June  2020, 9(2): 561-579. doi: 10.3934/eect.2020024

On a backward problem for two-dimensional time fractional wave equation with discrete random data

1. 

Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam

2. 

Department of Mathematics and Computer Science, University of Science, VNU-HCM, Ho Chi Minh City, Vietnam

3. 

Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

4. 

Faculty of Information Technology, Macau University of Science and Technology, Macau 999078, China

5. 

Faculty of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China

*Corresponding author: Tran Ngoc Thach

Received  February 2019 Revised  May 2019 Published  December 2019

This paper is concerned with a backward problem for a two- dimensional time fractional wave equation with discrete noise. In general, this problem is ill-posed, therefore the trigonometric method in nonparametric regression associated with Fourier truncation method is proposed to solve the problem. We also give some error estimates and convergence rates between the regularized solution and the sought solution under some assumptions.

Citation: Nguyen Huy Tuan, Tran Ngoc Thach, Yong Zhou. On a backward problem for two-dimensional time fractional wave equation with discrete random data. Evolution Equations & Control Theory, 2020, 9 (2) : 561-579. doi: 10.3934/eect.2020024
References:
[1]

P. AgarwalJ. J. Nieto and M. J. Luo, Extended Riemann-Liouville type fractional derivative operator with applications, Open Math., 15 (2017), 1667-1681.  doi: 10.1515/math-2017-0137.  Google Scholar

[2]

A. AngurajS. Kanjanadevi and J. J. Nieto, Mild solutions of Riemann-Liouville fractional differential equations with fractional impulses, Nonlinear Anal. Model. Control, 22 (2017), 753-764.   Google Scholar

[3] A. Atangana, Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology, Academic Press, London, 2018.   Google Scholar
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N. Bissantz and H. Holzmann, Statistical inference for inverse problems, Inverse Problems, 24 (2008), 17 pp. doi: 10.1088/0266-5611/24/3/034009.  Google Scholar

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L. Boyadjiev and Y. Luchko, Multi-dimensional a-fractional diffusion-wave equation and some properties of its fundamental solution, Comput. Math. Appl., 73 (2017), 2561-2572.  doi: 10.1016/j.camwa.2017.03.020.  Google Scholar

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L. Cavalier, Nonparametric statistical inverse problems, Inverse Problems, 24 (2008), 19 pp. doi: 10.1088/0266-5611/24/3/034004.  Google Scholar

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K. Diethelm, The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, 2004. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar

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X. L. Ding and J. J. Nieto, Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms, Fract. Calc. Appl. Anal., 21 (2018), 312-335.  doi: 10.1515/fca-2018-0019.  Google Scholar

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W. FanF. LiuX. Jiang and I. Turner, A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a twodimensional irregular convex domain, Fract. Calc. Appl. Anal., 20 (2017), 352-383.  doi: 10.1515/fca-2017-0019.  Google Scholar

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S. GuoL. Mei and Y. Li, An efficient Galerkin spectral method for two-dimensional fractional nonlinear reaction-diffusion-wave equation, Comput. Math. Appl., 74 (2017), 2449-2465.  doi: 10.1016/j.camwa.2017.07.022.  Google Scholar

[12]

S. Holm and S. Peter Nasholm, Comparison of Fractional Wave Equations for Power Law Attenuation in Ultrasound and Elastography, Ultrasound in Medicine and Biology, 40, (2014), 695–703. doi: 10.1016/j.ultrasmedbio.2013.09.033.  Google Scholar

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C. KönigF. Werner and T. Hohage, Convergence rates for exponentially ill-posed inverse problems with impulsive noise, SIAM J. Numer. Anal., 54 (2016), 341-360.  doi: 10.1137/15M1022252.  Google Scholar

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D. KumarJ. Singh and D. Baleanu, A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves, Math. Methods Appl. Sci., 40 (2017), 5642-5653.  doi: 10.1002/mma.4414.  Google Scholar

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Z. G. LiuA. J. Cheng and X. L. Li, A novel finite difference discrete scheme for the time fractional diffusion-wave equation, Appl. Numer. Math., 134 (2018), 17-30.  doi: 10.1016/j.apnum.2018.07.001.  Google Scholar

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F. Mainardi, Fractional Calculus in Wave Propagation Problems, Forum der Berliner Mathematischer Gesellschaft, 2011, arXiv: 1202.0261. Google Scholar

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B. Mair and F. H. Ruymgaart, Statistical estimation in Hilbert scale, SIAM J. Appl. Math., 56 (1996), 1424-1444.  doi: 10.1137/S0036139994264476.  Google Scholar

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J. Masoliver, Fractional telegrapher's equation from fractional persistent random walks, Phys. Rev. E, 93 (2016), 10 pp. doi: 10.1103/physreve.93.052107.  Google Scholar

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J. Masoliverdag and G. H. Weiss, Finite-velocity diffusion, Eur. J. Phys, 17 1996,190 pp. Google Scholar

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[22]

Z. S. RuanS. Zhang and S. C. Xiong, Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method, Evol. Equ. Control Theory, 7 (2018), 669-682.  doi: 10.3934/eect.2018032.  Google Scholar

[23]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[24]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[25]

T. Sandev, Z. Tomovski, L. A. J. Dubbeldam and A. Chechkin, Generalized diffusion-wave equation with memory kernel, J. Phys. A, 52 (2019), 22 pp.  Google Scholar

[26]

P. StrakaM. M. MeerschaertJ. R. McGough and Y. Zhou, Fractional wave equations with attenuation, Fract. Calc. Appl. Anal., 16 (2013), 262-272.  doi: 10.2478/s13540-013-0016-9.  Google Scholar

[27]

B. E. Treeby and B. T. Cox, Modeling power law absorption and dispersion in viscoelastic solids using a split-field and the fractional Laplacian, Acoustical Society of America, 127 (2014), 2741-2748.  doi: 10.1121/1.4894790.  Google Scholar

[28]

A. B. Tsybakov, Introduction to Nonparametric Estimation, Revised and extended from the 2004 French original. Translated by Vladimir Zaiats. Springer Series in Statistics. Springer, New York, 2009. doi: 10.1007/b13794.  Google Scholar

[29]

T. Wei and J. G. Wang, A modified quasi-boundary value method for the backward time-fractional diffusion problem, ESAIM Math. Model. Numer. Anal., 48 (2014), 603-621.  doi: 10.1051/m2an/2013107.  Google Scholar

[30]

T. Wei and Y. G. Zhang, The backward problem for a time-fractional diffusion-wave equation in a bounded domain, Comput. Math. Appl., 75 (2018), 3632-3648.  doi: 10.1016/j.camwa.2018.02.022.  Google Scholar

[31]

Y. ZhangM. M. Meerschaert and R. M. Neupauer, Backward fractional advection dispersion model for cantaminant source prediction, Water Resources Research, 52 (2016), 2462-2473.   Google Scholar

[32]

G. H. Zheng and T. Wei, Two regularization methods for solving a Riesz-Feller space-fractional backward diffusion problem, Inverse Problems, 26 (2010), 22 pp. doi: 10.1088/0266-5611/26/11/115017.  Google Scholar

show all references

References:
[1]

P. AgarwalJ. J. Nieto and M. J. Luo, Extended Riemann-Liouville type fractional derivative operator with applications, Open Math., 15 (2017), 1667-1681.  doi: 10.1515/math-2017-0137.  Google Scholar

[2]

A. AngurajS. Kanjanadevi and J. J. Nieto, Mild solutions of Riemann-Liouville fractional differential equations with fractional impulses, Nonlinear Anal. Model. Control, 22 (2017), 753-764.   Google Scholar

[3] A. Atangana, Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology, Academic Press, London, 2018.   Google Scholar
[4]

N. Bissantz and H. Holzmann, Statistical inference for inverse problems, Inverse Problems, 24 (2008), 17 pp. doi: 10.1088/0266-5611/24/3/034009.  Google Scholar

[5]

L. Boyadjiev and Y. Luchko, Multi-dimensional a-fractional diffusion-wave equation and some properties of its fundamental solution, Comput. Math. Appl., 73 (2017), 2561-2572.  doi: 10.1016/j.camwa.2017.03.020.  Google Scholar

[6]

L. Cavalier, Nonparametric statistical inverse problems, Inverse Problems, 24 (2008), 19 pp. doi: 10.1088/0266-5611/24/3/034004.  Google Scholar

[7]

K. Diethelm, The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, 2004. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar

[8]

X. L. Ding and J. J. Nieto, Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms, Fract. Calc. Appl. Anal., 21 (2018), 312-335.  doi: 10.1515/fca-2018-0019.  Google Scholar

[9]

R. L. Eubank, Nonparametric Regression and Spline Smoothing, Second edition. Statistics: Textbooks and Monographs, 157. Marcel Dekker, Inc., New York, 1999.  Google Scholar

[10]

W. FanF. LiuX. Jiang and I. Turner, A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a twodimensional irregular convex domain, Fract. Calc. Appl. Anal., 20 (2017), 352-383.  doi: 10.1515/fca-2017-0019.  Google Scholar

[11]

S. GuoL. Mei and Y. Li, An efficient Galerkin spectral method for two-dimensional fractional nonlinear reaction-diffusion-wave equation, Comput. Math. Appl., 74 (2017), 2449-2465.  doi: 10.1016/j.camwa.2017.07.022.  Google Scholar

[12]

S. Holm and S. Peter Nasholm, Comparison of Fractional Wave Equations for Power Law Attenuation in Ultrasound and Elastography, Ultrasound in Medicine and Biology, 40, (2014), 695–703. doi: 10.1016/j.ultrasmedbio.2013.09.033.  Google Scholar

[13]

C. KönigF. Werner and T. Hohage, Convergence rates for exponentially ill-posed inverse problems with impulsive noise, SIAM J. Numer. Anal., 54 (2016), 341-360.  doi: 10.1137/15M1022252.  Google Scholar

[14]

D. KumarJ. Singh and D. Baleanu, A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves, Math. Methods Appl. Sci., 40 (2017), 5642-5653.  doi: 10.1002/mma.4414.  Google Scholar

[15]

P. D. Lax, Functional Analysis, Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002.  Google Scholar

[16]

Z. G. LiuA. J. Cheng and X. L. Li, A novel finite difference discrete scheme for the time fractional diffusion-wave equation, Appl. Numer. Math., 134 (2018), 17-30.  doi: 10.1016/j.apnum.2018.07.001.  Google Scholar

[17]

F. Mainardi, Fractional Calculus in Wave Propagation Problems, Forum der Berliner Mathematischer Gesellschaft, 2011, arXiv: 1202.0261. Google Scholar

[18]

B. Mair and F. H. Ruymgaart, Statistical estimation in Hilbert scale, SIAM J. Appl. Math., 56 (1996), 1424-1444.  doi: 10.1137/S0036139994264476.  Google Scholar

[19]

J. Masoliver, Fractional telegrapher's equation from fractional persistent random walks, Phys. Rev. E, 93 (2016), 10 pp. doi: 10.1103/physreve.93.052107.  Google Scholar

[20]

J. Masoliverdag and G. H. Weiss, Finite-velocity diffusion, Eur. J. Phys, 17 1996,190 pp. Google Scholar

[21] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.   Google Scholar
[22]

Z. S. RuanS. Zhang and S. C. Xiong, Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method, Evol. Equ. Control Theory, 7 (2018), 669-682.  doi: 10.3934/eect.2018032.  Google Scholar

[23]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[24]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[25]

T. Sandev, Z. Tomovski, L. A. J. Dubbeldam and A. Chechkin, Generalized diffusion-wave equation with memory kernel, J. Phys. A, 52 (2019), 22 pp.  Google Scholar

[26]

P. StrakaM. M. MeerschaertJ. R. McGough and Y. Zhou, Fractional wave equations with attenuation, Fract. Calc. Appl. Anal., 16 (2013), 262-272.  doi: 10.2478/s13540-013-0016-9.  Google Scholar

[27]

B. E. Treeby and B. T. Cox, Modeling power law absorption and dispersion in viscoelastic solids using a split-field and the fractional Laplacian, Acoustical Society of America, 127 (2014), 2741-2748.  doi: 10.1121/1.4894790.  Google Scholar

[28]

A. B. Tsybakov, Introduction to Nonparametric Estimation, Revised and extended from the 2004 French original. Translated by Vladimir Zaiats. Springer Series in Statistics. Springer, New York, 2009. doi: 10.1007/b13794.  Google Scholar

[29]

T. Wei and J. G. Wang, A modified quasi-boundary value method for the backward time-fractional diffusion problem, ESAIM Math. Model. Numer. Anal., 48 (2014), 603-621.  doi: 10.1051/m2an/2013107.  Google Scholar

[30]

T. Wei and Y. G. Zhang, The backward problem for a time-fractional diffusion-wave equation in a bounded domain, Comput. Math. Appl., 75 (2018), 3632-3648.  doi: 10.1016/j.camwa.2018.02.022.  Google Scholar

[31]

Y. ZhangM. M. Meerschaert and R. M. Neupauer, Backward fractional advection dispersion model for cantaminant source prediction, Water Resources Research, 52 (2016), 2462-2473.   Google Scholar

[32]

G. H. Zheng and T. Wei, Two regularization methods for solving a Riesz-Feller space-fractional backward diffusion problem, Inverse Problems, 26 (2010), 22 pp. doi: 10.1088/0266-5611/26/11/115017.  Google Scholar

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