{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,15]],"date-time":"2025-04-15T04:00:38Z","timestamp":1744689638597,"version":"3.40.4"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,2,12]],"date-time":"2025-02-12T00:00:00Z","timestamp":1739318400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,2,12]],"date-time":"2025-02-12T00:00:00Z","timestamp":1739318400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Key-Area Research and Development Program of Guangdong Province","award":["2021B0101190003"],"award-info":[{"award-number":["2021B0101190003"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11571386"],"award-info":[{"award-number":["11571386"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University","award":["2020B1212060032"],"award-info":[{"award-number":["2020B1212060032"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,6]]},"DOI":"10.1007\/s40314-025-03124-y","type":"journal-article","created":{"date-parts":[[2025,2,12]],"date-time":"2025-02-12T18:54:06Z","timestamp":1739386446000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fourier regularization for solving sideways problem of a nonlinear time-fractional diffusion equation"],"prefix":"10.1007","volume":"44","author":[{"given":"Hanghang","family":"Wu","sequence":"first","affiliation":[]},{"given":"Hongqi","family":"Yang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,2,12]]},"reference":[{"key":"3124_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-0703-4","volume-title":"Advances on fractional inequalities","author":"GA Anastassiou","year":"2011","unstructured":"Anastassiou GA (2011) Advances on fractional inequalities. Springer, New York"},{"issue":"6","key":"3124_CR2","first-page":"1","volume":"2016","author":"ZB Bai","year":"2016","unstructured":"Bai ZB, Zhang S, Sun SJ, Yin C (2016) Monotone iterative method for fractional differential equations. Electron J Differential Equations 2016(6):1\u20138","journal-title":"Electron J Differential Equations"},{"issue":"1","key":"3124_CR3","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1029\/1999WR900295","volume":"36","author":"B Berkowitz","year":"2000","unstructured":"Berkowitz B, Scher H, Silliman SE (2000) Anomalous transport in laboratory-scale, heterogeneous porous media. Water Resour Res 36(1):149\u2013158","journal-title":"Water Resour Res"},{"issue":"11","key":"3124_CR4","doi-asserted-by":"publisher","first-page":"5642","DOI":"10.1016\/j.apm.2012.01.016","volume":"36","author":"H Cheng","year":"2012","unstructured":"Cheng H, Fu CL (2012) An iteration regularization for a time-fractional inverse diffusion problem. Appl Math Model 36(11):5642\u20135649","journal-title":"Appl Math Model"},{"issue":"7","key":"3124_CR5","doi-asserted-by":"publisher","first-page":"971","DOI":"10.1080\/17415977.2010.492519","volume":"18","author":"H Cheng","year":"2010","unstructured":"Cheng H, Feng XL, Fu CL (2010) A mollification regularization method for the Cauchy problem of an elliptic equation in a multi-dimensional case. Inverse Prbl Sci Eng 18(7):971\u2013982","journal-title":"Inverse Prbl Sci Eng"},{"issue":"2","key":"3124_CR6","first-page":"137","volume":"6","author":"ZL Deng","year":"2011","unstructured":"Deng ZL, Yang XM (2011) A quasi-boundary value regularization method for a fractional-diffusion inverse heat conduction problem. Int J Numer Methods Appl 6(2):137\u2013147","journal-title":"Int J Numer Methods Appl"},{"issue":"8","key":"3124_CR7","doi-asserted-by":"publisher","first-page":"1593","DOI":"10.1016\/j.matcom.2010.11.011","volume":"81","author":"ZL Deng","year":"2011","unstructured":"Deng ZL, Fu CL, Feng XL (2011) A mollification regularization method for stable analytic continuation. Math Comput Simul 81(8):1593\u20131608","journal-title":"Math Comput Simul"},{"key":"3124_CR8","volume-title":"Partial differential equations","author":"LC Evans","year":"1998","unstructured":"Evans LC (1998) Partial differential equations. American Mathematical Society, Providence"},{"issue":"4","key":"3124_CR9","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1007\/s10444-009-9136-5","volume":"33","author":"CL Fu","year":"2010","unstructured":"Fu CL, Qian Z (2010) Numerical pseudodifferential operator and Fourier regularization. Adv Comput Math 33(4):449\u2013470","journal-title":"Adv Comput Math"},{"issue":"1","key":"3124_CR10","doi-asserted-by":"publisher","first-page":"472","DOI":"10.1016\/j.jmaa.2006.08.040","volume":"331","author":"CL Fu","year":"2007","unstructured":"Fu CL, Xiong XT, Qian Z (2007) Fourier regularization for a backward heat equation. J Math Anal Appl 331(1):472\u2013480","journal-title":"J Math Anal Appl"},{"issue":"9","key":"3124_CR11","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/28\/9\/095002","volume":"28","author":"CL Fu","year":"2012","unstructured":"Fu CL, Zhang YX, Cheng H, Ma YJ (2012) The a posteriori Fourier method for solving ill-posed problems. Inverse Prob 28(9):095002","journal-title":"Inverse Prob"},{"issue":"3","key":"3124_CR12","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1007\/s11075-011-9498-x","volume":"59","author":"M Fuhry","year":"2012","unstructured":"Fuhry M, Reichel L (2012) A new Tikhonov regularization method. Numer Algor 59(3):433\u2013445","journal-title":"Numer Algor"},{"issue":"3","key":"3124_CR13","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/32\/3\/035001","volume":"32","author":"D Garmatter","year":"2016","unstructured":"Garmatter D, Haasdonk B, Harrach B (2016) A reduced basis Landweber method for nonlinear inverse problems. Inverse Prob 32(3):035001","journal-title":"Inverse Prob"},{"key":"3124_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.aml.2020.106448","volume":"107","author":"DND Hai","year":"2020","unstructured":"Hai DND, Trong DD (2020) Optimal error bound and truncation regularization method for a backward time-fractional diffusion problem in Hilbert scales. Appl Math Lett 107:1\u20139","journal-title":"Appl Math Lett"},{"issue":"12","key":"3124_CR15","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.100.128103","volume":"100","author":"BI Henry","year":"2008","unstructured":"Henry BI, Langlands TAM, Wearne SL (2008) Fractional cable models for spiny neuronal dendrites. Phys Rev Lett 100(12):128103","journal-title":"Phys Rev Lett"},{"key":"3124_CR16","volume-title":"Theory and applications of fractional differential equations","author":"A Kilbas","year":"2006","unstructured":"Kilbas A, Srivastava H, Trujillo J (2006) Theory and applications of fractional differential equations. Elsevier Science B.V, Amsterdam"},{"key":"3124_CR17","doi-asserted-by":"publisher","unstructured":"Liang YQ, Yang F, Li XX (2024) A hybrid regularization method for identifying the source term and the initial value simultaneously for fractional pseudo-parabolic equation with involution. Numer Algor. https:\/\/doi.org\/10.1007\/s11075-024-01944-3","DOI":"10.1007\/s11075-024-01944-3"},{"key":"3124_CR18","doi-asserted-by":"publisher","unstructured":"Liang YQ, Yang F, Li XX (2025) Two regularization methods for identifying the initial value of time-fractional telegraph equation. Comput Meth Appl Mat. https:\/\/doi.org\/10.1515\/cmam-2024-0159","DOI":"10.1515\/cmam-2024-0159"},{"key":"3124_CR19","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1016\/j.cam.2018.12.038","volume":"353","author":"SS Liu","year":"2019","unstructured":"Liu SS, Feng LX (2019) A posteriori regularization parameter choice rule for a modified kernel method for a time-fractional inverse diffusion problem. J Comput Appl Math 353:355\u2013366","journal-title":"J Comput Appl Math"},{"issue":"1\u20132","key":"3124_CR20","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1016\/S0378-4371(99)00503-8","volume":"278","author":"R Metzler","year":"2000","unstructured":"Metzler R, Klafter J (2000) Boundary value problems for fractional diffusion equations. Phys A 278(1\u20132):107\u2013125","journal-title":"Phys A"},{"issue":"1","key":"3124_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0370-1573(00)00070-3","volume":"339","author":"R Metzler","year":"2000","unstructured":"Metzler R, Klafter J (2000) The random walk\u2019s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339(1):1\u201377","journal-title":"Phys Rep"},{"issue":"2","key":"3124_CR22","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1515\/jiip-2018-0040","volume":"28","author":"TB Ngoc","year":"2020","unstructured":"Ngoc TB, Tuan NH, Kirane M (2020) Regularization of a sideways problem for a time-fractional diffusion equation with nonlinear source. J Inverse Ill-posed Prbl 28(2):211\u2013235","journal-title":"J Inverse Ill-posed Prbl"},{"key":"3124_CR23","volume-title":"The fractional calculus: theory and applications of differentiation and integration to abitrary order","author":"KB Oldham","year":"1974","unstructured":"Oldham KB, Spanier J (1974) The fractional calculus: theory and applications of differentiation and integration to abitrary order. Academic Press, New York"},{"key":"3124_CR24","volume-title":"Fractional differential equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny I (1999) Fractional differential equations. Academic Press, San Diego"},{"issue":"8","key":"3124_CR25","doi-asserted-by":"publisher","first-page":"1969","DOI":"10.1016\/j.cam.2009.09.031","volume":"233","author":"AL Qian","year":"2010","unstructured":"Qian AL, Xiong XT, Wu YJ (2010) On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation. J Comput Appl Math 233(8):1969\u20131979","journal-title":"J Comput Appl Math"},{"key":"3124_CR26","doi-asserted-by":"crossref","unstructured":"Qiao L, Yang F, Li XX (2024) Simultaneous identification of the unknown source term and initial value for the time fractional diffusion equation with local and nonlocal operators. Chaos Soliton Fract 189:115601","DOI":"10.1016\/j.chaos.2024.115601"},{"issue":"1","key":"3124_CR27","doi-asserted-by":"publisher","first-page":"749","DOI":"10.1016\/S0378-4371(02)01048-8","volume":"314","author":"M Raberto","year":"2002","unstructured":"Raberto M, Scalas E, Mainardi F (2002) Waiting-times and returns in high-frequency financial data: an empirical study. Phys A 314(1):749\u2013755","journal-title":"Phys A"},{"key":"3124_CR28","volume-title":"Fractional integrals and derivatives: theory and applications","author":"SG Samko","year":"1993","unstructured":"Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives: theory and applications. Gordon and Breach Science Press, New York"},{"issue":"7","key":"3124_CR29","doi-asserted-by":"publisher","first-page":"741","DOI":"10.1016\/j.aml.2013.02.006","volume":"26","author":"JG Wang","year":"2013","unstructured":"Wang JG, Zhou YB, Wei T (2013) A posteriori regularization parameter choice rule for the quasi-boundary value method for the backward time-fractional diffusion problem. Appl Math Lett 26(7):741\u2013747","journal-title":"Appl Math Lett"},{"issue":"18\u201319","key":"3124_CR30","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1016\/j.cam.2014.11.026","volume":"279","author":"JG Wang","year":"2015","unstructured":"Wang JG, Wei T, Zhou YB (2015) Optimal error bound and simplified Tikhonov regularization method for a backward problem for the time-fractional diffusion equation. J Comput Appl Math 279(18\u201319):277\u2013292","journal-title":"J Comput Appl Math"},{"issue":"1","key":"3124_CR31","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1016\/j.jmaa.2005.12.010","volume":"317","author":"XT Xiong","year":"2006","unstructured":"Xiong XT, Fu CL, Li HF (2006) Fourier regularization method of a sideways heat equation for determining surface heat flux. J Math Anal Appl 317(1):331\u2013348","journal-title":"J Math Anal Appl"},{"key":"3124_CR32","doi-asserted-by":"crossref","unstructured":"Yang F, Cao Y, Li XX (2024a) Identifying source term and initial value simultaneously for the time-fractional diffusion equation with Caputo-like hyper-Bessel operator. Fract Calc Appl Anal 27:2359\u20132396","DOI":"10.1007\/s13540-024-00304-1"},{"issue":"11","key":"3124_CR33","doi-asserted-by":"publisher","first-page":"3286","DOI":"10.1016\/j.apm.2010.02.020","volume":"34","author":"F Yang","year":"2010","unstructured":"Yang F, Fu CL (2010) A simplified Tikhonov regularization method for determining the heat source. Appl Math Model 34(11):3286\u20133299","journal-title":"Appl Math Model"},{"issue":"5\u20136","key":"3124_CR34","doi-asserted-by":"publisher","first-page":"1500","DOI":"10.1016\/j.apm.2014.08.010","volume":"39","author":"F Yang","year":"2015","unstructured":"Yang F, Fu CL (2015) The quasi-reversibility regularization method for identifying the unknown source for time fractional diffusion equation. Appl Math Model 39(5\u20136):1500\u20131512","journal-title":"Appl Math Model"},{"issue":"11","key":"3124_CR35","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3390\/math7111007","volume":"7","author":"F Yang","year":"2019","unstructured":"Yang F, Pu Q, Li XX (2019) The truncation regularization method for identifying the initial value on non-homogeneous time-fractional diffusion-wave equations. Mathematics 7(11):1\u201321","journal-title":"Mathematics"},{"key":"3124_CR36","doi-asserted-by":"crossref","unstructured":"Yang F, Xu JM, Li XX (2024b) Simultaneous inversion of the source term and initial value of the time fractional diffusion equation. Math Model Anal 29(2):193\u2013214","DOI":"10.3846\/mma.2024.18133"},{"key":"3124_CR37","unstructured":"Yang F, Yan LL, Liu H, Li XX (2025) Two regularization methods for identifying the unknown source of sobolev equation with fractional Laplacian. J Appl Anal Comput 15(1):198\u2013225"},{"key":"3124_CR38","doi-asserted-by":"crossref","unstructured":"Yang F, Zhang Y, Li XX (2024c) Landweber iterative method for an inverse source problem of time-space fractional diffusion-wave equation. Comput Meth Appl Mat 24(1):265\u2013278","DOI":"10.1515\/cmam-2022-0240"},{"issue":"2","key":"3124_CR39","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/25\/2\/025006","volume":"25","author":"CA Zarzer","year":"2009","unstructured":"Zarzer CA (2009) On Tikhonov regularization with non-convex sparsity constraints. Inverse Prob 25(2):025006","journal-title":"Inverse Prob"},{"key":"3124_CR40","doi-asserted-by":"crossref","unstructured":"Zhang CY, Yang F, Li XX (2024a) Two regularization methods for identifying the spatial source term problem for a space-time fractional diffusion-wave equation. Mathematics-Basel 12(2):231","DOI":"10.3390\/math12020231"},{"key":"3124_CR41","doi-asserted-by":"crossref","unstructured":"Zhang CY, Yang F, Li XX (2024b) Identifying the source term and the initial value simultaneously for Caputo-Hadamard fractional diffusion equation on spherically symmetric domain. Comput Appl Math 43(4):161","DOI":"10.1007\/s40314-024-02679-6"},{"issue":"2","key":"3124_CR42","doi-asserted-by":"publisher","first-page":"418","DOI":"10.1016\/j.jmaa.2011.01.067","volume":"378","author":"GH Zheng","year":"2011","unstructured":"Zheng GH, Wei T (2011) A new regularization method for solving a time-fractional inverse diffusion problem. J Math Anal Appl 378(2):418\u2013431","journal-title":"J Math Anal Appl"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03124-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03124-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03124-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,14]],"date-time":"2025-04-14T04:26:26Z","timestamp":1744604786000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03124-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,12]]},"references-count":42,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["3124"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03124-y","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2025,2,12]]},"assertion":[{"value":"19 October 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 December 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 January 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 February 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they are not aware of any financial or interpersonal conflicts that might have influenced the research presented in this article. The authors also state that there are no information conflicts, value conflicts, Conflict of interest, relational conflicts, or structural conflicts.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"170"}}