Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T16:32:10.819Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Inversion for the Initial Distribution in the Multi-Term Time-Fractional Diffusion Equation Using Final Observations

Part of: Miscellaneous topics - Partial differential equations Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  28 November 2017

Chunlong Sun ,
Gongsheng Li  and
Xianzheng Jia
Chunlong Sun
Affiliation:
School of Science, Shandong University of Technology, Zibo, Shandong 255049, China School of Mathematics, Southeast University, Nanjing, Jiangsu 210096, China
Gongsheng Li*
Affiliation:
School of Science, Shandong University of Technology, Zibo, Shandong 255049, China
Xianzheng Jia
Affiliation:
School of Science, Shandong University of Technology, Zibo, Shandong 255049, China
*
*Corresponding author. Email:ligs@sdut.edu.cn (G. S. Li)
Get access

Abstract

This article deals with numerical inversion for the initial distribution in the multi-term time-fractional diffusion equation using final observations. The inversion problem is of instability, but it is uniquely solvable based on the solution's expression for the forward problem and estimation to the multivariate Mittag-Leffler function. From view point of optimality, solving the inversion problem is transformed to minimizing a cost functional, and existence of a minimum is proved by the weakly lower semi-continuity of the functional. Furthermore, the homotopy regularization algorithm is introduced based on the minimization problem to perform numerical inversions, and the inversion solutions with noisy data give good approximations to the exact initial distribution demonstrating the efficiency of the inversion algorithm.

Keywords

Multi-term time-fractional diffusionmultivariate Mittag-Leffler functionbackward problemill-posednessnumerical inversion

MSC classification

Secondary: 35R11: Fractional partial differential equations 35R30: Inverse problems 65M06: Finite difference methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 6 , December 2017 , pp. 1525 - 1546
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Adams, E. E. and Gelhar, L. W., Field study of dispersion in a heterogeneous aquifer 2: Spatial moments analysis, Water Resources Research, 28 (1992), pp. 3293–3307.Google Scholar
[2] Bazhlekova, E., Properties of the fundamental and the impulse-response solutions of multi-term fractional differential equations, in: Kiryakova, V. (Eds.), Complex Analysis and Applications’ 13, Bulg. Acad. Sci. Sofia, (2013), pp. 55–64.Google Scholar
[3] Benson, D. A., The Fractional Advection-Dispersion Equation: Development and Application, University of Nevada, Reno, 1998.Google Scholar
[4] Berkowitz, B., Scher, H. and Silliman, S. E., Anomalous transport in laboratory-scale heterogeneous porous media, Water Resources Research, 36 (2000), pp. 149–158.CrossRefGoogle Scholar
[5] Caponetto, R., Dongola, G., Fortuna, L. and Petras, I., Fractional Order Systems: Modeling and Control Applications, World Scientific, Singapore, 2010.Google Scholar
[6] Cheng, J., Nakagawa, J., Yamamoto, M. and Yamazaki, T., Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Problems, 25 (2009), 115002.Google Scholar
[7] Courant, R. and Hilbert, D., Methods of Mathematical Physics, Interscience Publishers, New York, 1989.Google Scholar
[8] Daftardar-Gejji, V. and Bhalekar, S., Boundary value problems for multi-term fractional differential equations, J. Math. Anal. Appl., 345 (2008), pp. 754–765.Google Scholar
[9] Hatano, Y. and Hatano, N., Dispersive transport of ions in column experiments: an explanation of long-tailed profiles, Water Resources Research, 34 (1998), pp. 1027–1033.CrossRefGoogle Scholar
[10] Jia, X. Z., Li, G. S., Sun, C. L. and Du, D. H., Simultaneous inversion for a diffusion coefficient and a spatially dependent source term in the SFADE, Inverse Problems Sci. Eng., 24 (2016), pp. 832–859.Google Scholar
[11] Jiang, H., Liu, F., Turner, I. and Burrage, K., Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain, Comput. Math. Appl., 64 (2012), pp. 3377–3388.Google Scholar
[12] Jin, B. T., Lazarov, R., Liu, Y. K. and Zhou, Z., The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comput. Phys., 281 (2015), pp. 825–843.Google Scholar
[13] Jin, B. T. and Rundell, W., An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Problems, 28 (2012), 075010.Google Scholar
[14] Jin, B. T. and Rundell, W., A tutorial on inverse problems for anomalous diffusion processes, Inverse Problems, 31 (2015), 035003.Google Scholar
[15] Keisler, H. J., Elementary Calculus, An Infinitesimal Approach, Prindle, Weber & Schmidt, Boston & Massachusetts, 1986.Google Scholar
[16] Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.Google Scholar
[17] Kirsch, A., An Introduction to Mathematical Theory of Inverse Problems, Springer, New York, 1996.Google Scholar
[18] Li, G. S., Sun, C. L., Jia, X. Z. and Du, D. H., Numerical solution to the multi-term time fractional diffusion equation in a finite domain, Numer. Math. Theory Method Appl., 9 (2016), pp. 337–357.CrossRefGoogle Scholar
[19] Li, G. S., Zhang, D. L., Jia, X. Z. and Yamamoto, M., Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation, Inverse Problems, 29 (2013), 065014.Google Scholar
[20] Li, K. T. and Ma, Y. C., Hilbert Space Methods for Mathematics-Physics Equations: Part 2 (in Chinese), Xi’an, Xi’an Jiaotong University Press, 1992.Google Scholar
[21] Li, Z. Y., Imanuvilov, O. Y. and Yamamoto, M., Uniqueness in inverse boundary value problems for fractional diffusion equations, Inverse Problems, 32 (2016), 015004.CrossRefGoogle Scholar
[22] Li, Z. Y., Liu, Y. K. and Yamamoto, M., Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients, Appl. Math. Comput., 257 (2015), pp. 381–397.Google Scholar
[23] Li, Z. Y. and Yamamoto, M., Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation, Appl. Anal., 94 (2015), pp. 570–579.Google Scholar
[24] Liu, F., Meerschaert, M. M. and McGough, R. J. et al., Numerical methods for solving the multi-term time-fractional wave-diffusion equations, Fractional Calculus Appl. Anal., 16 (2013), pp. 9–25.CrossRefGoogle Scholar
[25] Liu, J. J. and Yamamoto, M., A backward problem for the time-fractional diffusion equation, Appl. Anal., 89 (2010), pp. 1769–1788.Google Scholar
[26] Liu, J. J., Yamamoto, M. and Yan, L., On the reconstruction of unknown time dependent boundary sources for time fractional diffusion process by distributing measurement, Inverse Problems, 32 (2016), 015009.CrossRefGoogle Scholar
[27] Luchko, Y., Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl., 351 (2009), pp. 218–223.Google Scholar
[28] Luchko, Y., Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Comput. Math. Appl., 59 (2010), pp. 1766–1772.Google Scholar
[29] Luchko, Y., Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl., 374 (2011), pp. 538–548.Google Scholar
[30] Luchko, Y., Rundell, W., Yamamoto, M. and Zuo, L. H., Uniqueness and reconstruction of an unknown semilinear term in a time-fractional reaction-diffusion equation, Inverse Problems, 29 (2013), 065019.Google Scholar
[31] Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.Google Scholar
[32] Miller, L. and Yamamoto, M., Coefficient inverse problem for a fractional diffusion equation, Inverse Problems, 29 (2013), 075013.Google Scholar
[33] Murio, D. A., Stable numerical solution of fractional-diffusion inverse heat conduction problem, Comput. Math. Appl., 53 (2007), pp. 1492–501.Google Scholar
[34] Podlubny, I., Fractional Differential Equations, Academic, San Diego, 1999.Google Scholar
[35] Sakamoto, K. and 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. 426–447.Google Scholar
[36] Tuan, V. K., Inverse problem for fractional diffusion equation, Fract. Calc. Appl. Anal., 14 (2011), pp. 31–55.CrossRefGoogle Scholar
[37] Wei, T., Li, X. L. and Li, Y. S., An inverse time-dependent source problem for a time-fractional diffusion equation, Inverse Problems, 32 (2016), 085003.Google Scholar
[38] Wei, T. and Wang, J. G., A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Appl. Numer. Math., 78 (2014), pp. 95–111.Google Scholar
[39] Wei, T. and Wang, J. G., A modified quasi-boundary value method for the backward time-fractional diffusion problem, ESAIM:M2AN, 48 (2014), pp. 603–621.CrossRefGoogle Scholar
[40] Yamamoto, M. and Zhang, Y., Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate, Inverse problems, 28 (2012), 105010.CrossRefGoogle Scholar
[41] Zhang, D. L., Li, G. S., Jia, X. Z. and Li, H. L., Simultaneous inversion for space-dependent diffusion coefficient and source magnitude in the time fractional diffusion equation, J. Math. Research, 5 (2013), pp. 65–78.Google Scholar
[42] Zhang, Z. D., An undetermined coefficient problem for a fractional diffusion equation, Inverse Problems, 32 (2016), 015011.CrossRefGoogle Scholar
[43] Zhang, Z. Q. and Wei, T., Identifying an unknown source in time-fractional diffusion equation by a truncation method, Appl. Math. Comput., 219 (2013), pp. 5972–5983.Google Scholar