{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:08:09Z","timestamp":1760058489090,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,5]],"date-time":"2025-04-05T00:00:00Z","timestamp":1743811200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fundamental Research Funds for the Central Universities","award":["JG2022-26","PHD2023-060","24CAFUC04034","ZJ2022-003","JG2022-27","GAMRC2021YB08"],"award-info":[{"award-number":["JG2022-26","PHD2023-060","24CAFUC04034","ZJ2022-003","JG2022-27","GAMRC2021YB08"]}]},{"name":"Sichuan Province Engineering Technology Research Center of General Aircraft Maintenance","award":["JG2022-26","PHD2023-060","24CAFUC04034","ZJ2022-003","JG2022-27","GAMRC2021YB08"],"award-info":[{"award-number":["JG2022-26","PHD2023-060","24CAFUC04034","ZJ2022-003","JG2022-27","GAMRC2021YB08"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, the inverse problem of identifying the source term of the time fractional diffusion-wave equation is studied. This problem is ill-posed, i.e., the solution (if it exists) does not depend on the measurable data. Under the priori bound condition, the condition stable result and the optimal error bound are all obtained. The fractional Landweber iterative regularization method is used to solve this inverse problem. Based on the priori regularization parameter selection rule and the posteriori regularization parameter selection rule, the error estimation between the regularization solution and the exact solution is obtained. Moreover, the error estimations are all order optimal. At the end, three numerical examples are given to prove the effectiveness and stability of this regularization method.<\/jats:p>","DOI":"10.3390\/sym17040554","type":"journal-article","created":{"date-parts":[[2025,4,9]],"date-time":"2025-04-09T12:05:56Z","timestamp":1744200356000},"page":"554","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Fractional Landweber Regularization Method for Identifying the Source Term of the Time Fractional Diffusion-Wave Equation"],"prefix":"10.3390","volume":"17","author":[{"given":"Zhenyu","family":"Liang","sequence":"first","affiliation":[{"name":"Key Laboratory of Photonic and Optical Detection in Civil Aviation, School of Science, Civil Aviation Flight University of China, Guanghan 618307, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qin","family":"Jiang","sequence":"additional","affiliation":[{"name":"Key Laboratory of Photonic and Optical Detection in Civil Aviation, School of Science, Civil Aviation Flight University of China, Guanghan 618307, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qingsong","family":"Liu","sequence":"additional","affiliation":[{"name":"Key Laboratory of Photonic and Optical Detection in Civil Aviation, School of Science, Civil Aviation Flight University of China, Guanghan 618307, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luopeng","family":"Xu","sequence":"additional","affiliation":[{"name":"Key Laboratory of Photonic and Optical Detection in Civil Aviation, School of Science, Civil Aviation Flight University of China, Guanghan 618307, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fan","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Science, Lanzhou University of Technology, Lanzhou 730050, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,5]]},"reference":[{"key":"ref_1","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Elsevier."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10915-019-01102-1","article-title":"Mathematical analysis and the local discontinuous galerkin method for caputo\u2013hadamard fractional partial differential equation","volume":"85","author":"Li","year":"2020","journal-title":"J. Sci. Comput."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00009-020-01605-4","article-title":"Finite difference methods for caputo\u2013hadamard fractional differential equations","volume":"17","author":"Gohar","year":"2020","journal-title":"Mediterr. J. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"79","DOI":"10.20852\/ntmsci.2016422308","article-title":"Boundary-value problems for fractional heat equation involving Caputo-Fabrizio derivative","volume":"4","author":"Kerbal","year":"2016","journal-title":"New Trends Math. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.apnum.2013.01.001","article-title":"Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation-ScienceDirect","volume":"68","author":"Wang","year":"2013","journal-title":"Appl. Numer. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"273","DOI":"10.4208\/eajam.310315.030715a","article-title":"Tikhonov regularization method for simultaneous inversion of the source term and initial data in a time-fractional diffusion equation","volume":"5","author":"Ruan","year":"2015","journal-title":"E. Asian J. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3219","DOI":"10.1007\/s12190-021-01656-0","article-title":"Landweber iteration method for simultaneous inversion of the source term and initial data in a time-fractional diffusion equation","volume":"68","author":"Wen","year":"2022","journal-title":"J. Appl. Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"A2976","DOI":"10.1137\/130910865","article-title":"The use of finite difference\/element approaches for solving the time-fractional subdiffusion equation","volume":"35","author":"Zeng","year":"2013","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1016\/j.jcp.2014.07.045","article-title":"Artificial boundary conditions and finite difference approximations for a time-fractional diffusion-wave equation on a two-dimensional unbounded spatial domain","volume":"276","author":"Brunner","year":"2014","journal-title":"J. Comput. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"113910","DOI":"10.1016\/j.cam.2021.113910","article-title":"Determining a time-dependent coefficient in a time-fractional diffusion-wave equation with the Caputo derivative by an additional integral condition","volume":"404","author":"Wei","year":"2022","journal-title":"J. Comput. Appl. Math."},{"key":"ref_11","first-page":"198","article-title":"Regularization Methods for identifying the unknown source of sobolev equation with fractional Laplacian","volume":"15","author":"Yang","year":"2025","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"035001","DOI":"10.1088\/0266-5611\/32\/3\/035001","article-title":"A reduced basis Landweber method for nonlinear inverse problems","volume":"32","author":"Garmatter","year":"2016","journal-title":"Inverse Probl."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Liang, Y.Q., Yang, F., and Li, X.X. (2025). Two Regularization Methods for Identifying the Initial Value of Time-Fractional Telegraph Equation. J. Appl. Anal. Comput.","DOI":"10.1515\/cmam-2024-0159"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/j.matcom.2006.09.005","article-title":"A modified Tikhonov regularization method for a spherically symmetric three-dimensional inverse heat conduction problem","volume":"75","author":"Cheng","year":"2007","journal-title":"Math. Comput. Simul."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Liang, Y.Q., Yang, F., and Li, X.X. (2024). A hybrid regularization method for identifying the source term and the initial value simultaneously for fractional pseudo-parabolic equation with involution. Numer. Algorithms.","DOI":"10.1007\/s11075-024-01944-3"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"741","DOI":"10.1016\/j.aml.2013.02.006","article-title":"A posteriori regularization parameter choice rule for the quasi-boundary value method for the backward time-fractional diffusion problem","volume":"26","author":"Wang","year":"2013","journal-title":"Appl. Math. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1051\/m2an\/2013107","article-title":"A modified quasi-boundary value method for the backward time-fractional diffusion problem","volume":"48","author":"Wei","year":"2014","journal-title":"Esaim-Math. Model. Num."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"107491","DOI":"10.1016\/j.aml.2021.107491","article-title":"On the convergence rate of an improved quasi-reversibility method for an inverse source problem of a nonlinear parabolic equation with nonlocal diffusion coefficient","volume":"5","author":"Wang","year":"2021","journal-title":"Appl. Math. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1515\/jip-2011-0035","article-title":"A Fourier truncated regularization method for a Cauchy problem of a semi-linear elliptic equation","volume":"22","author":"Zhang","year":"2014","journal-title":"J. Inverse-Ill Probl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"113896","DOI":"10.1016\/j.cam.2021.113896","article-title":"Truncated trust region method for nonlinear inverse problems and application in full-waveform inversion","volume":"404","author":"Yan","year":"2022","journal-title":"J. Comput. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1016\/j.jsv.2017.05.004","article-title":"A truncated generalized singular value decomposition algorithm for moving force identification with ill-posed problems","volume":"401","author":"Chen","year":"2017","journal-title":"J. Sound Vib."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/j.cam.2013.04.046","article-title":"An a posteriori parameter choice rule for the truncation regularization method for solving backward parabolic problems","volume":"255","author":"Zhang","year":"2014","journal-title":"J. Comput. Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1385","DOI":"10.1016\/0898-1221(89)90022-9","article-title":"The mollification method and the numerical solution of the inverse heat conduction problem by finite differences","volume":"17","author":"Murio","year":"1989","journal-title":"Comput. Math. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"288","DOI":"10.1023\/A:1025323525021","article-title":"Using numerical solutions in the fourier method for the Fokker-Planck-Kolmogorov equation in the analysis of first-order stochastic nonlinear systems for nonlinear detectors","volume":"46","author":"Nevolin","year":"2003","journal-title":"Radiophys. Quantum Electron."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"298628","DOI":"10.1155\/2011\/298628","article-title":"Mittag-Leffler Functions and Their Applications","volume":"2011","author":"Haubold","year":"2009","journal-title":"J. Appl. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1115","DOI":"10.1090\/S0002-9904-1948-09132-7","article-title":"The completely monotonic character of the Mittag-Leffler function","volume":"54","author":"Pollard","year":"1948","journal-title":"Bull. Amer. Math. Soc."},{"key":"ref_27","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. Funct. Anal. Optim."},{"key":"ref_28","first-page":"287","article-title":"Optimal stable approximations for the sideways heat equation","volume":"5","author":"Tautenhahn","year":"1997","journal-title":"J. Inverse Ill-Pose Probl."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Engl, H.W., Hanke, M., and Neubauer, A. (1996). Regularization of Inverse Problems, Springer.","DOI":"10.1007\/978-94-009-1740-8"},{"key":"ref_30","first-page":"449","article-title":"On optimal regularization methods for fractional differentiation","volume":"18","author":"Tautenhahn","year":"1999","journal-title":"J. Math. Anal. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1370","DOI":"10.1080\/01630563.2013.819515","article-title":"Conditional stability estimates for ill-posed PDE problems by using interpolation","volume":"34","author":"Tautenhahn","year":"2013","journal-title":"Numer. Funct. Anal. Optim."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/554\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:10:42Z","timestamp":1760029842000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/554"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,5]]},"references-count":31,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["sym17040554"],"URL":"https:\/\/doi.org\/10.3390\/sym17040554","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,4,5]]}}}