Regularization of nonlocal pseudo-parabolic equation with random noise

Authors

DOI:

https://doi.org/10.61383/ejam.20231119

Keywords:

Fractional Tikhonov, Regularization, Conformable time derivative, Discrete data, random noise

Abstract

In this paper, we consider an inverse problem for a time-fractional diffusion equation with the inhomogeneous source. These problems have many applications in engineering such as image processing, geophysics, biology. We get the result in random case as follows:

• This problem is ill-posed.

• We have used the nonlocal condition, instead of the final time condition.

• Using the IFT regularization method, constructing the regularized solution, the a-priori choice rule for the regularization parameter is discussed and yields the corresponding convergence rate.

• A numerical experiment is presented to illustrate the results in theory.

References

[1] R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), pp 161-208.

[2] R. Gorenflo, Mainardi, F., Scalas, E., Raberto, M., Fractional calculus and continuous-time finance III, in: The Diffusion Limit, Mathematical Finance, Springer-Verlag, New York, (2001) pp. 171-180.

[3] F. Mainardi, M. Raberto, R. Gorenflo, E. Scalas, Fractional calculus and continuous-time finance II: the waiting time distribution, Phys. A 287 (2000), pp. 468-481.

[4] E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance, Phys. A 284 (2000),pp. 376-384.

[5] L. Sabatelli, S. Keating, J. Dudley, P. Richmond, Waiting time distributions in financial markets, Eur. Phys. J. B. 27 (2002),pp. 273-275.

[6] S.B. Yuste, K. Lindenberg, Subdiffusion-limited reactions, Chem. Phys. 284 (2002), pp. 169-180.

[7] M.G. Hall, T.R. Barrick, From diffusion-weighted MRI to anomalous diffusion imaging, Magn. Reson. Med., 59, (2008), pp. 447-455.

[8] S.B. Yuste, L. Acedo, K. Lindenberg, Reaction front in anA + B→C reaction-subdiffusion process, Phys. Rev. E. 69 (2004) 036126.

[9] B. Berkowitz, H. Scher, S.E. Silliman, Anomalous transport in laboratory-scale, heterogeneous porous media, Water Resour. Res. 36 (2000), pp. 149-158.

[10] I.M. Sokolov, J. Klafter, From diffusion to anomalous diffusion: a century after Einsteins Brownian motion, Chaos 15 (2005), pp. 1-7

[11] N.H. Tuan, D. Baleanu, T.N. Thach, D. O’Regan, N.H. Can, Final value problem for nonlinear time fractional reaction-diffusion equation with discrete data. J. Comput. Appl. Math. 376 (2020), 112883, 25 pp.

[12] N.H. Tuan, D. Baleanu, T.N. Thach, D. O’Regan, & N.H. Can, Final value problem for nonlinear time fractional reaction-diffusion equation with discrete data. Journal of Computational and Applied Mathematics, 376, (2020) 112883.

[13] N.H. Tuan, T. Caraballo, On initial and terminal value problems for fractional non classical diffusion equations, Proceedings of the American Mathematical Society, 149(1), (2021), pp. 143-161.

[14] N.H. Tuan, V.V. Au, A.T. Nguyen, Mild solutions to a time-fractional Cauchy problem with nonlocal nonlinearity in Besov spaces. Arch. Math. (Basel) 118 (2022), no. 3, pp. 305-314.

[15] N.H. Tuan, M. Foondun, T.N. Thach, R. Wang, On backward problems for stochastic fractional reaction equations with standard and fractional Brownian motion. Bull. Sci. Math. 179, (2022), Paper No. 103158, 58 pp

[16] L. Cavalier, Nonparametric statistical inverse problems, Inverse Probl. 24 (3) (2008) 034004.

[17] N.H. Tuan, M. Kirane, B. Bin-Mohsin, Filter regularization for final value fractional diffusion problem with deterministic and random noise, Comput. Math. Appl. 74 (6) (2017),pp. 1340-1361.

[18] N.H. Tuan, T.N. Thach, Y. Zhou, On a backward problem for two-dimensional time fractional wave equation with discrete random data, Evol. Equ. Control Theory, 9(2), (2020), 561..

[19] T.N. Thach, N.H. Tuan, P. T. M. Tam, M.N. Minh, & N.H. Can, (2019). Identification of an inverse source problem for time-fractional diffusion equation with random noise, Mathematical Methods in the Applied Sciences, 42(1), (2019), pp. 204-218.

[20] N.H. Tuan, D. Baleanu, T.N. Thach, D. O’Regan, & N.H. Can, Final value problem for nonlinear time fractional reaction-diffusion equation with discrete data, Journal of Computational and Applied Mathematics, 376, (2020), 112883.

[21] N.H. Can, Y. Zhou, N.H. Tuan, T.N. Thach, Regularized solution approximation of a fractional pseudo-parabolic problem with a nonlinear source term and random data. Chaos Solitons Fractals 136 (2020), 109847, 14 pp.

[22] F.S. Ze, F.D. Hua, Instability analysis and regularization approximation to the forward/backward problems for fractional damped wave equations with random noise, Applied Numerical Mathematics, Available online 4 January 2023

[23] F.D. Hua, J.R. Wei, The Regularized Solution Approximation of Forward/Backward Problems for a Fractional Pseudo-Parabolic Equation with Random Noise, Acta Mathematica Scientia, Volume 43, (2023) pp. 324-348 (2023),

[24] H.T. Nguyen, D.L. Le, V.T. Nguyen, Regularized solution of an inverse source problem for a time fractional diffusion equation. Appl. Math. Model. (2016), 40, pp. 8244-8264.

[25] N.H. Tuan, Y. Zhou, L.D. Long, N.H Can, Identifying inverse source for fractional diffusion equation with Reimann-Liouville derivetive. Comput. Appl. Math. (2020), 39, 75.

[26] L.D. Long, N.H.Luc,Y. Zhou, H.C.Nguyen, Identification of Source term for the time-fractional duffusion-wave equation by Fractional Tikhonov method, Mathematics, (2019), 7, 934.

[27] T. Wei, X.L. Li, Y.S. Li, An inverse time-dependent source problem for a time-fractional diffusion equation. Inverse Probl. (2016), 32, 8.

[28] J.G. Wang, Y.B. Zhou, T. Wei, Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation. Appl. Numer. Math., (2013), 68, pp. 39-57.

[29] D. Bianchi, A. Buccini, M. Donatelli, & S. Serra-Capizzano, S, Iterated fractional Tikhonov regularization, Inverse Problems, 31(5),(2015) 055005.

[30] Y. Shuping, T.X. Xiang, N. Yan, Iterated fractional Tikhonov regularization method for solving the spherically symmetric backward time-fractional diffusion equation, Applied Numerical Mathematics 160 (2021), pp. 217-241.

[31] I. Podlubny, Fractional Differential Equations, Academic Press, London, 1999.

[32] K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (1) (2011), pp. 426-447.

[33] D. Bianchi, A. Buccini, M. Donatelli, Iterated fractional Tikhonov regularization, Inverse Probl. 31 (5) (2015) 055005.

[34] D. Gerth, E. Klann, R. Ramlau, On fractional Tikhonov regularization, J. Inverse Ill-Posed Probl. , 23 (6) (2015), pp 611-625.

[35] I. Podlubny, M. Kacenak, Mittag-Leffler function, The MATLAB routine, available at http://www.mathworks.com/matlabcentral/fileexchange, 2012.

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Published

2023 May 08

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Section

Research Article

How to Cite

[1]
“Regularization of nonlocal pseudo-parabolic equation with random noise”, Electron. J. Appl. Math., vol. 1, no. 1, pp. 40–61, May 2023, doi: 10.61383/ejam.20231119.

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