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On Tikhonov’s method and optimal error bound for inverse source problem for a time-fractional diffusion equation. (English) Zbl 1445.35328
Summary: We investigate the linear but ill-posed inverse problem of determining a multi-dimensional space-dependent heat source in a time-fractional diffusion equation. We show that the problem is ill-posed in the Hilbert scale $$\mathbb{H}^r(\mathbb{R}^n)$$ and establish global order optimal lower bound for the worst case error. Next, we use the Tikhonov regularization method to deal with this problem in the Hilbert scale $$\mathbb{H}^r(\mathbb{R}^n)$$. Locally optimal choices of parameters for the family of regularization operator in the Hilbert scales $$\mathbb{H}^r(\mathbb{R}^n)$$ are analyzed by a-priori and a-posteriori methods. Numerical implementations are presented to illustrate our theoretical findings.
##### MSC:
 35R30 Inverse problems for PDEs 35R25 Ill-posed problems for PDEs 35R11 Fractional partial differential equations 49K20 Optimality conditions for problems involving partial differential equations 35K15 Initial value problems for second-order parabolic equations
##### Keywords:
Tikhonov regularization method
##### Software:
Algorithm 760; FFTPACK; GitHub; ML; ML_fortran; QUADPACK
Full Text:
##### References:
 [1] Yuste, S. B.; Lindenberg, K., Subdiffusion-limited reactions, Chem. Phys., 284, 1-2, 169-180 (2002) [2] Metzler, R.; Klafter, J., Boundary value problems for fractional diffusion equations, Physica A, 278, 1-2, 107-125 (2000) [3] Metzler, R.; Klafter, J., The restaurant at the end of the random walk: recent development in the description of anomalous transport by fractional dynamics, J. Phys. A: Math. Gen., 37, R161-R208 (2004) · Zbl 1075.82018 [4] Yuste, S. B., Reaction front in an $$A + B \to C$$ reaction-subdiffusion process, Phys. Rev. E, 69, 3, Article 036126 pp. (2004) [5] Gorenflo, R.; Mainardi, F.; Scalas, E.; Raberto, M., Fractional calculus and continuous-time finance III: the diffusion limit, Math. Finance, 171-180 (2001) · Zbl 1138.91444 [6] Scalas, E.; Gorenflo, R.; Mainardi, F., Fractional calculus and continuous-time finace, Physica A, 284, 1-4, 376-384 (2000) [7] Jiang, H.; Liu, F.; Turner, I.; Burrage, K., Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equation equations in a finite domain, Comput. Math. Appl., 64, 3377-3388 (2012) · Zbl 1268.35124 [8] Jin, B.; Lazarov, R.; Zhou, Z., Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal., 51, 1, 445-466 (2013) · Zbl 1268.65126 [9] Li, Z.; Liu, Y.; Yamamoto, M., Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficient, Appl. Math. Comput., 257, 381-397 (2015) · Zbl 1338.35471 [10] Luchko, Y., Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl., 351, 1, 218-223 (2009) · Zbl 1172.35341 [11] Murio, D. A., Implicit finite difference approximation for time fractional diffusion equations, Comput. Math. Appl., 56, 4, 1138-1145 (2008) · Zbl 1155.65372 [12] Sakamoto, K.; Yamamoto, M., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382, 1, 426-447 (2011) · Zbl 1219.35367 [13] Cheng, J.; Nakagawa, J.; Yamamoto, M.; Yamazaki, T., Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Probl., 25, 11, Article 115002 pp. (2009), 16 · Zbl 1181.35322 [14] Rundell, W.; Xu, X.; Zuo, L., The determination of an unknown boundary condition in a fractional diffusion equation, Appl. Anal., 92, 7, 1511-1526 (2013) · Zbl 1302.35412 [15] Xiong, X. T.; Wang, J. X.; Li, M., An optimal method for fractional heat conduction problem backward in time, Appl. Anal., 91, 4, 823-840 (2012) · Zbl 1238.35179 [16] Xu, X.; Cheng, J.; Yamamoto, M., Carleman estimate for a fractional diffusion equation with half order and application, Appl. Anal., 90, 9, 1355-1371 (2011) · Zbl 1236.35199 [17] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier: Elsevier Amsterdam · Zbl 1092.45003 [18] Podlubny, I., (Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, Mathematics in Science and Engineering, vol. 198 (1999), Academic Press Inc.: Academic Press Inc. San Diego, CA) · Zbl 0924.34008 [19] Dou, F. F.; Fu, C. L.; Yang, F., Identifying an unknown source term in a heat equation, Inverse Probl. Sci. Eng., 17, 7, 901-913 (2009) · Zbl 1183.65116 [20] Wang, J. G.; Wei, T., Quasi-reversibility method to identify a space-dependent source for the time-fractional diffusion equation, Appl. Math. Model., 39, 20, 6139-6149 (2015) · Zbl 1443.35198 [21] Yang, F.; Fu, C. L., A mollification regularization method for the inverse spatial-dependent heat source problem, J. Comput. Appl. Math., 255, 555-567 (2014) · Zbl 1291.80010 [22] Trong, D. D.; Long, N. T.; Alain, P. N.D., Nonhomogeneous heat equation: Identification and regularization for the inhomogeneous term, J. Math. Anal. Appl., 312, 1, 93-104 (2005) · Zbl 1087.35095 [23] Tuan, N. H.; Long, L. D.; Thinh, N. V., Regularized solution of an inverse source problem for a time fractional diffusion equation, Appl. Math. Model., 40, 19-20, 8244-8264 (2016) · Zbl 07163012 [24] Wei, T.; Wang, J. G., A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Appl. Numer. Math., 78, 95-111 (2014) · Zbl 1282.65141 [25] Wei, T.; Li, X. L.; Li, Y. S., An inverse time-dependent source problem for a time-fractional diffusion equation, Inverse Probl., 32, 8, Article 085003 pp. (2016), 24 · Zbl 1351.65072 [26] Tautenhahn, U., Optimality for ill-posed problems under general source conditions, Numer. Funct. Anal. Optim., 19, 3-4, 377-398 (1998) · Zbl 0907.65049 [27] Seidman, T. I., “Optimal filtering” for some ill-posed problems, (Fitzgibbon, W.; Wheeler, M., Wave Propagation and Inversion (1992), SIAM), 108-123 · Zbl 0781.35069 [28] Seidman, T. I., Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33, 1, 162-170 (1996) · Zbl 0851.65066 [29] Wang, J. G.; Wei, T.; Zhou, Y. B., Optimal error bound and simplified Tikhonov regularization method for a backward problem for the time-fractional diffusion equation, J. Comput. Appl. Math., 279, 1, 277-292 (2015) · Zbl 1306.65260 [30] Zhao, J.; Liu, S.; Liu, T., An inverse problem for space-fractional backward diffusion problem, Math. Methods Appl. Sci., 37, 8, 1147-1158 (2014) · Zbl 06303262 [31] Engl, H. W.; Kunisch, K.; Neubauer, A., Convergence rates for Tikhonov regularization of nonlinear ill posed problems, Inverse Probl., 5, 523-540 (1989) · Zbl 0695.65037 [32] Groetsch, C. W., (The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Research Notes in Mathematics Series, vol. 105 (1984)) · Zbl 0545.65034 [33] Xiong, X.; Wang, J., A Tikhonov-type method for solving a multidimensional inverse heat source problem in an unbounded domain, J. Comput. Appl. Math., 236, 7, 1766-1774 (2012) · Zbl 1275.65058 [34] Kirsch, A., (An Introduction to the Mathematical Theory of Inverse Problems. An Introduction to the Mathematical Theory of Inverse Problems, Appl. Math. Sci., vol. 120 (1996), Springer: Springer New York) · Zbl 0865.35004 [35] Fujishiro, K.; Yamamoto, M., Approximate controllability for fractional diffusion equations by interior control, Appl. Anal., 93, 9, 1793-1810 (2014) · Zbl 1295.93009 [36] Reginska, T., Two parameter discrepancy principle for combined projection and Tikhonov regularization of ill-posed problems, J. Inverse Ill-Posed Probl., 21, 4, 561-577 (2013) · Zbl 1276.65030 [37] Trong, D. D.; Hai, D. N.D.; Minh, N. D., Optimal regularization for an unknown source of space-fractional diffusion equation, Appl. Math. Comput., 349, 184-206 (2019) · Zbl 1429.65221 [38] Bailey, D. H.; Swarztrauber, P. N., A fast method for the numerical evaluation of continuous fourier and laplace transforms, SIAM J. Sci. Comput., 15, 5, 1105-1110 (1994) · Zbl 0808.65143 [39] FFTPACK5: © 2004-2011, Computational Information Systems Laboratory, University Corporation for Atmospheric Research, Available at https://www2.cisl.ucar.edu/resources/legacy/fft5. [40] Trefethen, L. N.; Weideman, J. A.C., The exponentially convergent trapezoidal rule, SIAM Rev., 56, 3, 385-458 (2014) · Zbl 1307.65031 [41] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical Recipes in Fortran 90, Vol. 2 (1996), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0892.65001 [42] Garrappa, R., Numerical evaluation of two and three parameter Mittag-Leffler functions, SIAM J. Numer. Anal., 53, 3, 1350-1369 (2015), Matlab code: https://www.mathworks.com/matlabcentral/fileexchange/48154-the-mittag-leffler-function · Zbl 1331.33043 [43] T.Q. Viet, A Fortran package for evaluation of the Mittag-Leffler function and its derivative, https://github.com/tranqv/Mittag-Leffler-function-and-its-derivative. [44] Piessens, R.; de Doncker-Kapenga, E.; Überhuber, C. W.; Kahaner, D. K., QUADPACK: A Subroutine Package for Automatic Integration, Vol. 1 (2012), Springer Science & Business Media · Zbl 0508.65005 [45] Akima, H., Algorithm 760: rectangular-grid-data surface fitting that has the accuracy of a bicubic polynomial, ACM Trans. Math. Softw., 22, 3, 357-361 (1996) · Zbl 0884.65009 [46] Diethelm, K.; Ford, N. J.; Freed, A. D., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29, 1-4, 3-22 (2002) · Zbl 1009.65049 [47] Diethelm, K.; Ford, N. J.; Freed, A. D., Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36, 1, 31-52 (2004) · Zbl 1055.65098 [48] Stone, H. L., Iterative solution of implicit approximations of multidimensional partial differential equations, SIAM J. Numer. Anal., 5, 3, 530-558 (1968) · Zbl 0197.13304 [49] Ferziger, J. H.; Peric, M., Computational Methods for Fluid Dynamics, Vol. 3 (2002), Springer: Springer Berlin · Zbl 0998.76001 [50] Diethelm, K., The Analysis of Fractional Differential Equations: An Application-Oriented Exposition using Differential Operators of Caputo Type (2010), Springer Science & Business Media · Zbl 1215.34001
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