The Fractional Landweber method for identifying unknown source for the fractional elliptic equations

Authors

DOI:

https://doi.org/10.61383/ejam.20242489

Keywords:

Convergence estimates, Regularization method, Ill-posed problem, Nonlocal problem

Abstract

The article addresses the inverse problem of identifying an unknown source term in a fractional elliptic equation defined in a bounded domain. The approach to solving the problem under consideration, the Landweber fractional method is used. This method involves constructing a regularization algorithm. A posteriori and a priori lapses estimates are obtained, and final data with random data is regard.

References

[1] Han, Y., Xiong, X., & Xue, X., A fractional Landweber method for solving backward time-fractional diffusion problem, Computers & Mathematics with Applications, (2019) 78(1), 81-91.

[2] R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On The Solutions Of Fractional Differential Equations Via eraghty Type Hybrid Contractions, Appl. Comput. Math., 20(2021), No 2, 313-333.

[3] E. Karapinar, H. D. Binh, N. H. Luc, N. H. Can, On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems, Advances in Difference Equations (2021) 2021:70 DOI: https://doi.org/10.1186/s13662-021-03232-z

[4] R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions, Revista de la Real Academia de Ciencias Exactas, F´ısicas y Naturales. Serie A. Matem´aticas 115 (2021), no. 3, 1-16. DOI: https://doi.org/10.1007/s13398-021-01095-3

[5] Benchohra, M., Karapınar, E., Lazreg, J. E., & Salim, A., Impulsive Fractional Differential Equations with Retardation nd Anticipation In Fractional Differential Equations: New Advancements for Generalized Fractional Derivatives, Cham: Springer Nature Switzerland, 2023, pp. 109-155. DOI: https://doi.org/10.1007/978-3-031-34877-8_5

[6] Benaissa, A., Salim, A., Benchohra, M., & Karapınar E., Functional delay random semilinear differential equations, The Journal of Analysis (2023), 31(4), 2675-2686. DOI: https://doi.org/10.1007/s41478-023-00592-5

[7] Karapınar, E., & Cvetkovi ´c, M., An inevitable note on bipolar metric spaces, AIMS Mathematics (2024), 9(2), 3320-3331. DOI: https://doi.org/10.3934/math.2024162

[8] G. Jothilakshmi, B.S. Vadivoo, Y. Almalki, A. Debbouche, A. Controllability analysis of multiple fractional order integro-differential damping systems with impulsive interpretation, Journal of Computational and Applied Mathematics, 2022, pp.114-204. DOI: https://doi.org/10.1016/j.cam.2022.114204

[9] A. Debbouche, Time-partial differential equations: Modeling and simulation, Mathematical Methods in the Applied Sciences (2021), 44, no. 15, 11767-11767. DOI: https://doi.org/10.1002/mma.7621

[10] S. Guechi, D. Rajesh, D. Amar, M. Muslim., Analysis and Optimal Control of ϕ-Hilfer Fractional Semilinear Equations Involving Nonlocal Impulsive Conditions, Symmetry (2021) 13, no. 11, 20-84. DOI: https://doi.org/10.3390/sym13112084

[11] Meziani, M. S. E., Djemoui, S., & Boussetila N., Detection of source term in an abstract fractional subdivision model by the modified quasi-boundary value method with a priori and a posterior estimate, Eurasian journal of mathematical and computer applications (2023), Volume 11, Issue 1 98 –123. DOI: https://doi.org/10.32523/2306-6172-2023-11-1-98-123

[12] R. Sassane, N. Boussetila, F. Rebbani, A. Benrabah, Iterative regularization method for an abstract ill-posed generalized elliptic equation, Asian-European Journal of Mathematics (2021), Vol. 14, No. 5 2150069, (22 pages ). DOI: https://doi.org/10.1142/S1793557121500698

[13] Nguyen Huy Tuan, Erkan Nane, Inverse source problem for time-fractional diffusion with discrete random noise, Statistics and Probability Letters, 2016,http://dx.doi.org/10.1016/j.spl.2016.09.026. DOI: https://doi.org/10.1016/j.spl.2016.09.026

[14] Han, Y., Xiong, X., & Xue, X., A fractional Landweber method for solving backward time-fractional diffusion problem, Computers & Mathematics with Applications (2019), 78(1), 81-91.. DOI: https://doi.org/10.1016/j.camwa.2019.02.017

[15] N. D. Phuong, N. H. Luc, L. D. Long, Modified Quasi Boundary Value method for inverse source problem of the bi-parabolic equation, Advances in the Theory of Nonlinear Analysis and its Applications 4 (2020), No.3, 132–142. DOI: https://doi.org/10.31197/atnaa.752335

[16] X. X. Li, J. L. Lei, & F. Yang, An a posteriori Fourier regularization method for identifying the unknown source of the space-fractional diffusion equation, Journal of Inequalities and Applications (2014), (1), 1-13. DOI: https://doi.org/10.1186/1029-242X-2014-434

[17] F. Yang, Y. P. Ren, X. X. Li, & D. G. Li, Landweber iterative method for identifying a space-dependent source for the time-fractional diffusion equation, Boundary Value Problems (2017), no. 1, 1-19. DOI: https://doi.org/10.1186/s13661-017-0898-2

[18] N.D. Phuong, Note on a Allen-Cahn equation with Caputo-Fabrizio derivative, Results in Nonlinear Analysis 4 (2021), No.3, 179-185. DOI: https://doi.org/10.53006/rna.962068

[19] F. Yang, F. P. Ren, X.X. Li, The quasi-reversibility method for a final value problem of the time-fractional diffusion equation with inhomogeneous source, Mathematical Methods in the Applied Sciences (2018), 41(5), 1774-1795. DOI: https://doi.org/10.1002/mma.4705

[20] J. Kokila, M. T. Nair, Fourier truncation method for the non-homogeneous time fractional backward heat conduction problem, Inverse Problems in Science and Engineering (2019), 1-25. https://doi.org/10.1080/17415977.2019.1580707. DOI: https://doi.org/10.1080/17415977.2019.1580707

[21] X. B. Yan, T. Wei, Determine a Space-Dependent Source Term in a Time Fractional Diffusion-Wave Equation, Acta Appl. Math. (2020), 163-181. DOI: https://doi.org/10.1007/s10440-019-00248-2

Downloads

Published

2024 Dec 17

Issue

Section

Research Article

How to Cite

[1]
“The Fractional Landweber method for identifying unknown source for the fractional elliptic equations”, Electron. J. Appl. Math., vol. 2, no. 4, pp. 42–50, Dec. 2024, doi: 10.61383/ejam.20242489.

Similar Articles

1-10 of 23

You may also start an advanced similarity search for this article.