Fractional Tikhonov method for inverse source bi-parabolic: a priori parameter choice rule

Authors

Le Dinh Long Faculty of Math, FPT University HCM, Saigon Hi-Tech Park, Thu Duc City, Ho Chi Minh City, Vietnam

Keywords:

ill-posed, regularization method, Tikhonov method, Fractional Tikhonov method

Abstract

In this work, the unknown source function for the bi-parabolic is investigated. This problem is non-well-posed. Applying a Fractional Tikhonov method to construct the regularized solution. After that, we have test the estimation \(\|f_\gamma^\delta - f\|_{L_2} \to 0\), then \(\delta \to 0\), under a priori rule.

References

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

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, Físicas y Naturales. Serie A. Matemáticas 115, no. 3 (2021): 1-16.

M. Benchohra, E. Karapınar, J.E. Lazreg, & A. Salim, Impulsive Fractional Differential Equations with Retardation and Anticipation. In Fractional Differential Equations: New Advancements for Generalized Fractional Derivatives (2023), 109-155.

V.M. Bulavatsky, Fractional differential analog of biparabolic evolution equation and some its applications. Cybern Syst Anal. September 2016; 52(5) (2016):737-747.

V.M. Bulavatsky, Some nonlocal boundary-Value problems for the biparabolic evolution equation and its fractional-Differential analog. Cybern Syst Anal, ;55(5) (2019):796-804.

G. Fichera, Is the Fourier theory of heat propagation paradoxical. Rendiconti Del Circolo Matematico Di Palermo?. (1992);41:5-28.

V.L. Fushchich, A.S. Galitsyn , A.S. Polubinskii, A new mathematical model of heat conduction processes. Ukr Math J.;42, (1990) : 210-216.

L. Joseph, D. Preziosi, Heat waves. Rev Mod Phys. , 61 (1989) ; 41-73. DOI:10.1103/revmodphys.61.41

Andreas Kirsch An introduction to the Mathematical Theory of Inverse Problem Second Edition

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

D.H. Q. Nam, L.D. Long, D. O’Regan, T.B. Ngoc, N.H. Tuan, Identification of the right-hand side in a bi-parabolic equation with final data, Applicable Analysis, 101(4), (2022) 1157-1175.

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.

F. Zouyed, S. Djemoui, An Iterative Regularization Method for Identifying the Source Term in a Second Order Differential Equation, Hindawi Publishing Corporation, Mathematical Problems in Engineering, Volume 2015, 9 pages.