Identifying inverse source for diffusion equation with conformable time derivative by Fractional Tikhonov method

Year 2022, , 433 - 450, 30.12.2022

Ha Vo Thi Thanh Ngo Hung , Nguyen Duc Phuong

https://doi.org/10.31197/atnaa.1079951

Abstract

In this paper, we study inverse source for diffusion equation with conformable derivative:
$CoD_{t}^{(\gamma)}u - \Delta u = \Phi(t) \mathcal{F}(x)$, where $0<\gamma<1,~ (x,t) \in \Omega \times (0,T)$. We survey the following issues: The error estimate between the sought solution and the regularized solution under a priori parameter choice rule; The error estimate between the sought solution and the regularized solution under a posteriori \\ parameter choice rule; Regularization and ${\mathscr L}_{p}$ estimate by Truncation method.

Keywords

inverse source problem, Fractional diffusion equation, Inverse problem, inverse source problem, Regularization

Supporting Institution

Industrial University of Ho Chi Minh City, Vietnam under Grant named “Investigate some fractional partial differential equations”

Project Number

Grant No. 21/1CB03

References

• [1] A.R.Khalil, A.Yousef, M.Sababheh, A new definition of fractional derivetive, J. Comput. Appl. Math., 264 (2014), pp. 65-70.
• [2] A.Abdeljawad, R.P. Agarwal, E. Karapinar, P.S.Kumari, Solutions of he Nonlinear Integral Equation and Fractional Differential Equation Using the Technique of a Fixed Point with a Numerical Experiment in Extended b-Metric Space, Symmetry 2019, 11, 686.
• [3] B.Alqahtani, H. Aydi, E. Karapınar, V. Rakocevic, A Solution for Volterra Fractional Integral Equations by Hybrid Contractions. Mathematics 2019, 7, 694.
• [4] E. Karapınar, A.Fulga, M. Rashid, L.Shahid, H. Aydi, Large Contractions on Quasi-Metrics Spaces with a Application to Nonlinear Fractional Differential-Equations, Mathematics 2019, 7, 444.
• [5] E.Karapınar, Ho Duy Binh, Nguyen Hoang Luc, and Nguyen Huu Can, On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems, Advances in Difference Equations (2021) 2021:70.
• [6] A.Salim, B. Benchohra, E. Karapınar, J. E. Lazreg, Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations, Adv. Differ Equ. 2020, 601 (2020).
• [7] Lazreg, J. E., Abbas, S., Benchohra, M. and Karapınar, E. Impulsive Caputo Fabrizio fractional differential equations in b-metric spaces. Open Mathematics, 19(1), 363-372.
• [8] Jayshree PAT ˙ IL and Archana CHAUDHAR ˙ I and Mohammed ABDO and Basel HARDAN, Upper and Lower Solution method for Positive solution of generalized Caputo fractional differential equations, Advances in the Theory of Nonlinear Analysis and its Application, 4 (2020), 279-291.
• [9] S. Muthaiah, M. Murugesan, N.G. Thangaraj, Existence of Solutions for Nonlocal Boundary Value Problem of Hadamard Fractional Differential Equations, Advances in the Theory of Nonlinear Analysis and its Application, 3 (2019), 162-173.
• [10] E. Karapınar, H.D. Binh, N.H. Luc, N.H. Can, On continuity of the fractional derivative of the time-fractional semilinear pseudo parabolic systems, Adv. Difference Equ., (2021).
• [11] R.S. Adiguzel, U. Aksoy, E. Karapınar and I.M. Erhan, On the solution of a boundary value problem associated with a fractional differential equation, Mathematical Methods in the Applied Science, (2020).
• [12] H. Afshari and E. Karapınar, A discussion on the existence of positive solutions of the boundary value problems via-Hilfer fractional derivative on b-metric spaces, Advances in Difference Equations volume 2020, Article number: 616 (2020).
• [13] T.N. Thach and N.H. Tuan, Stochastic pseudo-parabolic equations with fractional derivative and fractional Brownian motion, Stochastic Analysis and Applications, 2021, 1-24.
• [14] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Application of Fractional differential equations. In North—Holland Mathematics Studies, Elsevier Science B.V.: Amsterdam, The Netherlands, 2006; Volume 204.
• [15] I. Podlubny, Fractional Differential Equations, Academic Press, USA, 1999.
• [16] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.
• [17] A. Jaiswad, D. Bahuguna, Semilinear Conformable Fractional Differential Equations in Banach spaces, Differ. Equ. Dyn. Sys., 27 (2019), no. 1-3, pp. 313-325.
• [18] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Volume 120 of Applied Mathematical Sciences, Springer, New-York, second edition.
• [19] F. Yang, C.L. Fu, The quasi-reversibility regularization method for identifying the unknown source for time fractional diffusion equation, Appl. Math. Model., 39(2015)1500-1512.
• [20] N.A. Triet and Vo Van Au and Le Dinh Long and D. Baleanu and N.H. Tuan, Regularization of a terminal value problem for time fractional diffusion equation, Math Meth Appl Sci, 2020.
• [21] N.D. PHUONG, Nguyen LUC, Le Dinh LONG, Modifined Quasi Boundary Value method for inverse source biparabolic, Advances in the Theory of Nonlinear Analysis and its Application 4 (2020) 132-142.
• [22] F. Yang, Y.P. Ren, X.X. Li, Landweber iterative method for identifying a spacedependent source for the time-fractional diffusion equation, Bound. Value Probl., 2017(1)(2017)163.
• [23] F. Yang, X. Liu, X.X. Li, Landweber iterative regularization method for identifying the unknown source of the time- fractional diffusion equation, Adv. Differ. Equ., 2017(1)(2017)388.
• [24] Y. Han, X. Xiong, X. Xue, A fractional Landweber method for solving backward time-fractional diffusion problem, Com- puters and Mathematics with Applications, Volume 78 (2019) 81–91.
• [25] H.T. Nguyen, Dinh Long Le, V.T. Nguyen, Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling 000 Vol (2016), pages 1–21, doi: 10.1016/j.apm.2016.04.009.
• [26] N.H. Tuan, L.D. Long, truncation method for an inverse source problem for space-time fractional diffusion equation, Electron. J. Differential Equations, Vol. 2017 (2017), No. 122, pp. 1-16, ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu.
• [27] D. Gerth and E. Klann, R. Ramlau and L. Reichel, On fractional Tikhonov regularization, Journal of Inverse and Ill-posed Problems, 2015.
• [28] X. Xiong, X. Xue, A fractional Tikhonov regularization method for identifying a space-dependent source in the time- fractional diffusion equation, Applied Mathematics and Computation 349 (2019) 292-303.
• [29] N. Duc Phuong, Note on a Allen-Cahn equation with Caputo-Fabrizio derivative, Results in Nonlinear Analysis, 2021.
• [30] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies), Elsevier Science Inc. New York, NY, USA.
• [31] L.D. Long, N.H. Luc, Y. Zhou and C. Nguyen, Identification of Source term for the Time-Fractional Diffusion-Wave Equation by Fractional Tikhonov Method, Mathematics 2019, 7, 934; doi:10.3390/math7100934.