Yıl 2021,
, 351 - 361, 30.09.2021
Bui Nghia
,
Nguyen Luc
,
Ho Binh
,
Le Dinh Long
Kaynakça
- I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Vol. 198 of Mathematics in Science and Engineering. Academic Press: San Diego, Calif, USA, 1990.
- F. Mainardi, Fractional diffusive waves in viscoelastic solids Nonlinear Waves in Solids, ed J L Wegner and F R Norwood (Fairfield, NJ: ASME/AMR), pp 93--7.
- R.R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Sol. B 133 (1986) 425--430.
- R. Metzler, J. Klafter, Boundary value problems for fractional diffusion equations, Phys. A
278 (2000) 107-125.
- T. Wei, Y. Zhang, The backward problem for a time-fractional diffusion- wave equation in a bounded domain, Comput. Math. Appl. 75 (2018), no. 10, 3632--3648.
- T. Wei, J. Xian, Variational method for a backward problem for a time-fractional diffusion equation, ESAIM Math. Model. Numer. Anal. 53 (2019), no. 4, 1223--1244.
- J. Xian, T. Wei, Determination of the initial data in a time-fractional diffusion-wave problem by a final time data, Comput. Math. Appl. 78 (2019), no. 8, 2525--2540.
- T. Wei, J.G. Wang, A modified quasi-boundary value method for the backward time-fractional diffusion problem, ESAIM Math. Model. Numer. Anal. 48 (2014), no. 2, 603--621.
- Yaozong Han, Xiangtuan Xiong, Xuemin Xue, A fractional Landweber method for solving backward time-fractional diffusion problem, Computers and Mathematics with Applications 78 (2019) 81--91.
- Fan Yang, Pan Zhang, Xiao-Xiao Li, Xin-Yi Wa, Tikhonov regularization method for identifying the space-dependent source for time-fractional diffusion equation on a columnar symmetric domain, Yang et al. Advances in Difference Equations (2020) 2020:128,
\url{https://doi.org/10.1186/s13662-020-2542-1}
- Huy Tuan Nguyen, Dinh Long Le, Van Thinh Nguyen, Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematics Modelling, Volume 40, Issues 19--20, October 2016, Pages 8244-8264.
- Fan Yang, Yu-Peng Ren, Xiao-Xiao Li, Landweber iteration regularization method for identifying unknown source on a columnar symmetric domain,
- Fan Yang, Qu Pu, Xiao-Xiao Li, The fractional Tikhonov regularization methods for identifying the initial value problem for a time-fractional diffusion equation, Journal of Computational and Applied
Mathematics, Volume 380, 15 December 2020, 112998, \url{https://doi.org/10.1016/j.cam.2020.112998}.
- Fan Yang, Qu Pu, Xiao-Xiao Li, The fractional Tikhonov regularization methods for identifying the initial value problem for a time-fractional diffusion equation, Journal of Computational and Applied Mathematics.
- Danh Hua Quoc Nam, Le Dinh Long, Donal ORegand, Tran Bao Ngoc and Nguyen Huy Tuan, Identification of the right-hand side in a bi-parabolic equation with final data, APPLICABLE ANALYSIS, \url{https://doi.org/10.1080/00036811.2020.1775817}
- A. Farcas, D. Lesnic, The boundary-element method for the determination of a heat source dependent on one variable, J. Eng. Math. 54, 375--388 (2006)
T. Johansson, D. Lesnic, Determination of a spacewise dependent heat source, J. Comput. Appl. Math. 209, 66-80
(2007).
- Wei Cheng, Chu Li Fu, Identifying an unknown source term in a spherically symmetric parabolic equation, Applied Mathematics Letters 26 (2013) 387--391.
- Li, G, Tan, Y, Cheng, J, Wang, X, Determining magnitude of groundwater pollution sources by data compatibility analysis, Inverse Probl. Sci. Eng. 14, 287-300 (2006).
- I.Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, vol 198, Academic Press Inc, San Diego, CA, 1990.
- Le Dinh Long, Identifying the unknown source of time fractional diffusion equation on a columnar symmetric domain, Bulletin of Mathematical Analysis and Applications, ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 13 Issue 1(2021), Pages 41--56.
- Jun-Gang Wang, Ting Wei, Quasi-reversibility method to identify a space-dependent source for the time-fractional diffusion equation, Applied Mathematical Modelling 39 (2015) 6139--6149.
- J.R. Cannon, Determination of an unknown heat source from overspecified boundary data, SIAM J. Numer. Anal. 5 (1968), pp. 275--286.
- J.R. Cannon and P. Duchateau, Structural identification of an unknown source term in a heat
equation, Inverse Probl. 14 (1998), pp. 535--551.
- J.R. Cannon and S.P. Esteva, An inverse problem for the heat equation, Inverse Probl. 2 (1986), pp. 395--403.
- P. Duchateau and W. Rundell, Unicity in an inverse problem for an unknown reaction term in a reation--diffusion--equation, J. Differ. Eqns. 59 (1985), pp. 155--164.
- F. Hettlich and W. Rundell, Identification of a discontinuous source in the heat equation,
Inverse Probl. 17 (2001), pp. 1465--1482.
- M. Micrzwiczak, JA Kolodziej, Application of the method of fundamental solutions and radical basis functions for inverse tranyient heat source problem, Commun. Comput. Phys. 181, 2035-2043 (2010).
- E. Klann, P. Maass, R. Ramlau, Two-step regularization methods for linear inverse problems, J. Inverse Ill-Posed Probl. 14 (6) (2006) 583--609.
- Wei Cheng, Yun-Jie Ma, Chu Li Fu, Identifying an unknown source term in radial heat conduction, Inverse Problems in Science and Engineering, Vol.20, No.3, April 2012, 335--349.
- A. Kirsch, An introduction to the mathematical theory of inverse problem, Berlin: Springer-Verlag; 1996.
- A. Salim, B. Benchohra, E. Karapinar, J.E. Lazreg,
Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations, Adv Differ Equ 2020, 601 (2020).
- E. Karapinar, T. Abdeljawad, F. Jarad, Applying new fixed point theorems on fractional and ordinary differential equations,
Advances in Difference Equations, 2019, 2019:421.
- R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On the solution of a boundary value problem associated with a fractional differential equation, Mathematical Methods in the Applied Sciences
https://doi.org/10.1002/mma.665.
- H. Afshari, E, Karapinar,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)
- H. Afshari, S. Kalantari, E. Karapinar, Solution of fractional differential equations via coupled fixed point, Electronic Journal of Differential Equations,Vol. 2015 (2015), No. 286, pp. 1--12.
- B. Alqahtani, H. Aydi, E. Karapinar, V. Rakocevic, A Solution for Volterra Fractional Integral Equations by Hybrid Contractions, Mathematics 2019, 7, 694.
- E. Karapinar, A. Fulga, M. Rashid, L. Shahid, H. Aydi, Large Contractions on Quasi-Metric Spaces with an Application to Nonlinear Fractional Differential-Equations, Mathematics 2019, 7, 444.
- A. Salim, B. Benchohra, E. Karapinar, J.E. Lazreg, Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations, Adv Differ Equ 2020, 601 (2020).
Regularization method for the problem of determining the source function using integral conditions
Yıl 2021,
, 351 - 361, 30.09.2021
Bui Nghia
,
Nguyen Luc
,
Ho Binh
,
Le Dinh Long
Öz
In this article, we deal with the inverse problem of identifying the unknown source of the time-fractional diffusion equation in a cylinder equation by A fractional Landweber method. This problem is ill-posed. Therefore, the regularization is required. The main result of this article is the error between the sought solution and its regularized under the selection of a priori parameter choice rule.
Kaynakça
- I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Vol. 198 of Mathematics in Science and Engineering. Academic Press: San Diego, Calif, USA, 1990.
- F. Mainardi, Fractional diffusive waves in viscoelastic solids Nonlinear Waves in Solids, ed J L Wegner and F R Norwood (Fairfield, NJ: ASME/AMR), pp 93--7.
- R.R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Sol. B 133 (1986) 425--430.
- R. Metzler, J. Klafter, Boundary value problems for fractional diffusion equations, Phys. A
278 (2000) 107-125.
- T. Wei, Y. Zhang, The backward problem for a time-fractional diffusion- wave equation in a bounded domain, Comput. Math. Appl. 75 (2018), no. 10, 3632--3648.
- T. Wei, J. Xian, Variational method for a backward problem for a time-fractional diffusion equation, ESAIM Math. Model. Numer. Anal. 53 (2019), no. 4, 1223--1244.
- J. Xian, T. Wei, Determination of the initial data in a time-fractional diffusion-wave problem by a final time data, Comput. Math. Appl. 78 (2019), no. 8, 2525--2540.
- T. Wei, J.G. Wang, A modified quasi-boundary value method for the backward time-fractional diffusion problem, ESAIM Math. Model. Numer. Anal. 48 (2014), no. 2, 603--621.
- Yaozong Han, Xiangtuan Xiong, Xuemin Xue, A fractional Landweber method for solving backward time-fractional diffusion problem, Computers and Mathematics with Applications 78 (2019) 81--91.
- Fan Yang, Pan Zhang, Xiao-Xiao Li, Xin-Yi Wa, Tikhonov regularization method for identifying the space-dependent source for time-fractional diffusion equation on a columnar symmetric domain, Yang et al. Advances in Difference Equations (2020) 2020:128,
\url{https://doi.org/10.1186/s13662-020-2542-1}
- Huy Tuan Nguyen, Dinh Long Le, Van Thinh Nguyen, Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematics Modelling, Volume 40, Issues 19--20, October 2016, Pages 8244-8264.
- Fan Yang, Yu-Peng Ren, Xiao-Xiao Li, Landweber iteration regularization method for identifying unknown source on a columnar symmetric domain,
- Fan Yang, Qu Pu, Xiao-Xiao Li, The fractional Tikhonov regularization methods for identifying the initial value problem for a time-fractional diffusion equation, Journal of Computational and Applied
Mathematics, Volume 380, 15 December 2020, 112998, \url{https://doi.org/10.1016/j.cam.2020.112998}.
- Fan Yang, Qu Pu, Xiao-Xiao Li, The fractional Tikhonov regularization methods for identifying the initial value problem for a time-fractional diffusion equation, Journal of Computational and Applied Mathematics.
- Danh Hua Quoc Nam, Le Dinh Long, Donal ORegand, Tran Bao Ngoc and Nguyen Huy Tuan, Identification of the right-hand side in a bi-parabolic equation with final data, APPLICABLE ANALYSIS, \url{https://doi.org/10.1080/00036811.2020.1775817}
- A. Farcas, D. Lesnic, The boundary-element method for the determination of a heat source dependent on one variable, J. Eng. Math. 54, 375--388 (2006)
T. Johansson, D. Lesnic, Determination of a spacewise dependent heat source, J. Comput. Appl. Math. 209, 66-80
(2007).
- Wei Cheng, Chu Li Fu, Identifying an unknown source term in a spherically symmetric parabolic equation, Applied Mathematics Letters 26 (2013) 387--391.
- Li, G, Tan, Y, Cheng, J, Wang, X, Determining magnitude of groundwater pollution sources by data compatibility analysis, Inverse Probl. Sci. Eng. 14, 287-300 (2006).
- I.Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, vol 198, Academic Press Inc, San Diego, CA, 1990.
- Le Dinh Long, Identifying the unknown source of time fractional diffusion equation on a columnar symmetric domain, Bulletin of Mathematical Analysis and Applications, ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 13 Issue 1(2021), Pages 41--56.
- Jun-Gang Wang, Ting Wei, Quasi-reversibility method to identify a space-dependent source for the time-fractional diffusion equation, Applied Mathematical Modelling 39 (2015) 6139--6149.
- J.R. Cannon, Determination of an unknown heat source from overspecified boundary data, SIAM J. Numer. Anal. 5 (1968), pp. 275--286.
- J.R. Cannon and P. Duchateau, Structural identification of an unknown source term in a heat
equation, Inverse Probl. 14 (1998), pp. 535--551.
- J.R. Cannon and S.P. Esteva, An inverse problem for the heat equation, Inverse Probl. 2 (1986), pp. 395--403.
- P. Duchateau and W. Rundell, Unicity in an inverse problem for an unknown reaction term in a reation--diffusion--equation, J. Differ. Eqns. 59 (1985), pp. 155--164.
- F. Hettlich and W. Rundell, Identification of a discontinuous source in the heat equation,
Inverse Probl. 17 (2001), pp. 1465--1482.
- M. Micrzwiczak, JA Kolodziej, Application of the method of fundamental solutions and radical basis functions for inverse tranyient heat source problem, Commun. Comput. Phys. 181, 2035-2043 (2010).
- E. Klann, P. Maass, R. Ramlau, Two-step regularization methods for linear inverse problems, J. Inverse Ill-Posed Probl. 14 (6) (2006) 583--609.
- Wei Cheng, Yun-Jie Ma, Chu Li Fu, Identifying an unknown source term in radial heat conduction, Inverse Problems in Science and Engineering, Vol.20, No.3, April 2012, 335--349.
- A. Kirsch, An introduction to the mathematical theory of inverse problem, Berlin: Springer-Verlag; 1996.
- A. Salim, B. Benchohra, E. Karapinar, J.E. Lazreg,
Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations, Adv Differ Equ 2020, 601 (2020).
- E. Karapinar, T. Abdeljawad, F. Jarad, Applying new fixed point theorems on fractional and ordinary differential equations,
Advances in Difference Equations, 2019, 2019:421.
- R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On the solution of a boundary value problem associated with a fractional differential equation, Mathematical Methods in the Applied Sciences
https://doi.org/10.1002/mma.665.
- H. Afshari, E, Karapinar,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)
- H. Afshari, S. Kalantari, E. Karapinar, Solution of fractional differential equations via coupled fixed point, Electronic Journal of Differential Equations,Vol. 2015 (2015), No. 286, pp. 1--12.
- B. Alqahtani, H. Aydi, E. Karapinar, V. Rakocevic, A Solution for Volterra Fractional Integral Equations by Hybrid Contractions, Mathematics 2019, 7, 694.
- E. Karapinar, A. Fulga, M. Rashid, L. Shahid, H. Aydi, Large Contractions on Quasi-Metric Spaces with an Application to Nonlinear Fractional Differential-Equations, Mathematics 2019, 7, 444.
- A. Salim, B. Benchohra, E. Karapinar, J.E. Lazreg, Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations, Adv Differ Equ 2020, 601 (2020).