# American Institute of Mathematical Sciences

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  June 2020 Early access  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 and Control Theory, 2020, 9 (2) : 561-579. doi: 10.3934/eect.2020024
##### References:
 [1] P. Agarwal, J. 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. [2] A. Anguraj, S. Kanjanadevi and J. J. Nieto, Mild solutions of Riemann-Liouville fractional differential equations with fractional impulses, Nonlinear Anal. Model. Control, 22 (2017), 753-764. [3] A. Atangana, Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology, Academic Press, London, 2018. [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. [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. [6] L. 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Anal., 20 (2017), 352-383.  doi: 10.1515/fca-2017-0019. [11] S. Guo, L. 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. [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. [13] C. König, F. 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. [14] D. Kumar, J. 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. [15] P. D. Lax, Functional Analysis, Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002. [16] Z. G. Liu, A. 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. [17] F. Mainardi, Fractional Calculus in Wave Propagation Problems, Forum der Berliner Mathematischer Gesellschaft, 2011, arXiv: 1202.0261. [18] B. Mair and F. H. Ruymgaart, Statistical estimation in Hilbert scale, SIAM J. Appl. Math., 56 (1996), 1424-1444.  doi: 10.1137/S0036139994264476. [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. [20] J. Masoliverdag and G. H. Weiss, Finite-velocity diffusion, Eur. J. Phys, 17 1996,190 pp. [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. [22] Z. S. Ruan, S. 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. [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. [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. [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. [26] P. Straka, M. M. Meerschaert, J. 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. [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. [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. [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. [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. [31] Y. Zhang, M. M. Meerschaert and R. M. Neupauer, Backward fractional advection dispersion model for cantaminant source prediction, Water Resources Research, 52 (2016), 2462-2473. [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.

show all references

##### References:
 [1] P. Agarwal, J. 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. [2] A. Anguraj, S. Kanjanadevi and J. J. Nieto, Mild solutions of Riemann-Liouville fractional differential equations with fractional impulses, Nonlinear Anal. Model. Control, 22 (2017), 753-764. [3] A. Atangana, Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology, Academic Press, London, 2018. [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. [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. [6] L. Cavalier, Nonparametric statistical inverse problems, Inverse Problems, 24 (2008), 19 pp. doi: 10.1088/0266-5611/24/3/034004. [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. [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. [9] R. L. Eubank, Nonparametric Regression and Spline Smoothing, Second edition. Statistics: Textbooks and Monographs, 157. Marcel Dekker, Inc., New York, 1999. [10] W. Fan, F. Liu, X. 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. [11] S. Guo, L. 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. [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. [13] C. König, F. 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. [14] D. Kumar, J. 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. [15] P. D. Lax, Functional Analysis, Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002. [16] Z. G. Liu, A. 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. [17] F. Mainardi, Fractional Calculus in Wave Propagation Problems, Forum der Berliner Mathematischer Gesellschaft, 2011, arXiv: 1202.0261. [18] B. Mair and F. H. Ruymgaart, Statistical estimation in Hilbert scale, SIAM J. Appl. Math., 56 (1996), 1424-1444.  doi: 10.1137/S0036139994264476. [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. [20] J. Masoliverdag and G. H. Weiss, Finite-velocity diffusion, Eur. J. Phys, 17 1996,190 pp. [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. [22] Z. S. Ruan, S. 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. [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. [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. [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. [26] P. Straka, M. M. Meerschaert, J. 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. [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. [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. [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. [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. [31] Y. Zhang, M. M. Meerschaert and R. M. Neupauer, Backward fractional advection dispersion model for cantaminant source prediction, Water Resources Research, 52 (2016), 2462-2473. [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.
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