Unknown source identification problem for space-time fractional diffusion equation: optimal error bound analysis and regularization method

References

• Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Progr Fract Differ Appl. 2015;1(2):73–85.
   Google Scholar
• Losada J, Nieto JJ. Properties of a new fractional derivative without singular kernel. Progr Fract Differ Appl. 2015;1(2):87–92.
   Google Scholar
• Tuan NH, Zhou Y. Well-posedness of an initial value problem for fractional diffusion equation with Caputo-Fabrizio derivative. Chaos Soliton Fract. 2020;138:109966.
   Google Scholar
• Zheng XC, Wang H, Fu HF. Well-posedness of fractional differential equations with variable-order Caputo-Fabrizio derivative. Chaos Soliton Fract. 2020;138:109966.
   Google Scholar
• Al-Salti N, Karimov E, Kerbal S. Boundary-value problems for fractional heat equation involving Caputo-Fabrizio derivative. New Trends Math Sci. 2016;4(4):79–89.
   Google Scholar
• Baleanu D, Jajarmi A, Mohammadi H, et al. A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative. Chaos Soliton Fract. 2020;134:109705.
   Google Scholar
• Al-khedhairi A. Dynamical analysis and chaos synchronization of a fractional-order novel financial model based on Caputo-Fabrizio derivative. Eur Phys J Plus. 2019;134(10):532.
   Google Scholar
• Al-Salti N, Karimov E, Sadarangani K. On a differential equation with Caputo-Fabrizio fractional derivative of order 1<β≤2 and application to mass-spring-damper system. Progr Fract Differ Appl. 2016;2(4):257–263.
   Google Scholar
• Shi JK, Chen MH. A second-order accurate scheme for two-dimensional space fractional diffusion equations with time Caputo-Fabrizio fractional derivative. Appl Numer Math. 2020;151:246–262.
   Google Scholar
• Liu ZG, Cheng AJ, Li XL. A second order Crank-Nicolson scheme for fractional Cattaneo equation based on new fractional derivative. Appl Math Comput. 2017;311:361–374.
   Google Scholar
• Wang JG, Wei T, Zhou YB. Tikhonov regularization method for a backward problem for the time-fractional diffusion equation. Appl Math Model. 2013;37(18–19):8518–8532.
   Google Scholar
• Yang F, Zhang P, Li XX, et al. Tikhonov regularization method for identifying the space-dependent source for time-fractional diffusion equation on a columnar symmetric domain. Adv Differ Equ. 2020;2020:128.
   Google Scholar
• Yang F, Fu CL, Li XX. A modified tikhonov regularization method for the cauchy problem of Laplace equation. Acta Math Sci. 2015;35(6):1339–1348.
   Google Scholar
• Xiong XT, Xue X. A fractional Tikhonov regularization method for identifying a space-dependent source in the time-fractional diffusion equation. Appl Math Comput. 2019;349:292–303.
   Google Scholar
• Yang F, Pu Q, Li XX. The fractional Tikhonov regularization methods for identifying the initial value problem for a time-fractional diffusion equation. J Comput Appl Math. 2020;380:112998.
   Google Scholar
• Wang JG, Wei T, Zhou Y. Optimal error bound and simplified Tikhonov regularization method for a backward problem for the time-fractional diffusion equation. J Comput Appl Math. 2015;279(18–19):277–292.
   Google Scholar
• Feng XL, Eldén L. Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method. Inverse Probl. 2014;30(1):015005.
   Google Scholar
• Yang F, Wang N, Li XX, et al. A quasi-boundary regularization method for identifying the initial value of time-fractional diffusion equation on spherically symmetric domain. Inverse Ill-posed Probl. 2019;27(5):609–621.
   Google Scholar
• Yang F, Sun YR, Li XX, et al. The quasi-boundary regularization value method for identifying the initial value of heat equation on a columnar symmetric domain. Numer Algor. 2019;82(2):623–639.
   Google Scholar
• Yang F, Zhang P, Li XX. The truncation method for the Cauchy problem of the inhomogeneous Helmholtz equation. Appl Anal. 2019;98(5):991–1004.
   Google Scholar
• Yang F, Pu Q, Li XX, et al. The truncation regularization method for identifying the initial value on non-homogeneous time-fractional diffusion-wave equations. Mathematics. 2019;7:1007.
   Google Scholar
• Wei T, Wang JG. A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation. Appl Numer Math. 2014;78:95–111.
   Google Scholar
• Yang F, Fu CL, Li XX. A mollification regularization method for unknown source in time-fractional diffusion equation. Int J Comput Math. 2014;91(7):1516–1534.
   Google Scholar
• Zhang HW, Qin HH, Wei T. A quasi-reversibility regularization method for the Cauchy problem of the Helmholtz equation. Int J Comput Math. 2011;88(4):839–850.
   Google Scholar
• Yang F, Fu CL. The quasi-reversibility regularization method for identifying the unknown source for time fractional diffusion equation. Appl Math Model. 2015;39(5–6):1500–1512.
   Google Scholar
• Yang F, Ren YP, Li XX. Landweber iteration regularization method for identifying unknown source on a columnar symmetric domain. Inverse Probl Sci Eng. 2018;26(8):1109–1129.
   Google Scholar
• Yang F, Liu X, Li XX. Landweber iteration regularization method for identifying unknown source of the modified Helmholtz equation. Bound Value Probl. 2017;2017(1):1–16.
   Google Scholar
• Yang F, Zhang Y, Li XX. Landweber iteration regularization method for identifying the initial value problem of the time-space fractional diffusion-wave equation. Numer Algor. 2020;83:1509–1530.
   Google Scholar
• Xiong XT, Fu CL, Li HF. Fourier regularization method of a sideways heat equation for determining surface heat flux. J Math Anal Appl. 2006;317(1):331–348.
   Google Scholar
• Li XX, Lei JL, Yang F. An a posteriori fourier regularization method for identifying the unknown source of the space-fractional diffusion equation. J Inequal Appl. 2014;2014(1):1–13.
   Google Scholar
• Yang F, Fu CL, Li XX, et al. The Fourier regularization method for identifying the unknown source for the modified Helmholtz equation. Acta Math Sci. 2014;34(4):1040–1047.
   Google Scholar
• Yang F, Fan P, Li XX, et al. Fourier truncation regularization method for a time-fractional backward diffusion problem with a nonlinear source. Mathematics. 2019;7(9):865.
   Google Scholar
• Liu SS, Feng LX. A posteriori regularization parameter choice rule for a modified kernel method for a time-fractional inverse diffusion problem. J Comput Appl Math. 2019;353:355–366.
   Google Scholar
• Liu SS, Feng LX. Optimal error bound and modified kernel method for a space-fractional backward diffusion problem. Adv Differ Equ. 2018;2018:268.
   Google Scholar
• Liu SS, Feng LX. A modified kernel method for a time-fractional inverse diffusion problem. Adv Differ Equ. 2015;2015:342.
   Google Scholar
• Yang F, Fu JL, Li XX. A potential-free field inverse Schrödinger problem: optimal error bound analysis and regularization method. Inverse Probl Sci Eng. 2020;28(9):1209–1252.
   Google Scholar
• Zhang ZQ, Ma YJ. A modified kernel method for numerical analytic continuation. Inverse Probl Sci Eng. 2013;21(5):840–853.
   Google Scholar
• Tautenhahn U. Optimality for ill-posed problems under general source conditions. Numer Funct Anal Optim. 1998;19(3–4):377–398.
   Google Scholar
• Ivanchov MI. The inverse problem of determining the heat source power for a parabolic equation under arbitrary boundary conditions. J Math Sci. 1998;88(3):432–436.
   Google Scholar
• Schr o¨er T, Tautenhahn U. On the optimal regularization methods for solving linear ill-posed problems. Z Anal Anwend. 1994;13(4):697–710.
   Google Scholar
• Tautenhahn U. Optimal stable approximations for the sideways heat equation. J Inverse Ill-Posed Probl. 1997;5(3):287–307.
   Google Scholar
• Vainikko G. On the optimality of methods for ill-posed problems. Z Anal Anwend. 1987;6(4):351–362.
   Google Scholar
• Hohage T. Regularization of exponentially ill-posed problems. Numer Funct Anal Optim. 2000;21(3–4):439–464.
   Google Scholar
• Engl HW, Hanke M, Neubauer A. Regularization of inverse problem. Dordrecht: Kluwer Academic Publishers; 1996.
   Google Scholar
• Tautenhahn U, Gorenflo R. On optimal regularization methods for fractional differentiation. J Anal Appl. 1999;18(2):449–467.
   Google Scholar
• Tautenhahn U. Optimal stable solution of Cauchy problems for elliptic equations. Z Anal Anwend. 1996;15:961–984.
   Google Scholar
• Hofmann B, Tautenhahn U, H a¨mrik U. Conditional stability estimates for ill-posed PDE problem by using interpolation. Preprint 2011–16, 2011. Available from: http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-72654.
   Google Scholar
• Atangana A, Alqahtani RT. Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation. Adv Differ Equ. 2016;2016(1):156.
   Google Scholar