{"id":"https://openalex.org/W3161474364","doi":"https://doi.org/10.1007/s10915-021-01509-9","title":"Backward Difference Formulae: The Energy Technique for Subdiffusion Equation","display_name":"Backward Difference Formulae: The Energy Technique for Subdiffusion Equation","publication_year":2021,"publication_date":"2021-05-09","ids":{"openalex":"https://openalex.org/W3161474364","doi":"https://doi.org/10.1007/s10915-021-01509-9","mag":"3161474364"},"language":"en","primary_location":{"id":"doi:10.1007/s10915-021-01509-9","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s10915-021-01509-9","pdf_url":null,"source":{"id":"https://openalex.org/S151550017","display_name":"Journal of Scientific Computing","issn_l":"0885-7474","issn":["0885-7474","1573-7691"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Scientific Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100602647","display_name":"Minghua Chen","orcid":"https://orcid.org/0000-0002-1327-7688"},"institutions":[{"id":"https://openalex.org/I76214153","display_name":"Lanzhou University","ror":"https://ror.org/01mkqqe32","country_code":"CN","type":"education","lineage":["https://openalex.org/I76214153"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Minghua Chen","raw_affiliation_strings":["School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People\u2019s Republic of China","School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People's Republic of China"],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People\u2019s Republic of China","institution_ids":["https://openalex.org/I76214153"]},{"raw_affiliation_string":"School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People's Republic of China","institution_ids":["https://openalex.org/I76214153"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5002472609","display_name":"Yu Fan","orcid":"https://orcid.org/0000-0002-7473-6104"},"institutions":[{"id":"https://openalex.org/I76214153","display_name":"Lanzhou University","ror":"https://ror.org/01mkqqe32","country_code":"CN","type":"education","lineage":["https://openalex.org/I76214153"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Fan Yu","raw_affiliation_strings":["School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People\u2019s Republic of China","School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People's Republic of China"],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People\u2019s Republic of China","institution_ids":["https://openalex.org/I76214153"]},{"raw_affiliation_string":"School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People's Republic of China","institution_ids":["https://openalex.org/I76214153"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5002445122","display_name":"Zhi Zhou","orcid":"https://orcid.org/0000-0002-3439-0313"},"institutions":[{"id":"https://openalex.org/I14243506","display_name":"Hong Kong Polytechnic University","ror":"https://ror.org/0030zas98","country_code":"HK","type":"education","lineage":["https://openalex.org/I14243506"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Zhi Zhou","raw_affiliation_strings":["Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, People\u2019s Republic of China","Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, People's Republic of China"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, People\u2019s Republic of China","institution_ids":["https://openalex.org/I14243506"]},{"raw_affiliation_string":"Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, People's Republic of China","institution_ids":["https://openalex.org/I14243506"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5100602647"],"corresponding_institution_ids":["https://openalex.org/I76214153"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":null,"fwci":0.453,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.59107463,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"87","issue":"3","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9994999766349792,"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":0.9994999766349792,"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9965000152587891,"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"}},{"id":"https://openalex.org/T11565","display_name":"Thermoelastic and Magnetoelastic Phenomena","score":0.98580002784729,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8493273258209229},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.6619156002998352},{"id":"https://openalex.org/keywords/heat-equation","display_name":"Heat equation","score":0.6297410130500793},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.6162671446800232},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5257031917572021},{"id":"https://openalex.org/keywords/term","display_name":"Term (time)","score":0.5020556449890137},{"id":"https://openalex.org/keywords/diffusion-equation","display_name":"Diffusion equation","score":0.4546414613723755},{"id":"https://openalex.org/keywords/energy","display_name":"Energy (signal processing)","score":0.430378794670105},{"id":"https://openalex.org/keywords/work","display_name":"Work (physics)","score":0.4175894260406494},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.38913455605506897},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.15346169471740723},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.07655557990074158}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8493273258209229},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.6619156002998352},{"id":"https://openalex.org/C202787564","wikidata":"https://www.wikidata.org/wiki/Q6510488","display_name":"Heat equation","level":2,"score":0.6297410130500793},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.6162671446800232},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5257031917572021},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.5020556449890137},{"id":"https://openalex.org/C571446","wikidata":"https://www.wikidata.org/wiki/Q6510183","display_name":"Diffusion equation","level":3,"score":0.4546414613723755},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.430378794670105},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.4175894260406494},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.38913455605506897},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.15346169471740723},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.07655557990074158},{"id":"https://openalex.org/C2780378061","wikidata":"https://www.wikidata.org/wiki/Q25351891","display_name":"Service (business)","level":2,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C136264566","wikidata":"https://www.wikidata.org/wiki/Q159810","display_name":"Economy","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s10915-021-01509-9","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s10915-021-01509-9","pdf_url":null,"source":{"id":"https://openalex.org/S151550017","display_name":"Journal of Scientific Computing","issn_l":"0885-7474","issn":["0885-7474","1573-7691"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Scientific Computing","raw_type":"journal-article"},{"id":"pmh:oai:ir.lzu.edu.cn/:262010/451869","is_oa":false,"landing_page_url":"http://ir.lzu.edu.cn/handle/262010/451869","pdf_url":null,"source":{"id":"https://openalex.org/S4406923049","display_name":"Lanzhou University Institutional Repository","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":"","raw_type":"Journal article (JA)"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/7","score":0.8399999737739563,"display_name":"Affordable and clean energy"}],"awards":[{"id":"https://openalex.org/G1068403663","display_name":null,"funder_award_id":"25300818","funder_id":"https://openalex.org/F4320321592","funder_display_name":"Research Grants Council, University Grants Committee"},{"id":"https://openalex.org/G7986504235","display_name":null,"funder_award_id":"11601206","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320321592","display_name":"Research Grants Council, University Grants Committee","ror":"https://ror.org/00djwmt25"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":42,"referenced_works":["https://openalex.org/W99776242","https://openalex.org/W575708337","https://openalex.org/W648869842","https://openalex.org/W1509923322","https://openalex.org/W1963580797","https://openalex.org/W1979313496","https://openalex.org/W1979551969","https://openalex.org/W1980593821","https://openalex.org/W1989611874","https://openalex.org/W1990730790","https://openalex.org/W1991982189","https://openalex.org/W2000486121","https://openalex.org/W2003437637","https://openalex.org/W2020095085","https://openalex.org/W2023413246","https://openalex.org/W2026646621","https://openalex.org/W2035327124","https://openalex.org/W2041540906","https://openalex.org/W2046883902","https://openalex.org/W2052045706","https://openalex.org/W2057119166","https://openalex.org/W2073659542","https://openalex.org/W2082621526","https://openalex.org/W2088789193","https://openalex.org/W2091244859","https://openalex.org/W2121533457","https://openalex.org/W2147268354","https://openalex.org/W2274070693","https://openalex.org/W2343137341","https://openalex.org/W2601627861","https://openalex.org/W2764314153","https://openalex.org/W2787959293","https://openalex.org/W2962970195","https://openalex.org/W3000427044","https://openalex.org/W3038383063","https://openalex.org/W3042666828","https://openalex.org/W3099415531","https://openalex.org/W3103086007","https://openalex.org/W3104959316","https://openalex.org/W3148434010","https://openalex.org/W4239469613","https://openalex.org/W4240596267"],"related_works":["https://openalex.org/W2389517732","https://openalex.org/W346881876","https://openalex.org/W3099809193","https://openalex.org/W2084108921","https://openalex.org/W4388240777","https://openalex.org/W2607759403","https://openalex.org/W2106562951","https://openalex.org/W4308910509","https://openalex.org/W2988063917","https://openalex.org/W2898594924"],"abstract_inverted_index":null,"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
