli, S., Roman, H. E., Fractional diffusion equation and relaxation in complex viscoelastic materials, Physica A, 191 (1992), pp. 449453.CrossRefGoogle Scholar
[9]Hatano, Y., Hatano, N., Dispersive transport of ions in column experiments: an explanation of long-tailed profiles, Water Resources Research, 34 (1998), pp. 10271033.Google Scholar
[10]Jiang, H., Liu, F., Turner, I., Burrage, K., Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain, Computers and Mathematics with Applications, 64 (2012), pp. 33773388.Google Scholar
[11]Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.Google Scholar
[12]Li, G. S., Gu, W. J., Jia, X. Z., Numerical inversions for space-dependent diffusion coefficient in the time fractional diffusion equation, Journal of Inverse and Ill-Posed Problems, 20 (2012), pp. 339366.CrossRefGoogle Scholar
[13]Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K., Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Applied Mathematics and Computation, 191 (2007), pp. 1220.Google Scholar
[14]Liu, F., Zhuang, P., Burrage, K., Numerical methods and analysis for a class of fractional advection-dispersion models, Computers and Mathematics with Applications, 64 (2012), pp. 29903007.CrossRefGoogle Scholar
[15]Liu, F., Meerschaert, M. M., McGough, R. J., Zhuang, P., Liu, Q., Numerical methods for solving the multi-term time-fractional wave-diffusion equation, Fractional Calculus and Applied Analysis, 16 (2013), pp. 925.Google Scholar
[16]Lorenzo, C. F., Hartley, T. T., Variable order and distributed order fractional operators, Nonlinear Dynam., 29 (2002), pp. 5798.Google Scholar
[17]Luchko, Y., Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl., 351 (2009), pp. 218223.Google Scholar
[18]Luchko, Y., Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Computers and Mathematics with Applications, 59 (2010), pp. 17661772.Google Scholar
[19]Luchko, Y., Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl., 374 (2011), pp. 538548.Google Scholar
[20]Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.Google Scholar
[21]Meerschaert, M. M., Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, Journal of Computional and Applied Mathematics, 172 (2004), pp. 6577.Google Scholar
[22]Pedro, H. T. C., Kobayashi, M. H., Pereira, J. M. C., Coimbra, C. F. M., Variable order modelling of diffusive-convective effects on the oscillatory flow past a sphere, J. Vib. Control, 14 (2008), pp. 16591672.Google Scholar
[23]Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.Google Scholar
[24]Sakamoto, K., Yamamoto, M., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), pp. 426447.Google Scholar
[25]Zhou, L., Selim, H. M., Application of the fractional advection-dispersion equations in porous media, Soil. Sci. Soc. Am. J., 67 (2003), pp. 10791084.Google Scholar