Quasi solution of a backward space fractional diffusion equation

References

[1] K. A. Ames and J. F. Epperson, A kernel-based method for the approximate solution of backward parabolic problems, SIAM J. Numer. Anal. 34 (1997), no. 4, 1357–1390. 10.1137/S0036142994276785Search in Google Scholar

[2] Y. J. Choi and S. K. Chung, Finite element solutions for the space fractional diffusion equation with a nonlinear source term, Abstr. Appl. Anal. 2012 (2012), Article ID 596184. 10.1155/2012/596184Search in Google Scholar

[3] G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well-posed problems, Electron. J. Differential Equations 1994 (1994), Paper No. 08. Search in Google Scholar

[4] W. Deng, Finite element method for the space and time fractional Fokker–Planck equation, SIAM J. Numer. Anal. 47 (2008/09), no. 1, 204–226. 10.1137/080714130Search in Google Scholar

[5] F. F. Dou and Y. C. Hon, Fundamental kernel-based method for backward space-time fractional diffusion problem, Comput. Math. Appl. 71 (2016), no. 1, 356–367. 10.1016/j.camwa.2015.11.023Search in Google Scholar

[6] P. DuChateau, R. Thelwell and G. Butters, Analysis of an adjoint problem approach to the identification of an unknown diffusion coefficient, Inverse Problems 20 (2004), no. 2, 601–625. 10.1088/0266-5611/20/2/019Search in Google Scholar

[7] V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differential Equations 22 (2006), no. 3, 558–576. 10.1002/num.20112Search in Google Scholar

[8] L. B. Feng, P. Zhuang, F. Liu, I. Turner and Y. T. Gu, Finite element method for space-time fractional diffusion equation, Numer. Algorithms 72 (2016), no. 3, 749–767. 10.1007/s11075-015-0065-8Search in Google Scholar

[9] N. J. Ford, J. Xiao and Y. Yan, A finite element method for time fractional partial differential equations, Fract. Calc. Appl. Anal. 14 (2011), no. 3, 454–474. 10.2478/s13540-011-0028-2Search in Google Scholar

[10] C.-L. Fu, X.-T. Xiong and Z. Qian, Fourier regularization for a backward heat equation, J. Math. Anal. Appl. 331 (2007), no. 1, 472–480. 10.1016/j.jmaa.2006.08.040Search in Google Scholar

[11] P. C. Hansen, The truncated SVD as a method for regularization, BIT 27 (1987), no. 4, 534–553. 10.1007/BF01937276Search in Google Scholar

[12] P. C. Hansen, Analysis of discrete ill-posed problems by means of the 𝖫-curve, SIAM Rev. 34 (1992), no. 4, 561–580. 10.1137/1034115Search in Google Scholar

[13] D. N. Hào, A mollification method for ill-posed problems, Numer. Math. 68 (1994), no. 4, 469–506. 10.1007/s002110050073Search in Google Scholar

[14] A. Hasanov and Z. Liu, An inverse coefficient problem for a nonlinear parabolic variational inequality, Appl. Math. Lett. 21 (2008), no. 6, 563–570. 10.1016/j.aml.2007.06.007Search in Google Scholar

[15] A. Hasanov and J. L. Mueller, A numerical method for backward parabolic problems with non-selfadjoint elliptic operators, Appl. Numer. Math. 37 (2001), no. 1–2, 55–78. 10.1016/S0168-9274(00)00025-8Search in Google Scholar

[16] K. Höllig, C. Apprich and A. Streit, Introduction to the Web-method and its applications, Adv. Comput. Math. 23 (2005), no. 1–2, 215–237. 10.1007/s10444-004-1811-ySearch in Google Scholar

[17] J. Jia, J. Peng, J. Gao and Y. Li, Backward problem for a time-space fractional diffusion equation, Inverse Probl. Imaging 12 (2018), no. 3, 773–799. 10.3934/ipi.2018033Search in Google Scholar

[18] X. Li and C. Xu, Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys. 8 (2010), no. 5, 1016–1051. 10.4208/cicp.020709.221209aSearch in Google Scholar

[19] J. J. Liu and M. Yamamoto, A backward problem for the time-fractional diffusion equation, Appl. Anal. 89 (2010), no. 11, 1769–1788. 10.1080/00036810903479731Search in Google Scholar

[20] N. S. Mera, The method of fundamental solutions for the backward heat conduction problem, Inverse Probl. Sci. Eng. 13 (2005), no. 1, 65–78. 10.1080/10682760410001710141Search in Google Scholar

[21] T. Nguyen Huy, D. D. Trong, D. N. D. Hai and D. D. X. Thanh, A Riesz–Feller space-fractional backward diffusion problem with a time-dependent coefficient: Regularization and error estimates, Math. Methods Appl. Sci. 40 (2017), no. 11, 4040–4064. 10.1002/mma.4284Search in Google Scholar

[22] C. Ren, X. Xu and S. Lu, Regularization by projection for a backward problem of the time-fractional diffusion equation, J. Inverse Ill-Posed Probl. 22 (2014), no. 1, 121–139. 10.1515/jip-2011-0021Search in Google Scholar

[23] A. H. Salehi Shayegan and A. Zakeri, A numerical method for determining a quasi solution of a backward time-fractional diffusion equation, Inverse Probl. Sci. Eng. 26 (2018), no. 8, 1130–1154. 10.1080/17415977.2017.1384826Search in Google Scholar

[24] A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-posed Problems, V. H. Winston & Sons, New York, 1977. Search in Google Scholar

[25] J.-G. Wang, T. Wei and Y.-B. Zhou, Tikhonov regularization method for a backward problem for the time-fractional diffusion equation, Appl. Math. Model. 37 (2013), no. 18–19, 8518–8532. 10.1016/j.apm.2013.03.071Search in Google Scholar

[26] X.-T. Xiong, C.-L. Fu and Z. Qian, Two numerical methods for solving a backward heat conduction problem, Appl. Math. Comput. 179 (2006), no. 1, 370–377. 10.1016/j.amc.2005.11.114Search in Google Scholar

[27] A. Zakeri, A. H. Salehi Shayegan and S. Sakaki, Application of sinc-Galerkin method for solving a nonlinear inverse parabolic problem, Trans. A. Razmadze Math. Inst. 171 (2017), no. 3, 411–423. 10.1016/j.trmi.2017.05.003Search in Google Scholar