A modified fractional Landweber method for a backward problem for the inhomogeneous time-fractional diffusion equation in a cylinder

References

• O. Abu Arqub, Numerical solutions of systems of first-order, two-point BVPs based on the reproducing kernel algorithm, Calcolo, 55 (2018), pp. 1–28.
   Web of Science ®Google Scholar
• O. Abu Arqub, Application of residual power series method for the solution of time-fractional Schrodinger equations in one-dimensional space, Fundamenta Informaticae 166 (2019), pp. 87–110.
   Web of Science ®Google Scholar
• O. Abu Arqub and M. Al-Smadi, An adaptive numerical approach for the solutions of fractional advection-diffusion and dispersion equations in singular case under Riesz's derivative operator, Phys. A540 (2020), pp. 123257. 13 pp.
   Web of Science ®Google Scholar
• O. Abu Arqub and N. Shawagfeh, Application of reproducing kernel algorithm for solving Dirichlet time-fractional diffusion-Gordon types equations in porous media, J. Porous Media 22 (2019), pp. 411–434.
   Web of Science ®Google Scholar
• H. Egger and A. Neubauer, Preconditioning Landweber iteration in Hilbert scales, Numer. Math. 101 (2005), pp. 643–662.
   Web of Science ®Google Scholar
• D. Gerth, E. Klann, R. Ramlau, and L. Reichel, On fractional Tikhonov regularization, J. Inverse Ill-Posed Probl. 23(6) (2015), pp. 611–625.
   Web of Science ®Google Scholar
• Y. Han, X. Xiong, and X. Xue, A fractional Landweber method for solving backward time-fractional diffusion problem, Comput. Math. Appl. 78 (2019), pp. 81–91.
   Web of Science ®Google Scholar
• M.E. Hochstenbach and L. Reichel, Fractional Tikhonov regularization for linear discrete ill-posed problems, BIT Numer. Math. 51(1) (2011), pp. 197–215.
   Web of Science ®Google Scholar
• M. Jiang and G. Wang, Convergence studies on iterative algorithms for image reconstruction, IEEE Trans. Med. Imaging 22(5) (2003), pp. 569–579.
   PubMed Web of Science ®Google Scholar
• B. Kaltenbacher, A. Neubauer, and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, De Gruyter, Berlin, 2008.
   Google Scholar
• E. Klann, P. Maass, and R. Ramlau, Two-step regularization methods for linear inverse problems, J. Inverse Ill-Posed Probl. 14(6) (2006), pp. 583–607.
   Google Scholar
• E. Klann and R. Ramlau, Regularization by fractional filter methods and data smoothing, Inverse Probl. 24(2) (2008), pp. 045005.
   Web of Science ®Google Scholar
• L. Landweber, An iteration formula for Fredholm integral equations of the first kind, Am. J. Math. 73(3) (1951), pp. 615–624.
   Google Scholar
• C. Li, W. Deng, and Y. Wu, Numerical analysis and physical simulations for the time fractional radial diffusion equation, Comput. Math. Appl. 62 (2011), pp. 1024–1037.
   Web of Science ®Google Scholar
• A. Neubauer, On Landweber iteration for nonlinear ill-posed problems in Hilbert scales, Numerische Mathematik 85(2) (2000), pp. 309–328.
   Web of Science ®Google Scholar
• I. Podlubny, Fractional Differential Equations, Acad Press, 1999. San Diego, California 92101-4495, USA.
   Google Scholar
• I. Podlubny and M. Kacenak, Mittag-Leffler function, The MATLAB routine, 2012, Available at http://www.mathworks.com/matlabcentral/fileexchange.
   Google Scholar
• H. Pollard, The completely monotonic character of the mittag-leffler function Eα(−x), Bull. Am. Math. Soc.54(12) (1948), pp. 1115–1117.
   Web of Science ®Google Scholar
• N. H. Tuan, L. D. Long, V. T. Nguyen, and T. Tran, On a final value problem for the time-fractional diffusion equation with inhomogeneous source, Inverse Prob. Sci. Eng. 25(9) (2017), pp. 1367–1395.
   Web of Science ®Google Scholar
• N.H. Tuan, L.D. Long, and S. Tatar, Tikhonov regularization method for a backward problem for the inhomogeneous time-fractional diffusion equation, Appl. Anal. 97(5) (2018), pp. 842–863.
   Web of Science ®Google Scholar
• J. Wang, T. Wei, and Y. Zhou, Tikhonov regularization method for a backward problem for the time-fractional diffusion equation, Appl. Math. Model. 37(18–19) (2013), pp. 8518–8532.
   Web of Science ®Google Scholar
• X. Xiong and X. Xue, A fractional Tikhonov regularization method for identifying a space-dependent source in the time-fractional diffusion equation, Appl. Math. Comput. 349 (2019), pp. 292–301.
   Web of Science ®Google Scholar
• X. Xiong, X. Xue, and Z. Qian, A modified iterative regularization method for ill-posed problems, Appl. Numer. Math. 122 (2017), pp. 108–128.
   Web of Science ®Google Scholar
• F. Zhang and X. Jiang, Analytical solutions for a time-fractional axisymmetric diffusion C wave equation with a source term, Nonlinear Anal.: Real World Appl. 12 (2011), pp. 1841–1849.
   Web of Science ®Google Scholar