A Priori Strategy for Pseudo-Parabolic Equations by Hybrid Regularization Method

Ngo Ngoc Hung; Dinh Long Le

doi:10.61383/ejam.202533103

Authors

Ngo Ngoc Hung Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam https://orcid.org/0000-0002-4380-0257
Le Dinh Long Faculty of Information Technology, Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam https://orcid.org/0000-0001-8805-4588

DOI:

https://doi.org/10.61383/ejam.202533103

Keywords:

time-fractional diffusion equation, inverse source term, initial value problem, hybrid regularization method, priori estimation

Abstract

In this study, we address the simultaneous recovery of the source term and the initial value in a time-fractional diffusion equation, a problem inherently characterized by its ill-posed nature. To overcome this challenge, we propose a hybrid regularization method, offering robust solutions and providing precise estimations for both the source term and the initial value.

References

[1] V T T Ha, Reconstruct an unknown source term on the right hand side of the time-fractional diffusion equation with Caputo-Hadamard derivative, Advances in the Theory of Nonlinear Analysis and Its Application, 7(5), (2023) 28-43, 10.17762/atnaa.v7.i5.324.

[2] G Yazid, R Mahdi, D Zoubir, A Sequential Differential Problem With Caputo and Riemann Liouville Derivatives Involving Convergent Series, Advances in the Theory of Nonlinear Analysis and its Applications 7 (2023) No. 2, 319--335.

[3] N H Tuan, N M Hai, N D Phuong, On the nonlinear Volterra equation with conformable derivative, Advances in the Theory of Nonlinear Analysis and its Applications 7 (2023) No. 2, 292-302.

[4] Y Q Liang, F Yang, X X Li, A hybrid regularization method for identifying the source term and the initial value simultaneously for fractional pseudo-parabolic equation with involution, Numerical Algorithms, (2024) 1-33.

[5] F F Dou, Y C Hon, Fundamental kernel-based method for backward space–time fractional diffusion problem, Comput Math Appl; 71(1), (2016), 356–67.

[6] X L Feng, X Y Yuan, M X Zhao, Numerical methods for the forward and backward problems of a time-space fractional diffusion equation, Calcolo 61(16), 2024.

[7] J L Wang, X T Xiong, X X Cao, Fractional Tikhonov regularization method for a time-fractional backward heat equation with a fractional Laplacian, J Partial Differ Equ, 31(4), (2018) 333–42.

[8] J X Jia, J G Peng, J H Gao, Backward problem for a time-space fractional diffusion equation, Inverse Probl Imaging, 12(3), (2018) 773–99.

[9] F Yang, Q Pu, X X Li, The fractional landweber method for identifying the space source term problem for time-space fractional diffusion equation, Numer Algorithms 2021;87(3):1229–55.

[10] S Z Jiang, Y J Wu, Recovering space-dependent source for a time-space fractional diffusion wave equation by fractional Landweber method, Inverse Probl Sci Eng 2021; 29(5/8):990–1011.

[11] Y S Li, T Wei, An inverse time-dependent source problem for a time-space fractional diffusion equation, Appl Math Comput 2018; 336:257–71.

[12] J G Wang, T Wei, Y B Zhou, Tikhonov regularization method for a backward problem for the time-fractional diffusion equation, Applied Mathematical Modelling, (2013) 37(18–19):8518–8532.

[13] Y K Liu, W Rundell, M Yamamoto, Strong maximum principle for fractional diffusion equations and an application to an inverse source problem, Fractional Calculus and Applied Analysis, (2016) 19(4):888–906.

[14] M I Ismailov, M Cicek, Inverse source problem for a time-fractional diffusion equation with nonlocal boundary conditions, Applied Mathematical Modelling, 40(7–8), (2016):4891–4899.

[15] Z Q Zhang, T Wei, Identifying an unknown source in time-fractional diffusion equation by a truncation method, Applied Mathematics and Computation, 219(11), (2013), 5972–5983.

[16] X T Xiong, H B Guo, X H Li, An inverse problem for a fractional diffusion equation, Journal of Computational and Applied Mathematics, 236(17):4474–4484, 2012.

[17] J J Liu, M Yamamoto, A backward problem for the time-fractional diffusion equation, Applicable Analysis, 89(11):1769–1788, 2010. https://doi.org/10.1080/00036810903479731.

[18] J G Wang, T Wei, Y B Zhou, Optimal error bound and simplified Tikhonov regularization method for a backward problem for the time-fractional diffusion equation, Journal of Computational and Applied Mathematics, 279(18–19):277–292, 2015.

[19] T Wei, J G Wang, A modified quasi-boundary value method for the backward time-fractional diffusion problem, ESAIM: Mathematical Modelling and Numerical Analysis, 48(2):603–621, 2014.

[20] M Yang, J J Liu, Solving a final value fractional diffusion problem by boundary condition regularization, Applied Numerical Mathematics, 66(1):45–58, 2013.

[21] J J Liu, M Yamamoto, A backward problem for the time-fractional diffusion equation, Appl Anal, 89(11):1769–88, 2010.

[22] T Wei, X L Li, Y S Li, An inverse time-dependent source problem for a time-fractional diffusion equation, Inverse Problems, 32(8) (2016):085003, DOI: 10.1088/0266-5611/32/8/085003.

[23] H Cheng, J Gao, P Zhu, Optimal results for a time-fractional inverse diffusion problem under the Hölder type source condition, Bull Iranian Math Soc, 41(4) (2015):825–34.

[24] Y J Jiang, J T Ma, High-order finite element methods for time-fractional partial differential equations, J Comput Appl Math, 235(11) (2011):3285–90.

[25] Z Y Li, Y Luchko, M Yamamoto, Analyticity of solutions to a distributed order time-fractional diffusion equation and its application to an inverse problem, Comput Math Appl; 73(6) (2017):1041–52.

[26] H Chen, S J Lu, W P Chen, Finite difference/spectral approximations for the distributed order time fractional reaction–diffusion equation on an unbounded domain, J Comput Phys 315 (2016):84–97.