{"id":"https://openalex.org/W1963114151","doi":"https://doi.org/10.1080/01630569608816721","title":"Open problems in the discretization of volterra integral equations","display_name":"Open problems in the discretization of volterra integral equations","publication_year":1996,"publication_date":"1996-01-01","ids":{"openalex":"https://openalex.org/W1963114151","doi":"https://doi.org/10.1080/01630569608816721","mag":"1963114151"},"language":"en","primary_location":{"id":"doi:10.1080/01630569608816721","doi":"https://doi.org/10.1080/01630569608816721","is_oa":false,"landing_page_url":"https://doi.org/10.1080/01630569608816721","pdf_url":null,"source":{"id":"https://openalex.org/S27114296","display_name":"Numerical Functional Analysis and Optimization","issn_l":"0163-0563","issn":["0163-0563","1532-2467"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","dongbi-b","jufo-1","norway-1","poland-70","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Functional Analysis and Optimization","raw_type":"journal-article"},"volume":null,"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/A5031788253","display_name":"Hermann Brunner","orcid":"https://orcid.org/0000-0002-0089-3481","observed_orcids":["https://orcid.org/0000-0002-0089-3481"]},"institutions":[{"id":"https://openalex.org/I130438778","display_name":"Memorial University of Newfoundland","ror":"https://ror.org/04haebc03","country_code":"CA","type":"education","lineage":["https://openalex.org/I130438778"]}],"countries":["CA"],"is_corresponding":true,"raw_author_name":"Hermann Brunner","raw_affiliation_strings":["Dept. of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland, Canada","Dept. of Mathematics and Statistics, Memorial University of Newfoundland, St. John's , Newfoundland, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dept. of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland, Canada","institution_ids":["https://openalex.org/I130438778"]},{"raw_affiliation_string":"Dept. of Mathematics and Statistics, Memorial University of Newfoundland, St. John's , Newfoundland, Canada","institution_ids":["https://openalex.org/I130438778"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5031788253"],"corresponding_institution_ids":["https://openalex.org/I130438778"],"apc_list":null,"apc_paid":null,"fwci":0.4505,"has_fulltext":false,"cited_by_count":10,"citation_normalized_percentile":{"value":0.57528332,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"17","issue":"7-8","first_page":"717","last_page":"736"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9750999808311462,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9750999808311462,"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":0.01209999993443489,"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.003800000064074993,"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/volterra-integral-equations","display_name":"Volterra integral equations","score":0.9990000128746033},{"id":"https://openalex.org/keywords/discretization","display_name":"discretization","score":0.8759999871253967},{"id":"https://openalex.org/keywords/collocation-method","display_name":"collocation method","score":0.9179999828338623},{"id":"https://openalex.org/keywords/piecewise-polynomial-spaces","display_name":"piecewise polynomial spaces","score":0.9639999866485596},{"id":"https://openalex.org/keywords/weakly-singular-kernels","display_name":"weakly singular kernels","score":0.9940000176429749},{"id":"https://openalex.org/keywords/delay-integral-equations","display_name":"delay integral equations","score":0.7509999871253967},{"id":"https://openalex.org/keywords/extrapolation","display_name":"extrapolation","score":0.8889999985694885},{"id":"https://openalex.org/keywords/graded-meshes","display_name":"graded meshes","score":0.9860000014305115},{"id":"https://openalex.org/keywords/variable-delay","display_name":"variable delay","score":0.7269999980926514},{"id":"https://openalex.org/keywords/open-problems","display_name":"open problems","score":0.9390000104904175}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8903658390045166},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.7570512294769287},{"id":"https://openalex.org/C21965488","wikidata":"https://www.wikidata.org/wiki/Q2632613","display_name":"Volterra integral equation","level":3,"score":0.7243040800094604},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.6792297959327698},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.6169779300689697},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5219980478286743},{"id":"https://openalex.org/C2126413","wikidata":"https://www.wikidata.org/wiki/Q1779479","display_name":"Collocation method","level":4,"score":0.5045777559280396},{"id":"https://openalex.org/C192209626","wikidata":"https://www.wikidata.org/wiki/Q190909","display_name":"Focus (optics)","level":2,"score":0.4994344711303711},{"id":"https://openalex.org/C132459708","wikidata":"https://www.wikidata.org/wiki/Q744069","display_name":"Extrapolation","level":2,"score":0.4783018231391907},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.47752347588539124},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.46414342522621155},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.43640366196632385},{"id":"https://openalex.org/C80023036","wikidata":"https://www.wikidata.org/wiki/Q5147531","display_name":"Collocation (remote sensing)","level":2,"score":0.4349361062049866},{"id":"https://openalex.org/C92571104","wikidata":"https://www.wikidata.org/wiki/Q888122","display_name":"Singular integral","level":3,"score":0.43047255277633667},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.4295917749404907},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.186576247215271},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.06904542446136475},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","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/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/01630569608816721","doi":"https://doi.org/10.1080/01630569608816721","is_oa":false,"landing_page_url":"https://doi.org/10.1080/01630569608816721","pdf_url":null,"source":{"id":"https://openalex.org/S27114296","display_name":"Numerical Functional Analysis and Optimization","issn_l":"0163-0563","issn":["0163-0563","1532-2467"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","dongbi-b","jufo-1","norway-1","poland-70","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Functional Analysis and Optimization","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":30,"referenced_works":["https://openalex.org/W24071957","https://openalex.org/W79444176","https://openalex.org/W1606942750","https://openalex.org/W1970806880","https://openalex.org/W1974584296","https://openalex.org/W1978848077","https://openalex.org/W1986410683","https://openalex.org/W1986683394","https://openalex.org/W1995504856","https://openalex.org/W1998424577","https://openalex.org/W2000770285","https://openalex.org/W2007785426","https://openalex.org/W2012163315","https://openalex.org/W2012810094","https://openalex.org/W2022987917","https://openalex.org/W2026110563","https://openalex.org/W2038360943","https://openalex.org/W2046022096","https://openalex.org/W2061881346","https://openalex.org/W2081442517","https://openalex.org/W2087310436","https://openalex.org/W2088977327","https://openalex.org/W2093095086","https://openalex.org/W2111471396","https://openalex.org/W2152605913","https://openalex.org/W2383499743","https://openalex.org/W2481497112","https://openalex.org/W2911553120","https://openalex.org/W4241034009","https://openalex.org/W4253848581"],"related_works":["https://openalex.org/W2078318892","https://openalex.org/W2791210196","https://openalex.org/W1987365920","https://openalex.org/W2124845280","https://openalex.org/W322101810","https://openalex.org/W2384941625","https://openalex.org/W2889856026","https://openalex.org/W2009058751","https://openalex.org/W1989122168","https://openalex.org/W2131278000"],"abstract_inverted_index":{"The":[0],"open":[1],"problems":[2],"(and":[3],"conjectures)":[4],"discussed":[5],"in":[6,10,29],"this":[7],"paper":[8],"arise":[9],"the":[11],"discretization":[12],"of":[13,37],"Volterra":[14],"integral":[15],"equations,":[16],"including":[17],"equations":[18,48],"with":[19,49],"weakly":[20],"singular":[21],"kernels":[22],"and":[23,44,47],"delay":[24],"arguments,":[25],"by":[26],"collocation":[27],"methods":[28],"piecewise":[30],"polynomial":[31],"spaces.":[32],"They":[33],"focus":[34],"on":[35,42],"questions":[36],"stability":[38],"versus":[39],"accuracy;":[40],"extrapolation":[41],"regular":[43],"graded":[45],"meshes;":[46],"certain":[50],"variable":[51],"delays":[52]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2017,"cited_by_count":2}],"updated_date":"2026-10-12T09:40:04.860316","created_date":"2025-10-10T00:00:00"}