{"id":"https://openalex.org/W2592672261","doi":"https://doi.org/10.1080/00207160.2017.1299862","title":"Convergence and stability of split-step theta methods with variable step-size for stochastic pantograph differential equations","display_name":"Convergence and stability of split-step theta methods with variable step-size for stochastic pantograph differential equations","publication_year":2017,"publication_date":"2017-03-27","ids":{"openalex":"https://openalex.org/W2592672261","doi":"https://doi.org/10.1080/00207160.2017.1299862","mag":"2592672261"},"language":"en","primary_location":{"id":"doi:10.1080/00207160.2017.1299862","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160.2017.1299862","pdf_url":null,"source":{"id":"https://openalex.org/S124867444","display_name":"International Journal of Computer Mathematics","issn_l":"0020-7160","issn":["0020-7160","1026-7425","1029-0265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computer Mathematics","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/A5080746874","display_name":"Yu Xiao","orcid":"https://orcid.org/0000-0003-0269-0474"},"institutions":[{"id":"https://openalex.org/I204983213","display_name":"Harbin Institute of Technology","ror":"https://ror.org/01yqg2h08","country_code":"CN","type":"education","lineage":["https://openalex.org/I204983213"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yu Xiao","raw_affiliation_strings":["Department of Mathematics, Harbin Institute of Technology, Harbin, China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Harbin Institute of Technology, Harbin, China","institution_ids":["https://openalex.org/I204983213"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5044553525","display_name":"Mahmoud A. Eissa","orcid":"https://orcid.org/0000-0001-6522-7546"},"institutions":[{"id":"https://openalex.org/I63601056","display_name":"Menoufia University","ror":"https://ror.org/05sjrb944","country_code":"EG","type":"education","lineage":["https://openalex.org/I63601056"]},{"id":"https://openalex.org/I204983213","display_name":"Harbin Institute of Technology","ror":"https://ror.org/01yqg2h08","country_code":"CN","type":"education","lineage":["https://openalex.org/I204983213"]}],"countries":["CN","EG"],"is_corresponding":true,"raw_author_name":"Mahmoud A. Eissa","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Menoufia University, Menoufia, Egypt","Department of Mathematics, Harbin Institute of Technology, Harbin, China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Menoufia University, Menoufia, Egypt","institution_ids":["https://openalex.org/I63601056"]},{"raw_affiliation_string":"Department of Mathematics, Harbin Institute of Technology, Harbin, China","institution_ids":["https://openalex.org/I204983213"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5101914761","display_name":"Boping Tian","orcid":"https://orcid.org/0000-0001-5267-6764"},"institutions":[{"id":"https://openalex.org/I204983213","display_name":"Harbin Institute of Technology","ror":"https://ror.org/01yqg2h08","country_code":"CN","type":"education","lineage":["https://openalex.org/I204983213"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Boping Tian","raw_affiliation_strings":["Department of Mathematics, Harbin Institute of Technology, Harbin, China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Harbin Institute of Technology, Harbin, China","institution_ids":["https://openalex.org/I204983213"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5044553525"],"corresponding_institution_ids":["https://openalex.org/I204983213","https://openalex.org/I63601056"],"apc_list":null,"apc_paid":null,"fwci":1.0783,"has_fulltext":false,"cited_by_count":13,"citation_normalized_percentile":{"value":0.75013922,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"95","issue":"5","first_page":"939","last_page":"960"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9933000206947327,"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":0.9933000206947327,"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":0.9925000071525574,"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/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9839000105857849,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8090730905532837},{"id":"https://openalex.org/keywords/pantograph","display_name":"Pantograph","score":0.6774102449417114},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.6572722792625427},{"id":"https://openalex.org/keywords/variable","display_name":"Variable (mathematics)","score":0.6046191453933716},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.5926281213760376},{"id":"https://openalex.org/keywords/stochastic-differential-equation","display_name":"Stochastic differential equation","score":0.53604656457901},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.508105993270874},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.4979109764099121},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.42735588550567627},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.36273640394210815},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1104271411895752}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8090730905532837},{"id":"https://openalex.org/C20756127","wikidata":"https://www.wikidata.org/wiki/Q722757","display_name":"Pantograph","level":2,"score":0.6774102449417114},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6572722792625427},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.6046191453933716},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.5926281213760376},{"id":"https://openalex.org/C51955184","wikidata":"https://www.wikidata.org/wiki/Q1545585","display_name":"Stochastic differential equation","level":2,"score":0.53604656457901},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.508105993270874},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.4979109764099121},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.42735588550567627},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.36273640394210815},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1104271411895752},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","level":1,"score":0.0},{"id":"https://openalex.org/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/00207160.2017.1299862","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160.2017.1299862","pdf_url":null,"source":{"id":"https://openalex.org/S124867444","display_name":"International Journal of Computer Mathematics","issn_l":"0020-7160","issn":["0020-7160","1026-7425","1029-0265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computer Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2504671062","display_name":null,"funder_award_id":"71350005","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4360350797","display_name":null,"funder_award_id":"JSKFJ201604120014","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8892121633","display_name":null,"funder_award_id":"91646106","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":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W133249462","https://openalex.org/W1978255949","https://openalex.org/W1978275259","https://openalex.org/W1983176089","https://openalex.org/W1984965123","https://openalex.org/W1990781511","https://openalex.org/W1993539434","https://openalex.org/W2013859220","https://openalex.org/W2016694818","https://openalex.org/W2022113217","https://openalex.org/W2035460619","https://openalex.org/W2039261336","https://openalex.org/W2050667572","https://openalex.org/W2061230119","https://openalex.org/W2070845660","https://openalex.org/W2076024467","https://openalex.org/W2078703821","https://openalex.org/W2084827643","https://openalex.org/W2086468288","https://openalex.org/W2088945003","https://openalex.org/W2095272332","https://openalex.org/W2123141825","https://openalex.org/W2129849411","https://openalex.org/W2140333923","https://openalex.org/W2168111113","https://openalex.org/W2360293356","https://openalex.org/W2492155337","https://openalex.org/W2530178561"],"related_works":["https://openalex.org/W2573375108","https://openalex.org/W4252833549","https://openalex.org/W2378529747","https://openalex.org/W4387011711","https://openalex.org/W2372731345","https://openalex.org/W2057677727","https://openalex.org/W2360526382","https://openalex.org/W2379657325","https://openalex.org/W2136457064","https://openalex.org/W2787802451"],"abstract_inverted_index":{"In":[0,48,106],"this":[1,49],"paper,":[2],"we":[3,51],"are":[4,19,43,71,134],"interested":[5],"in":[6],"numerical":[7,38,69,123,132],"methods":[8,39,56,70,94,124,148],"with":[9,27,40,57,95,125],"variable":[10,58,96],"step-size":[11,42,59,97],"for":[12,60,98,154],"stochastic":[13,22],"pantograph":[14],"differential":[15,24],"equations":[16,25],"(SPDEs).":[17],"SPDEs":[18],"very":[20],"special":[21],"delay":[23],"(SDDEs)":[26],"unbounded":[28],"memory.":[29],"The":[30,62,117],"problem":[31],"of":[32,67,121,140,146],"computer":[33],"memory":[34],"hold,":[35],"when":[36],"the":[37,46,68,83,92,103,108,122,138,141,151],"constant":[41],"applied":[44],"to":[45,102,136],"SPDEs.":[47,61],"work,":[50],"construct":[52],"split-step":[53],"theta":[54,153],"(SS\u03b8)":[55],"boundedness":[63],"and":[64,78,85,143,157],"strong":[65,109],"convergence":[66],"investigated":[72],"under":[73,114],"a":[74,79],"local":[75],"Lipschitz":[76],"condition":[77,81],"coupled":[80],"on":[82,150],"drift":[84],"diffusion":[86],"coefficients.":[87],"It":[88],"is":[89,112,127],"proved":[90],"that,":[91],"SS\u03b8":[93,147],"\u03b8\u2208[12,1]":[99],"converge":[100],"strongly":[101],"exact":[104],"solution.":[105],"addition,":[107],"order":[110],"0.5":[111],"given":[113],"mild":[115],"assumptions.":[116],"mean-square":[118],"stability":[119],"(MS-Stability)":[120],"\u03b8\u2208(12,1]":[126],"given.":[128],"Finally,":[129],"some":[130],"illustrative":[131],"examples":[133],"presented":[135],"show":[137],"efficiency":[139],"methods,":[142],"how":[144],"MS-Stability":[145],"depends":[149],"parameter":[152],"both":[155],"linear":[156],"nonlinear":[158],"models.":[159]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":3},{"year":2018,"cited_by_count":4}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}