{"id":"https://openalex.org/W4318984986","doi":"https://doi.org/10.3934/nhm.2023023","title":"Error estimate of BDF2 scheme on a Bakhvalov-type mesh for a singularly perturbed Volterra integro-differential equation","display_name":"Error estimate of BDF2 scheme on a Bakhvalov-type mesh for a singularly perturbed Volterra integro-differential equation","publication_year":2023,"publication_date":"2023-01-01","ids":{"openalex":"https://openalex.org/W4318984986","doi":"https://doi.org/10.3934/nhm.2023023"},"language":"en","primary_location":{"id":"doi:10.3934/nhm.2023023","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3934/nhm.2023023","pdf_url":null,"source":{"id":"https://openalex.org/S21752972","display_name":"Networks and Heterogeneous Media","issn_l":"1556-1801","issn":["1556-1801","1556-181X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Networks and Heterogeneous Media","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"http://dx.doi.org/10.3934/nhm.2023023","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5052372054","display_name":"Libin Liu","orcid":"https://orcid.org/0000-0001-5156-4752"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Li-Bin Liu","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5052131241","display_name":"Yige Liao","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yige Liao","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5087519125","display_name":"Guangqing Long","orcid":"https://orcid.org/0000-0002-0039-8974"},"institutions":[{"id":"https://openalex.org/I4210151929","display_name":"Nanning Normal University","ror":"https://ror.org/04dx82x73","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210151929"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Guangqing Long","raw_affiliation_strings":["School of Mathematics and Statistics, Nanning Normal University, Nanning 530100, China"],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Nanning Normal University, Nanning 530100, China","institution_ids":["https://openalex.org/I4210151929"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5087519125"],"corresponding_institution_ids":["https://openalex.org/I4210151929"],"apc_list":null,"apc_paid":null,"fwci":1.2406,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.72277973,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"18","issue":"2","first_page":"547","last_page":"561"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12727","display_name":"Differential Equations and Numerical Methods","score":1.0,"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"}},"topics":[{"id":"https://openalex.org/T12727","display_name":"Differential Equations and Numerical Methods","score":1.0,"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/T12589","display_name":"Differential Equations and Boundary Problems","score":0.9664000272750854,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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":0.9625999927520752,"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":0.8467060327529907},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.791754424571991},{"id":"https://openalex.org/keywords/backward-differentiation-formula","display_name":"Backward differentiation formula","score":0.5810073614120483},{"id":"https://openalex.org/keywords/variable-coefficient","display_name":"Variable coefficient","score":0.572971761226654},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5156437158584595},{"id":"https://openalex.org/keywords/variable","display_name":"Variable (mathematics)","score":0.5062922239303589},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.5040825605392456},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4785669147968292},{"id":"https://openalex.org/keywords/linear-multistep-method","display_name":"Linear multistep method","score":0.477450430393219},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.47737690806388855},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.474933922290802},{"id":"https://openalex.org/keywords/perturbation","display_name":"Perturbation (astronomy)","score":0.4684358239173889},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.4547837972640991},{"id":"https://openalex.org/keywords/derivative","display_name":"Derivative (finance)","score":0.4299248158931732},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.4217788279056549},{"id":"https://openalex.org/keywords/singular-perturbation","display_name":"Singular perturbation","score":0.4167519211769104},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.26975691318511963},{"id":"https://openalex.org/keywords/differential-algebraic-equation","display_name":"Differential algebraic equation","score":0.10263875126838684},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.0649556815624237}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8467060327529907},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.791754424571991},{"id":"https://openalex.org/C152522140","wikidata":"https://www.wikidata.org/wiki/Q795746","display_name":"Backward differentiation formula","level":5,"score":0.5810073614120483},{"id":"https://openalex.org/C3018795394","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Variable coefficient","level":2,"score":0.572971761226654},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5156437158584595},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.5062922239303589},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.5040825605392456},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4785669147968292},{"id":"https://openalex.org/C65889100","wikidata":"https://www.wikidata.org/wiki/Q1462003","display_name":"Linear multistep method","level":5,"score":0.477450430393219},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.47737690806388855},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.474933922290802},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.4684358239173889},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.4547837972640991},{"id":"https://openalex.org/C111771559","wikidata":"https://www.wikidata.org/wiki/Q66295","display_name":"Derivative (finance)","level":2,"score":0.4299248158931732},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.4217788279056549},{"id":"https://openalex.org/C165160513","wikidata":"https://www.wikidata.org/wiki/Q7524249","display_name":"Singular perturbation","level":2,"score":0.4167519211769104},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.26975691318511963},{"id":"https://openalex.org/C186219872","wikidata":"https://www.wikidata.org/wiki/Q955889","display_name":"Differential algebraic equation","level":4,"score":0.10263875126838684},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0649556815624237},{"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/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C106159729","wikidata":"https://www.wikidata.org/wiki/Q2294553","display_name":"Financial economics","level":1,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3934/nhm.2023023","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3934/nhm.2023023","pdf_url":null,"source":{"id":"https://openalex.org/S21752972","display_name":"Networks and Heterogeneous Media","issn_l":"1556-1801","issn":["1556-1801","1556-181X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Networks and Heterogeneous Media","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:84d23c5ab59b4c4ea5a7f0185b0a3e86","is_oa":true,"landing_page_url":"https://doaj.org/article/84d23c5ab59b4c4ea5a7f0185b0a3e86","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Networks and Heterogeneous Media, Vol 18, Iss 2, Pp 547-561 (2023)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3934/nhm.2023023","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3934/nhm.2023023","pdf_url":null,"source":{"id":"https://openalex.org/S21752972","display_name":"Networks and Heterogeneous Media","issn_l":"1556-1801","issn":["1556-1801","1556-181X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Networks and Heterogeneous Media","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":25,"referenced_works":["https://openalex.org/W1967786809","https://openalex.org/W1975007709","https://openalex.org/W1978865609","https://openalex.org/W1984691419","https://openalex.org/W2005280306","https://openalex.org/W2035159203","https://openalex.org/W2042831535","https://openalex.org/W2043033533","https://openalex.org/W2090708196","https://openalex.org/W2142397770","https://openalex.org/W2328071665","https://openalex.org/W2340358137","https://openalex.org/W2417461243","https://openalex.org/W2917228799","https://openalex.org/W2917303770","https://openalex.org/W2924092691","https://openalex.org/W2944225007","https://openalex.org/W3026890537","https://openalex.org/W3118266091","https://openalex.org/W3151605865","https://openalex.org/W3164559554","https://openalex.org/W3165874937","https://openalex.org/W3199556142","https://openalex.org/W3204480260","https://openalex.org/W4312224081"],"related_works":["https://openalex.org/W2142285866","https://openalex.org/W2015517699","https://openalex.org/W3091448100","https://openalex.org/W2281892811","https://openalex.org/W1974047522","https://openalex.org/W2912618248","https://openalex.org/W2060587726","https://openalex.org/W2358336335","https://openalex.org/W2950222062","https://openalex.org/W2300816247"],"abstract_inverted_index":{"&lt;abstract&gt;&lt;p&gt;A":[0],"singularly":[1],"perturbed":[2],"Volterra":[3],"integro-differential":[4],"problem":[5],"is":[6,14,26,47,53],"considered.":[7],"The":[8],"variable":[9],"two-step":[10],"backward":[11],"differentiation":[12],"formula":[13,25],"used":[15,27],"to":[16,28,59],"approximate":[17],"the":[18,23,30,34,40,50,66,75],"first-order":[19],"derivative":[20],"term":[21],"and":[22,36],"trapezoidal":[24],"discretize":[29],"integral":[31],"term.":[32],"Then,":[33],"stability":[35],"convergence":[37],"analysis":[38],"of":[39],"proposed":[41,51],"numerical":[42,72],"method":[43],"are":[44],"proved.":[45],"It":[46],"shown":[48],"that":[49],"scheme":[52],"second-order":[54],"uniformly":[55],"convergent":[56],"with":[57],"respect":[58],"perturbation":[60],"parameter":[61],"$":[62,64],"\\varepsilon":[63],"in":[65],"discrete":[67],"maximum":[68],"norm.":[69],"Finally,":[70],"a":[71],"experiment":[73],"verifies":[74],"theoretical":[76],"results.&lt;/p&gt;&lt;/abstract&gt;":[77]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1}],"updated_date":"2026-03-15T09:29:46.208133","created_date":"2025-10-10T00:00:00"}
