{"id":"https://openalex.org/W4362558810","doi":"https://doi.org/10.3390/sym15040853","title":"The Quasi-Boundary Regularization Method for Recovering the Initial Value in a Nonlinear Time\u2013Space Fractional Diffusion Equation","display_name":"The Quasi-Boundary Regularization Method for Recovering the Initial Value in a Nonlinear Time\u2013Space Fractional Diffusion Equation","publication_year":2023,"publication_date":"2023-04-03","ids":{"openalex":"https://openalex.org/W4362558810","doi":"https://doi.org/10.3390/sym15040853"},"language":"en","primary_location":{"id":"doi:10.3390/sym15040853","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15040853","pdf_url":"https://www.mdpi.com/2073-8994/15/4/853/pdf?version=1680515461","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/15/4/853/pdf?version=1680515461","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5040331392","display_name":"Dun-Gang Li","orcid":null},"institutions":[{"id":"https://openalex.org/I22716506","display_name":"Lanzhou University of Technology","ror":"https://ror.org/03panb555","country_code":"CN","type":"education","lineage":["https://openalex.org/I22716506"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Dun-Gang Li","raw_affiliation_strings":["School of Science, Lanzhou University of Technology, Lanzhou 730050, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Science, Lanzhou University of Technology, Lanzhou 730050, China","institution_ids":["https://openalex.org/I22716506"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5053517624","display_name":"Yong-Gang Chen","orcid":"https://orcid.org/0000-0002-7597-3371"},"institutions":[{"id":"https://openalex.org/I4210162190","display_name":"China University of Petroleum, East China","ror":"https://ror.org/05gbn2817","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210162190"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yong-Gang Chen","raw_affiliation_strings":["School of Science, China University of Petroleum, Qingdao 266580, China"],"raw_orcid":"https://orcid.org/0000-0002-7597-3371","affiliations":[{"raw_affiliation_string":"School of Science, China University of Petroleum, Qingdao 266580, China","institution_ids":["https://openalex.org/I4210162190"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5068743115","display_name":"Yin-Xia Gao","orcid":null},"institutions":[{"id":"https://openalex.org/I22716506","display_name":"Lanzhou University of Technology","ror":"https://ror.org/03panb555","country_code":"CN","type":"education","lineage":["https://openalex.org/I22716506"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yin-Xia Gao","raw_affiliation_strings":["School of Science, Lanzhou University of Technology, Lanzhou 730050, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Science, Lanzhou University of Technology, Lanzhou 730050, China","institution_ids":["https://openalex.org/I22716506"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5040886615","display_name":"Fan Yang","orcid":"https://orcid.org/0000-0002-8266-2280"},"institutions":[{"id":"https://openalex.org/I22716506","display_name":"Lanzhou University of Technology","ror":"https://ror.org/03panb555","country_code":"CN","type":"education","lineage":["https://openalex.org/I22716506"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Fan Yang","raw_affiliation_strings":["School of Science, Lanzhou University of Technology, Lanzhou 730050, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Science, Lanzhou University of Technology, Lanzhou 730050, China","institution_ids":["https://openalex.org/I22716506"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5082762334","display_name":"Jian-Ming Xu","orcid":null},"institutions":[{"id":"https://openalex.org/I22716506","display_name":"Lanzhou University of Technology","ror":"https://ror.org/03panb555","country_code":"CN","type":"education","lineage":["https://openalex.org/I22716506"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jian-Ming Xu","raw_affiliation_strings":["School of Science, Lanzhou University of Technology, Lanzhou 730050, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Science, Lanzhou University of Technology, Lanzhou 730050, China","institution_ids":["https://openalex.org/I22716506"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100458643","display_name":"Xiaoxiao Li","orcid":"https://orcid.org/0000-0001-8210-8338"},"institutions":[{"id":"https://openalex.org/I22716506","display_name":"Lanzhou University of Technology","ror":"https://ror.org/03panb555","country_code":"CN","type":"education","lineage":["https://openalex.org/I22716506"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xiao-Xiao Li","raw_affiliation_strings":["School of Science, Lanzhou University of Technology, Lanzhou 730050, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Science, Lanzhou University of Technology, Lanzhou 730050, China","institution_ids":["https://openalex.org/I22716506"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5040886615"],"corresponding_institution_ids":["https://openalex.org/I22716506"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.2128,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.45103791,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":95},"biblio":{"volume":"15","issue":"4","first_page":"853","last_page":"853"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10514","display_name":"Numerical methods in engineering","score":0.9976000189781189,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.996399998664856,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/regularization","display_name":"Regularization (linguistics)","score":0.7433708906173706},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7231981754302979},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.7158101797103882},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5905054807662964},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.5739163160324097},{"id":"https://openalex.org/keywords/inverse-problem","display_name":"Inverse problem","score":0.5664319396018982},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5488090515136719},{"id":"https://openalex.org/keywords/banach-space","display_name":"Banach space","score":0.44883862137794495},{"id":"https://openalex.org/keywords/banach-fixed-point-theorem","display_name":"Banach fixed-point theorem","score":0.4348212480545044},{"id":"https://openalex.org/keywords/well-posed-problem","display_name":"Well-posed problem","score":0.41870272159576416},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4101943373680115},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.09045928716659546},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.0841589868068695}],"concepts":[{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.7433708906173706},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7231981754302979},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.7158101797103882},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5905054807662964},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.5739163160324097},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.5664319396018982},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5488090515136719},{"id":"https://openalex.org/C132954091","wikidata":"https://www.wikidata.org/wiki/Q194397","display_name":"Banach space","level":2,"score":0.44883862137794495},{"id":"https://openalex.org/C2780549194","wikidata":"https://www.wikidata.org/wiki/Q220680","display_name":"Banach fixed-point theorem","level":3,"score":0.4348212480545044},{"id":"https://openalex.org/C106214006","wikidata":"https://www.wikidata.org/wiki/Q1455898","display_name":"Well-posed problem","level":2,"score":0.41870272159576416},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4101943373680115},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.09045928716659546},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0841589868068695},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym15040853","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15040853","pdf_url":"https://www.mdpi.com/2073-8994/15/4/853/pdf?version=1680515461","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:41615dc4763c4d8cb3197b3c742dce59","is_oa":false,"landing_page_url":"https://doaj.org/article/41615dc4763c4d8cb3197b3c742dce59","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 15, Iss 4, p 853 (2023)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/15/4/853/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym15040853","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym15040853","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15040853","pdf_url":"https://www.mdpi.com/2073-8994/15/4/853/pdf?version=1680515461","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4362558810.pdf"},"referenced_works_count":33,"referenced_works":["https://openalex.org/W849899646","https://openalex.org/W1968325374","https://openalex.org/W1974058933","https://openalex.org/W1991508118","https://openalex.org/W1998214447","https://openalex.org/W2021735811","https://openalex.org/W2043288970","https://openalex.org/W2046740511","https://openalex.org/W2051245043","https://openalex.org/W2063879575","https://openalex.org/W2081196614","https://openalex.org/W2092440717","https://openalex.org/W2112722438","https://openalex.org/W2124536171","https://openalex.org/W2144458158","https://openalex.org/W2182721444","https://openalex.org/W2211069905","https://openalex.org/W2420651851","https://openalex.org/W2605247189","https://openalex.org/W2625182672","https://openalex.org/W2902204684","https://openalex.org/W2908871135","https://openalex.org/W2911872586","https://openalex.org/W2925053715","https://openalex.org/W2974288973","https://openalex.org/W3003229185","https://openalex.org/W3098690120","https://openalex.org/W3104478817","https://openalex.org/W3136086822","https://openalex.org/W4210736437","https://openalex.org/W6685764665","https://openalex.org/W6760731482","https://openalex.org/W6773376715"],"related_works":["https://openalex.org/W2018624941","https://openalex.org/W1965313012","https://openalex.org/W2028308512","https://openalex.org/W2931773519","https://openalex.org/W2371498728","https://openalex.org/W2024067282","https://openalex.org/W1965977581","https://openalex.org/W2061980133","https://openalex.org/W2374214022","https://openalex.org/W2980301926"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3],"consider":[4],"the":[5,10,15,24,30,38,41,46,57,61,65,69,82,85],"inverse":[6],"problem":[7,13,42],"for":[8],"identifying":[9],"initial":[11],"value":[12],"of":[14,23,34,40,84],"time\u2013space":[16,99],"fractional":[17,100],"nonlinear":[18,101],"diffusion":[19,102],"equation.":[20],"The":[21,49],"uniqueness":[22],"solution":[25,67,71],"is":[26,43,53,72],"proved":[27],"by":[28],"taking":[29],"fixed":[31],"point":[32],"theorem":[33],"Banach":[35],"compression,":[36],"and":[37,60,68,107],"ill-posedness":[39],"analyzed":[44],"through":[45],"exact":[47,70],"solution.":[48],"quasi-boundary":[50,86],"regularization":[51,66,87],"method":[52,91],"chosen":[54,79],"to":[55,80,95],"solve":[56,96],"ill-posed":[58],"problem,":[59],"error":[62],"estimate":[63],"between":[64],"given.":[73],"Moreover,":[74],"several":[75],"numerical":[76],"examples":[77],"are":[78],"prove":[81],"effectiveness":[83],"method.":[88],"Finally,":[89],"our":[90],"can":[92],"be":[93],"used":[94],"high":[97],"dimensional":[98],"equation,":[103],"especially":[104],"in":[105],"cylindrical":[106],"spherical":[108],"symmetric":[109],"regions.":[110]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2023-04-06T00:00:00"}