{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T05:17:46Z","timestamp":1773465466023,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,25]],"date-time":"2022-09-25T00:00:00Z","timestamp":1664064000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11961044"],"award-info":[{"award-number":["11961044"]}]},{"name":"National Natural Science Foundation of China","award":["21JR7RA214"],"award-info":[{"award-number":["21JR7RA214"]}]},{"name":"Lan Zhou University of Technology","award":["11961044"],"award-info":[{"award-number":["11961044"]}]},{"name":"Lan Zhou University of Technology","award":["21JR7RA214"],"award-info":[{"award-number":["21JR7RA214"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we consider the inverse problem for the time-fractional Schr\u00f6dinger equation. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. We use the fractional Landweber iterative regularization method to solve this inverse problem and obtain the regularization solution. Under a priori and a posteriori regularization parameter choices, the error estimates are all obtained, respectively.<\/jats:p>","DOI":"10.3390\/sym14102010","type":"journal-article","created":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T01:23:16Z","timestamp":1664414596000},"page":"2010","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Fractional Landweber Iterative Regularization Method for Solving the Inverse Problem of Time-Fractional Schr\u00f6dinger Equation"],"prefix":"10.3390","volume":"14","author":[{"given":"Yinxia","family":"Gao","sequence":"first","affiliation":[{"name":"School of Science, Lanzhou University of Technology, Lanzhou 730050, China"}]},{"given":"Dungang","family":"Li","sequence":"additional","affiliation":[{"name":"School of Science, Lanzhou University of Technology, Lanzhou 730050, China"}]},{"given":"Fan","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Science, Lanzhou University of Technology, Lanzhou 730050, China"}]},{"given":"Xiaoxiao","family":"Li","sequence":"additional","affiliation":[{"name":"School of Science, Lanzhou University of Technology, Lanzhou 730050, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"012102","DOI":"10.1063\/1.4973368","article-title":"Path integrals, supersymmetric quantum mechanics, and the atiyah-singer index theorem for twisted dirac","volume":"58","author":"Fine","year":"2017","journal-title":"J. Math. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3339","DOI":"10.1063\/1.1769611","article-title":"Time fractional Schr\u00f6dinger equation","volume":"45","author":"Naber","year":"2004","journal-title":"J. Math. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/0378-4754(92)90118-Z","article-title":"The Schr\u00f6dinger equation","volume":"34","author":"Berezin","year":"1992","journal-title":"Math. Comput. Simulat."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1537","DOI":"10.1088\/0266-5611\/18\/6\/307","article-title":"Uniqueness and stability in an inverse problem for the Schr\u00f6dinger equation","volume":"18","author":"Baudouin","year":"2007","journal-title":"Inverse Prob."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1063\/1.528578","article-title":"Fractional diffusion and wave equation","volume":"30","author":"Schneider","year":"1989","journal-title":"J. Math. Phys."},{"key":"ref_6","first-page":"154","article-title":"Fractional derivative anomalous diffusion equation modeling prime number distribution","volume":"18","author":"Sun","year":"2015","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1016\/j.cnsns.2016.09.006","article-title":"A caputo fractional derivative of a function with respect to another function","volume":"44","author":"Ricardo","year":"2017","journal-title":"Commun. Nonlinear. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3232","DOI":"10.1016\/j.na.2010.07.003","article-title":"Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative","volume":"73","author":"Wei","year":"2010","journal-title":"Nonlinear. Anal."},{"key":"ref_9","first-page":"2029","article-title":"Analog system design problem formulation by optimum control theory","volume":"84","author":"Zemliak","year":"2001","journal-title":"Ieice T. Fund. Electr."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1007\/s00220-018-3240-7","article-title":"The initial value problem for the euler equations of incompressible fluids viewed as a concave maximization problem","volume":"364","author":"Brenier","year":"2018","journal-title":"Commun. Math. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1134\/S1990478911040053","article-title":"On analytical methods in the theory of inverse problems for hyperbolic equations","volume":"5","author":"Anikonov","year":"2011","journal-title":"J. Appl. Ind. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/0898-1221(91)90134-P","article-title":"Inverse problems of potential theory and flows in porous media with time-dependent free boundary","volume":"22","author":"Etingof","year":"1991","journal-title":"Comput. Math. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"085003","DOI":"10.1088\/0266-5611\/32\/8\/085003","article-title":"An inverse time-dependent source problem for a time-fractional diffusion equation","volume":"32","author":"Wei","year":"2018","journal-title":"Inverse Prob."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1093\/imamci\/dnt009","article-title":"Reconstructing the potential for the one-dimensional Schr\u00f6dinger equation from boundary measurements","volume":"31","author":"Sergei","year":"2014","journal-title":"Ima J. Math. Control. I."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1016\/j.jcp.2014.09.006","article-title":"Stable and efficient momentum-space solutions of the time-dependent Schr\u00f6dinger equation for one-dimensional atoms in strong laser fields","volume":"279","year":"2014","journal-title":"J. Comput. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1530","DOI":"10.1007\/BF02634513","article-title":"Spectral problem of the radial Schr\u00f6dinger equation with confining power potentials","volume":"113","author":"Faustov","year":"1997","journal-title":"Theor. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1090\/S0273-0979-1988-15662-5","article-title":"Book Review: Ill-posed problems of mathematical physics and analysis","volume":"19","author":"Knops","year":"1988","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1025","DOI":"10.1016\/j.enganabound.2004.03.001","article-title":"BEM solution for the cauchy problem associated with helmholtz-type equations by the landweber method","volume":"28","author":"Marin","year":"2004","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"911","DOI":"10.1006\/jmaa.1995.1335","article-title":"Convergence criteria of iterative methods based on landweber iteration for solving nonlinear problems","volume":"194","author":"Scherzer","year":"1995","journal-title":"J. Math. Anal. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/j.camwa.2019.02.017","article-title":"A fractional landweber method for solving backward time-fractional diffusion problem","volume":"78","author":"Han","year":"2019","journal-title":"Comput. Math. Appl."},{"key":"ref_21","first-page":"1","article-title":"Recovering space-dependent source for a time-space fractional diffusion wave equation by fractional Landweber method","volume":"29","author":"Jiang","year":"2020","journal-title":"Inverse Probl. Sci. En."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1016\/j.jmaa.2006.08.040","article-title":"Fourier regularization for a backward heat equation","volume":"31","author":"Fu","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"728","DOI":"10.1016\/j.cam.2009.01.008","article-title":"Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation","volume":"230","author":"Dou","year":"2009","journal-title":"J. Comput. Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1209","DOI":"10.1080\/17415977.2019.1700243","article-title":"A potential-free field inverse Schr\u00f6dinger problem: Optimal error bound analysis and regularization method","volume":"28","author":"Yang","year":"2020","journal-title":"Inverse Probl. Sci. En."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"13723","DOI":"10.1002\/mma.7654","article-title":"Three regularization methods for identifying the initial value of homogeneous anomalous secondary diffusion equation","volume":"44","author":"Yang","year":"2021","journal-title":"Math. Method. Appl. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1969","DOI":"10.1016\/j.cam.2009.09.031","article-title":"On a quasi-reversibility regularization method for a cauchy problem of the helmholtz equation","volume":"233","author":"Qian","year":"2010","journal-title":"J. Comput. Appl. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1228","DOI":"10.1016\/j.camwa.2010.06.004","article-title":"The method of simplified Tikhonov regularization for dealing with the inverse time-dependent heat source problem","volume":"60","author":"Yang","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"111127","DOI":"10.1016\/j.chaos.2021.111127","article-title":"A fractional Tikhonov regularization method for an inverse backward and source problems in the time-space fractional diffusion equations","volume":"150","author":"Djennadi","year":"2021","journal-title":"Chaos Soliton. Fract."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1007\/BF02365265","article-title":"The inverse problem of determining the heat source power for a parabolic equation under arbitrary boundary conditions","volume":"88","author":"Ivanchov","year":"1998","journal-title":"J. Math. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1080\/01630560008816965","article-title":"Regularization of exponentially ill-posed problems","volume":"21","author":"Thorsten","year":"2000","journal-title":"Numer. Func. Anal. Opt."},{"key":"ref_31","first-page":"57","article-title":"Partly smooth regularization of inverse problems","volume":"5303","author":"Vaiter","year":"2014","journal-title":"Inverse Probl. Imag."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"961","DOI":"10.4171\/zaa\/740","article-title":"Optimal stable solution of cauchy problems for elliptic equations","volume":"15","author":"Tautenhahn","year":"1996","journal-title":"Z. Anal. Anwend."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1080\/01630569808816834","article-title":"Optimality for ill-posed problems under general source conditions","volume":"19","author":"Tautenhahn","year":"2007","journal-title":"Numer. Func. Anal. 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