{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T16:43:06Z","timestamp":1740156186816,"version":"3.37.3"},"reference-count":29,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,3]],"date-time":"2023-04-03T00:00:00Z","timestamp":1680480000000},"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"]},{"name":"Natural Science Foundation of Gansu Province","award":["21JR7RA214"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"In this paper, we consider the inverse problem for identifying the initial value problem of the time\u2013space fractional nonlinear diffusion equation. The uniqueness of the solution is proved by taking the fixed point theorem of Banach compression, and the ill-posedness of the problem is analyzed through the exact solution. The quasi-boundary regularization method is chosen to solve the ill-posed problem, and the error estimate between the regularization solution and the exact solution is given. Moreover, several numerical examples are chosen to prove the effectiveness of the quasi-boundary regularization method. Finally, our method can be used to solve high dimensional time\u2013space fractional nonlinear diffusion equation, especially in cylindrical and spherical symmetric regions.<\/jats:p>","DOI":"10.3390\/sym15040853","type":"journal-article","created":{"date-parts":[[2023,4,4]],"date-time":"2023-04-04T06:30:15Z","timestamp":1680589815000},"page":"853","source":"Crossref","is-referenced-by-count":0,"title":["The Quasi-Boundary Regularization Method for Recovering the Initial Value in a Nonlinear Time\u2013Space Fractional Diffusion Equation"],"prefix":"10.3390","volume":"15","author":[{"given":"Dun-Gang","family":"Li","sequence":"first","affiliation":[{"name":"School of Science, Lanzhou University of Technology, Lanzhou 730050, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7597-3371","authenticated-orcid":false,"given":"Yong-Gang","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Science, China University of Petroleum, Qingdao 266580, China"}]},{"given":"Yin-Xia","family":"Gao","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":"Jian-Ming","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Science, Lanzhou University of Technology, Lanzhou 730050, China"}]},{"given":"Xiao-Xiao","family":"Li","sequence":"additional","affiliation":[{"name":"School of Science, Lanzhou University of Technology, Lanzhou 730050, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1002\/pssb.2221330150","article-title":"The realization of the generalized transfer equation in a medium with fractal geometry","volume":"133","author":"Nigmatullin","year":"1986","journal-title":"Phys. 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