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Fractional Tikhonov method for an inverse time-fractional diffusion problem in 2-dimensional space. (English) Zbl 1431.65160

Summary: In this paper, we present a new fractional Tikhonov regularization method for solving an inverse problem for a time-fractional diffusion equation which is highly ill-posed in the two-dimensional setting. Fractional Tikhonov regularization method not only retains the advantage of classical Tikhonov method, but also overcomes the effect of over-smoothing of classical Tikhonov method. We give the selection of regularization parameters of the new method and the corresponding error estimation. Furthermore, numerical results show that the fractional Tikhonov method outperforms the classical one.

MSC:

65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
35R30 Inverse problems for PDEs
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[1] Chen, W.; Ye, Lj; Sun, Hg, Fractional diffusion equations by the Kansa method, Comput. Math. Appl., 59, 1614-1620 (2010) · Zbl 1189.35356
[2] Yu, Z.; Lin, J., Numerical research on the coherent structure in the viscoelastic second-order mixing layers, Appl. Math. Mech., 19, 671-677 (1998) · Zbl 0915.73034
[3] Laskin, N.; Lambadaris, I.; Harmantzis, Fc; Devetsikiotis, M., Fractional Lévy motion and its application to network traffic modeling, Comput. Netw., 40, 3, 363-375 (2002)
[4] Scher, H.; Montroll, Ew, Anomalous transit-time dispersion in amorphous, Phys. Rev. B., 12, 6, 2455-2477 (1975)
[5] Szabo, Tl; Wu, J., A model for longitudinal and shear wave propagation in viscoelastic media, J. Acoust. Soc. Am., 107, 5, 2437-2446 (2000)
[6] Gorenflo, R.; Mainardi, F.; Moretti, D.; Pagnini, G.; Paradisi, P., Discrete random walk models for space-time fractional diffusion, Chem. Phys., 284, 521-541 (2002)
[7] Sokolov, Im; Klafter, J.; Blumen, A., Fractional kinetics, Phys. Today., 55, 48-54 (2002)
[8] Mendes, Rv, A fractional calculus interpretation of the fractional volatility model, Nonlinear Dyn., 55, 395-399 (2009) · Zbl 1187.91231
[9] Agrawal, Op, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dyn., 29, 145-155 (2002) · Zbl 1009.65085
[10] Chavez, A., Fractional diffusion equation to describe Lévy flights, Phys. Lett. A., 239, 13-16 (1998) · Zbl 1026.82524
[11] Liu, F.; Anh, V.; Turner, I., Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166, 209-219 (2004) · Zbl 1036.82019
[12] Meerschaert, Mm; Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172, 65-77 (2004) · Zbl 1126.76346
[13] Berntsson, F., A spectral method for solving the sideways heat equation, Inverse Probl., 15, 891-906 (1999) · Zbl 0934.35201
[14] Eldén, L.; Berntsson, F.; Regiéska, T., Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. Sci. Comput., 21, 6, 2187-2205 (2000) · Zbl 0959.65107
[15] Cheng, J.; Nakagawa, J.; Yamamoto, M.; Yamazaki, T., Uniqueness in an inverse problem for one-dimensional fractional diffusion equation, Inverse Probl., 16, 115002 (2009) · Zbl 1181.35322
[16] Zheng, Gh; Wei, T., Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation, J. Comput. Appl. Math., 233, 2631-2640 (2010) · Zbl 1186.65128
[17] Bondarenko, An; Ivaschenko, Ds, Numerical methods for solving inverse problems for time fractional diffusion equation with variable coefficient, J. Inverse Ill Posed Probl., 17, 419-440 (2009) · Zbl 1205.26009
[18] Murio, Da, Stable numerical solution of a fractional-diffusion inverse heat conduction problem, Comput. Math. Appl., 53, 1492-1501 (2007) · Zbl 1152.65463
[19] Qian, Z., Optimal modified method for a fractional-diffusion inverse heat conduction problem, Inverse Probl. Sci. Eng., 18, 521-533 (2010) · Zbl 1195.65125
[20] Cheng, H.; Fu, Cl, An iteration regularization for a time-fractional inverse diffusion problem, Appl. Math. Model., 36, 5642-5649 (2012) · Zbl 1254.65100
[21] Xiong, Xt; Zhou, Q.; Hon, Yc, An inverse problem for fractional diffusion equation in 2-dimensional case: stability analysis and regularization, J. Math. Anal. Appl., 393, 185-199 (2012) · Zbl 1245.35144
[22] Li, M.; Xiong, Xt, On a fractional backward heat conduction problem: application to deblurring, Comput. Math. Appl., 64, 2594-2602 (2012) · Zbl 1268.65127
[23] Klann, E.; Maass, P.; Ramlau, R., Two-step regularization methods for linear inverse problems, J. Inverse Ill Posed Probl., 14, 583-609 (2006) · Zbl 1107.65045
[24] Klann, E.; Ramlau, R., Regularization by fractional filter methods and data smoothing, Inverse Probl., 24, 045005 (2008) · Zbl 1141.47009
[25] Hochstenbach, Me; Reichel, L., Fractional Tikhonov regularization for linear discrete ill-posed problems, BIT Numer. Math., 51, 197-215 (2011) · Zbl 1215.65075
[26] Gerth, D.; Klann, E.; Ramlau, R.; Reichel, L., On fractional Tikhonov regularization, J. Inverse Ill Posed Probl., 23, 611-625 (2015) · Zbl 1327.65075
[27] Bianchi, D.; Buccini, A.; Donatelli, M.; Serra-Capizzano, S., Iterated fractional Tikhonov regularization, Inverse Probl., 31, 055005 (2015) · Zbl 1434.65081
[28] Qian, Z.; Feng, Xl, A fractional Tikhonov method for solving a Cauchy problem of Helmholtz equation, Appl. Anal., 96, 1656-1668 (2017) · Zbl 1379.65074
[29] Podlubny, I., Fractional Differential Equations (1999), San Diego: Academic Press, San Diego · Zbl 0918.34010
[30] Gorenflo, R.; Rutman, R.; Rusev, P.; Dimovski, I.; Kiryakova, V., On ultraslow and on intermediate processes, Transform Methods and Special Functions (1995), Singapore: SCT Publishers, Singapore · Zbl 0923.34005
[31] Xiong, Xt; Li, Jm; Wen, J., Some novel linear regularization methods for a deblurring problem, Inverse Probl Imaging, 11, 403-426 (2017) · Zbl 1359.35235
[32] Xiong, Xt, A regularization method for a Cauchy problem of the Helmholtz equation, J. Comput. Appl. Math., 233, 1723-1732 (2010) · Zbl 1186.65127
[33] Engl, Hw; Hanke, M.; Neubauer, A., Regularization of Inverse Problem (1996), Boston: Kluwer Academic, Boston
[34] Xiong, X.T.: Regularization theory and algorithm for some inverse problems for parabolic differential equations. Ph.D. Dissertation, Lanzhou University (2007) (in Chinese)
[35] Qian, Z.; Fu, Cl, Regularization strategy for a two-dimensional inverse heat conduction problem, Inverse Probl., 23, 1053-1068 (2007) · Zbl 1118.35073
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