{"id":"https://openalex.org/W2026950697","doi":"https://doi.org/10.1090/s0025-5718-2013-02756-1","title":"Uniform $l^{1} $ convergence in the Crank-Nicolson method of a linear integro-differential equation for viscoelastic rods and plates","display_name":"Uniform $l^{1} $ convergence in the Crank-Nicolson method of a linear integro-differential equation for viscoelastic rods and plates","publication_year":2013,"publication_date":"2013-09-05","ids":{"openalex":"https://openalex.org/W2026950697","doi":"https://doi.org/10.1090/s0025-5718-2013-02756-1","mag":"2026950697"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-2013-02756-1","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2013-02756-1","pdf_url":"https://www.ams.org/mcom/2014-83-286/S0025-5718-2013-02756-1/S0025-5718-2013-02756-1.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.ams.org/mcom/2014-83-286/S0025-5718-2013-02756-1/S0025-5718-2013-02756-1.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100778599","display_name":"Da Xu","orcid":"https://orcid.org/0000-0002-6283-2123"},"institutions":[{"id":"https://openalex.org/I173759888","display_name":"Hunan Normal University","ror":"https://ror.org/053w1zy07","country_code":"CN","type":"education","lineage":["https://openalex.org/I173759888"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Da Xu","raw_affiliation_strings":["Department of Mathematics, Hunan Normal University, Changsha 410081, Hunan, People\u00e2\u0080\u0099s Republic of China","Department of Mathematics, Hunan Normal University, Changsha 410081, Hunan, People's Republic of China,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Hunan Normal University, Changsha 410081, Hunan, People\u00e2\u0080\u0099s Republic of China","institution_ids":["https://openalex.org/I173759888"]},{"raw_affiliation_string":"Department of Mathematics, Hunan Normal University, Changsha 410081, Hunan, People's Republic of China,","institution_ids":["https://openalex.org/I173759888"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5100778599"],"corresponding_institution_ids":["https://openalex.org/I173759888"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":12,"citation_normalized_percentile":{"value":0.08436404,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"83","issue":"286","first_page":"735","last_page":"769"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9991000294685364,"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":0.9991000294685364,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9991000294685364,"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9987999796867371,"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/mathematics","display_name":"Mathematics","score":0.8818504214286804},{"id":"https://openalex.org/keywords/trapezoidal-rule","display_name":"Trapezoidal rule","score":0.7869589328765869},{"id":"https://openalex.org/keywords/integro-differential-equation","display_name":"Integro-differential equation","score":0.5948864221572876},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5505481362342834},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.5441088080406189},{"id":"https://openalex.org/keywords/crank\u2013nicolson-method","display_name":"Crank\u2013Nicolson method","score":0.485115647315979},{"id":"https://openalex.org/keywords/viscoelasticity","display_name":"Viscoelasticity","score":0.44701895117759705},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.44077038764953613},{"id":"https://openalex.org/keywords/quadrature","display_name":"Quadrature (astronomy)","score":0.4321981966495514},{"id":"https://openalex.org/keywords/numerical-integration","display_name":"Numerical integration","score":0.2577001452445984},{"id":"https://openalex.org/keywords/riccati-equation","display_name":"Riccati equation","score":0.07125169038772583}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8818504214286804},{"id":"https://openalex.org/C78727","wikidata":"https://www.wikidata.org/wiki/Q833293","display_name":"Trapezoidal rule","level":3,"score":0.7869589328765869},{"id":"https://openalex.org/C70985411","wikidata":"https://www.wikidata.org/wiki/Q2380676","display_name":"Integro-differential equation","level":4,"score":0.5948864221572876},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5505481362342834},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.5441088080406189},{"id":"https://openalex.org/C180931078","wikidata":"https://www.wikidata.org/wiki/Q588725","display_name":"Crank\u2013Nicolson method","level":3,"score":0.485115647315979},{"id":"https://openalex.org/C186541917","wikidata":"https://www.wikidata.org/wiki/Q910483","display_name":"Viscoelasticity","level":2,"score":0.44701895117759705},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.44077038764953613},{"id":"https://openalex.org/C62869609","wikidata":"https://www.wikidata.org/wiki/Q28137","display_name":"Quadrature (astronomy)","level":2,"score":0.4321981966495514},{"id":"https://openalex.org/C127349201","wikidata":"https://www.wikidata.org/wiki/Q753445","display_name":"Numerical integration","level":2,"score":0.2577001452445984},{"id":"https://openalex.org/C45473103","wikidata":"https://www.wikidata.org/wiki/Q851503","display_name":"Riccati equation","level":3,"score":0.07125169038772583},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","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/C119599485","wikidata":"https://www.wikidata.org/wiki/Q43035","display_name":"Electrical engineering","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1090/s0025-5718-2013-02756-1","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2013-02756-1","pdf_url":"https://www.ams.org/mcom/2014-83-286/S0025-5718-2013-02756-1/S0025-5718-2013-02756-1.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-2013-02756-1","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2013-02756-1","pdf_url":"https://www.ams.org/mcom/2014-83-286/S0025-5718-2013-02756-1/S0025-5718-2013-02756-1.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G51475283","display_name":null,"funder_award_id":"11271123","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6513204074","display_name":null,"funder_award_id":"10971062","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"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2026950697.pdf","grobid_xml":"https://content.openalex.org/works/W2026950697.grobid-xml"},"referenced_works_count":37,"referenced_works":["https://openalex.org/W70807113","https://openalex.org/W386282068","https://openalex.org/W1480637530","https://openalex.org/W1733789441","https://openalex.org/W1972645181","https://openalex.org/W1979313496","https://openalex.org/W1979551969","https://openalex.org/W1982170880","https://openalex.org/W1988932741","https://openalex.org/W1991982189","https://openalex.org/W2007598418","https://openalex.org/W2025020152","https://openalex.org/W2026110563","https://openalex.org/W2031029580","https://openalex.org/W2039099273","https://openalex.org/W2039174185","https://openalex.org/W2046976085","https://openalex.org/W2049135749","https://openalex.org/W2053796073","https://openalex.org/W2054602319","https://openalex.org/W2070438803","https://openalex.org/W2073659542","https://openalex.org/W2083955243","https://openalex.org/W2086095236","https://openalex.org/W2088789193","https://openalex.org/W2090055904","https://openalex.org/W2092552789","https://openalex.org/W2312420987","https://openalex.org/W2327572656","https://openalex.org/W2476108055","https://openalex.org/W2490352015","https://openalex.org/W2525313324","https://openalex.org/W2561739634","https://openalex.org/W3148434010","https://openalex.org/W4230923404","https://openalex.org/W4256329948","https://openalex.org/W6835301874"],"related_works":["https://openalex.org/W4221057967","https://openalex.org/W4229512832","https://openalex.org/W2618840297","https://openalex.org/W2059356455","https://openalex.org/W2027038046","https://openalex.org/W1982234448","https://openalex.org/W2386326552","https://openalex.org/W2525607688","https://openalex.org/W1981373189","https://openalex.org/W4283066982"],"abstract_inverted_index":{"We":[0],"study":[1],"the":[2,17,37,41,47,61,68,80],"numerical":[3],"approximation":[4],"of":[5,20],"a":[6,33,51,54],"certain":[7],"Volterra":[8],"integro-differential":[9],"equation":[10,27],"in":[11,16,30,50,79],"Hilbert":[12],"space":[13],"which":[14],"arises":[15],"linear":[18],"theory":[19],"isotropic":[21],"viscoelastic":[22],"rods":[23],"and":[24],"plates.":[25],"The":[26,71],"is":[28,44,64,74],"discretized":[29],"time":[31,42],"using":[32],"method":[34],"based":[35],"on":[36],"trapezoidal":[38,48,62],"rule:":[39],"while":[40],"derivative":[43],"approximated":[45],"by":[46],"rule":[49],"two-step":[52],"method,":[53],"convolution":[55],"quadrature":[56],"rule,":[57,63],"constructed":[58],"again":[59],"from":[60],"used":[65],"to":[66,76],"approximate":[67],"integral":[69],"term.":[70],"resulting":[72],"scheme":[73],"shown":[75],"be":[77],"convergence":[78],"$l_{t}^{1}(0,\\infty":[81],";H)\\bigcap":[82],"l_{t}^{\\infty":[83],"}(0,\\infty":[84],";":[85],"H)$":[86],"norm.":[87]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":3}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}