{"id":"https://openalex.org/W7143413835","doi":"https://doi.org/10.55452/1998-6688-2026-23-1-220-230","title":"ASYMPTOTIC CONVERGENCE OF THE CAUCHY PROBLEM SOLUTION FOR AN INTEGRO\u2013DIFFERENTIAL EQUATION WITH A SMALL PARAMETER","display_name":"ASYMPTOTIC CONVERGENCE OF THE CAUCHY PROBLEM SOLUTION FOR AN INTEGRO\u2013DIFFERENTIAL EQUATION WITH A SMALL PARAMETER","publication_year":2026,"publication_date":"2026-03-29","ids":{"openalex":"https://openalex.org/W7143413835","doi":"https://doi.org/10.55452/1998-6688-2026-23-1-220-230"},"language":"en","primary_location":{"id":"doi:10.55452/1998-6688-2026-23-1-220-230","doi":"https://doi.org/10.55452/1998-6688-2026-23-1-220-230","is_oa":true,"landing_page_url":"https://doi.org/10.55452/1998-6688-2026-23-1-220-230","pdf_url":"https://vestnik.kbtu.edu.kz/jour/article/download/2516/716","source":{"id":"https://openalex.org/S4220651845","display_name":"Herald of Kazakh-British technical university","issn_l":"1998-6688","issn":["1998-6688","2959-8109"],"is_oa":true,"is_in_doaj":true,"is_core":false,"listed_in":["doaj"],"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":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Herald of the Kazakh-British Technical University","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://vestnik.kbtu.edu.kz/jour/article/download/2516/716","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5019551907","display_name":"N.U. Bukanay","orcid":"https://orcid.org/0009-0009-2206-2302","observed_orcids":["https://orcid.org/0009-0009-2206-2302"]},"institutions":[{"id":"https://openalex.org/I185571130","display_name":"Al-Farabi Kazakh National University","ror":"https://ror.org/03q0vrn42","country_code":"KZ","type":"education","lineage":["https://openalex.org/I185571130"]}],"countries":["KZ"],"is_corresponding":false,"raw_author_name":"N. U. Bukanay","raw_affiliation_strings":["Al-Farabi Kazakh National University"],"raw_orcid":"https://orcid.org/0009-0009-2206-2302","affiliations":[{"raw_affiliation_string":"Al-Farabi Kazakh National University","institution_ids":["https://openalex.org/I185571130"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5103699447","display_name":"A. E. Mirzakulova","orcid":"https://orcid.org/0000-0001-6445-6371","observed_orcids":["https://orcid.org/0000-0001-6445-6371"]},"institutions":[{"id":"https://openalex.org/I185571130","display_name":"Al-Farabi Kazakh National University","ror":"https://ror.org/03q0vrn42","country_code":"KZ","type":"education","lineage":["https://openalex.org/I185571130"]}],"countries":["KZ"],"is_corresponding":false,"raw_author_name":"A. E. Mirzakulova","raw_affiliation_strings":["Al-Farabi Kazakh National University"],"raw_orcid":"https://orcid.org/0000-0001-6445-6371","affiliations":[{"raw_affiliation_string":"Al-Farabi Kazakh National University","institution_ids":["https://openalex.org/I185571130"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5067096607","display_name":"A. T. Assanova","orcid":"https://orcid.org/0000-0001-8697-8920","observed_orcids":["https://orcid.org/0000-0001-8697-8920"]},"institutions":[{"id":"https://openalex.org/I4210151480","display_name":"Institute of Mathematics and Mathematical Modeling","ror":"https://ror.org/05xx3wf87","country_code":"KZ","type":"government","lineage":["https://openalex.org/I4210151480"]}],"countries":["KZ"],"is_corresponding":false,"raw_author_name":"A. T. Assanova","raw_affiliation_strings":["Institute of Mathematics and Mathematical Modeling"],"raw_orcid":"https://orcid.org/0000-0001-8697-8920","affiliations":[{"raw_affiliation_string":"Institute of Mathematics and Mathematical Modeling","institution_ids":["https://openalex.org/I4210151480"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.941,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.77338945,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":95},"biblio":{"volume":"23","issue":"1","first_page":"220","last_page":"230"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12727","display_name":"Singular Perturbation, Asymptotics and Difference Schemes","score":0.9839000105857849,"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":"Singular Perturbation, Asymptotics and Difference Schemes","score":0.9839000105857849,"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":0.011599999852478504,"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/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.003599999938160181,"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/cauchy-problem","display_name":"Cauchy problem","score":0.8880000114440918},{"id":"https://openalex.org/keywords/integro-differential-equations","display_name":"integro-differential equations","score":0.9750000238418579},{"id":"https://openalex.org/keywords/small-parameter","display_name":"small parameter","score":0.9570000171661377},{"id":"https://openalex.org/keywords/asymptotic-convergence","display_name":"asymptotic convergence","score":0.972000002861023},{"id":"https://openalex.org/keywords/higher-order-derivatives","display_name":"higher-order derivatives","score":0.4909999966621399}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8439000248908997},{"id":"https://openalex.org/C153635880","wikidata":"https://www.wikidata.org/wiki/Q1412436","display_name":"Cauchy problem","level":3,"score":0.6345999836921692},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6341999769210815},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.5795000195503235},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.4968999922275543},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.4097000062465668},{"id":"https://openalex.org/C7321624","wikidata":"https://www.wikidata.org/wiki/Q205","display_name":"Infinity","level":2,"score":0.4058000147342682},{"id":"https://openalex.org/C49344536","wikidata":"https://www.wikidata.org/wiki/Q726441","display_name":"Cauchy distribution","level":2,"score":0.3982999920845032},{"id":"https://openalex.org/C22219631","wikidata":"https://www.wikidata.org/wiki/Q2434565","display_name":"Homogeneous differential equation","level":5,"score":0.39419999718666077},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.387800008058548},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.3698999881744385},{"id":"https://openalex.org/C185184677","wikidata":"https://www.wikidata.org/wiki/Q1366795","display_name":"Exact differential equation","level":4,"score":0.36660000681877136},{"id":"https://openalex.org/C54067925","wikidata":"https://www.wikidata.org/wiki/Q3607962","display_name":"Elliptic partial differential equation","level":3,"score":0.3635999858379364},{"id":"https://openalex.org/C111615704","wikidata":"https://www.wikidata.org/wiki/Q2627459","display_name":"Hyperbolic partial differential equation","level":3,"score":0.3580999970436096},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.35089999437332153},{"id":"https://openalex.org/C106947605","wikidata":"https://www.wikidata.org/wiki/Q3278575","display_name":"Cauchy boundary condition","level":4,"score":0.34540000557899475},{"id":"https://openalex.org/C64057670","wikidata":"https://www.wikidata.org/wiki/Q4381442","display_name":"First-order partial differential equation","level":3,"score":0.3434999883174896},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.33980000019073486},{"id":"https://openalex.org/C70985411","wikidata":"https://www.wikidata.org/wiki/Q2380676","display_name":"Integro-differential equation","level":4,"score":0.3240000009536743},{"id":"https://openalex.org/C155237215","wikidata":"https://www.wikidata.org/wiki/Q7894123","display_name":"Universal differential equation","level":5,"score":0.3174999952316284},{"id":"https://openalex.org/C82006148","wikidata":"https://www.wikidata.org/wiki/Q913764","display_name":"Cauchy's integral formula","level":4,"score":0.3010999858379364},{"id":"https://openalex.org/C70915906","wikidata":"https://www.wikidata.org/wiki/Q1058681","display_name":"Differential operator","level":2,"score":0.29789999127388},{"id":"https://openalex.org/C76835800","wikidata":"https://www.wikidata.org/wiki/Q859102","display_name":"Cauchy's convergence test","level":5,"score":0.29510000348091125},{"id":"https://openalex.org/C188198153","wikidata":"https://www.wikidata.org/wiki/Q1613840","display_name":"Limiting","level":2,"score":0.29089999198913574},{"id":"https://openalex.org/C13982400","wikidata":"https://www.wikidata.org/wiki/Q1129902","display_name":"Linear differential equation","level":3,"score":0.28839999437332153},{"id":"https://openalex.org/C155940","wikidata":"https://www.wikidata.org/wiki/Q7886967","display_name":"Uniqueness theorem for Poisson's equation","level":3,"score":0.28360000252723694},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.2703000009059906},{"id":"https://openalex.org/C520416788","wikidata":"https://www.wikidata.org/wiki/Q5419229","display_name":"Exact solutions in general relativity","level":2,"score":0.26579999923706055},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.2614000141620636},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.259799987077713}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.55452/1998-6688-2026-23-1-220-230","doi":"https://doi.org/10.55452/1998-6688-2026-23-1-220-230","is_oa":true,"landing_page_url":"https://doi.org/10.55452/1998-6688-2026-23-1-220-230","pdf_url":"https://vestnik.kbtu.edu.kz/jour/article/download/2516/716","source":{"id":"https://openalex.org/S4220651845","display_name":"Herald of Kazakh-British technical university","issn_l":"1998-6688","issn":["1998-6688","2959-8109"],"is_oa":true,"is_in_doaj":true,"is_core":false,"listed_in":["doaj"],"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":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Herald of the Kazakh-British Technical University","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.55452/1998-6688-2026-23-1-220-230","doi":"https://doi.org/10.55452/1998-6688-2026-23-1-220-230","is_oa":true,"landing_page_url":"https://doi.org/10.55452/1998-6688-2026-23-1-220-230","pdf_url":"https://vestnik.kbtu.edu.kz/jour/article/download/2516/716","source":{"id":"https://openalex.org/S4220651845","display_name":"Herald of Kazakh-British technical university","issn_l":"1998-6688","issn":["1998-6688","2959-8109"],"is_oa":true,"is_in_doaj":true,"is_core":false,"listed_in":["doaj"],"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":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Herald of the Kazakh-British Technical University","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7143413835.pdf","grobid_xml":"https://content.openalex.org/works/W7143413835.grobid-xml"},"referenced_works_count":10,"referenced_works":["https://openalex.org/W1967183859","https://openalex.org/W2073097401","https://openalex.org/W2168752276","https://openalex.org/W4225551248","https://openalex.org/W4390876426","https://openalex.org/W4391966646","https://openalex.org/W4392359523","https://openalex.org/W4393342854","https://openalex.org/W4405885397","https://openalex.org/W4411839884"],"related_works":["https://openalex.org/W2891853201","https://openalex.org/W2998942741","https://openalex.org/W2005177264","https://openalex.org/W1985383922","https://openalex.org/W4411839884","https://openalex.org/W3004741258","https://openalex.org/W2077242229","https://openalex.org/W4400354908","https://openalex.org/W2916009102","https://openalex.org/W2151159707"],"abstract_inverted_index":{"The":[0,73,207],"Cauchy":[1,74],"problem":[2,75,132,139,147,182],"at":[3],"point":[4],"0,":[5],"where":[6],"the":[7,13,17,30,33,36,47,59,67,94,97,109,113,117,145,148,157,177,180,192,195,210],"characteristic":[8,68,98],"equation":[9,20,79,99,119],"built":[10],"according":[11],"to":[12,39,112],"homogeneous":[14],"part":[15,111],"of":[16,32,35,49,86,96,116,179],"linear":[18],"integro-differential":[19],"under":[21],"consideration":[22],"has":[23,41],"both":[24],"positive":[25],"and":[26,29,120,166,185,197,202],"negative":[27],"roots,":[28],"order":[31,154],"derivative":[34],"integral":[37,110],"uo":[38],"2,":[40],"never":[42],"been":[43],"examined":[44],"before":[45],"in":[46,84],"theory":[48],"equations":[50],"with":[51,80,140],"small":[52,82,142],"parameters.":[53],"It":[54],"is":[55,90,133,152,162,183,188],"well":[56],"known":[57],"that":[58],"problem\u2019s":[60],"solution":[61,178],"may":[62],"veer":[63],"towards":[64],"infinity":[65],"when":[66,93],"equation\u2019s":[69],"roots":[70,95],"are":[71,100],"opposite.":[72,101],"for":[76,135,176],"a":[77,81,136,141,163],"differential":[78,118,150,159],"parameter":[83],"front":[85],"two":[87],"higherorder":[88],"derivatives":[89],"still":[91],"uncertain":[92],"This":[102,161],"can":[103],"be":[104],"solved":[105],"analytically":[106],"by":[107,204],"adding":[108],"right-hand":[114],"side":[115],"treating":[121],"it":[122],"as":[123],"an":[124,130,173,205],"integrodifferential":[125],"equation.":[126],"In":[127,144,170],"this":[128,171],"paper,":[129],"unperturbed":[131,146,181,198],"constructed":[134],"given":[137],"perturbed":[138,196],"parameter.":[143],"external":[149],"operator":[151],"one":[153],"lower":[155],"than":[156],"internal":[158],"operator.":[160],"non-standard":[164],"case":[165],"requires":[167],"special":[168],"consideration.":[169],"regard,":[172],"analytical":[174],"formula":[175],"obtained,":[184],"further":[186],"analysis":[187],"carried":[189],"out.":[190],"Moreover,":[191],"interrelation":[193],"between":[194],"problems":[199],"was":[200,213],"established":[201],"illustrated":[203],"example.":[206],"theorem":[208],"on":[209],"limiting":[211],"transition":[212],"also":[214],"formulated.":[215]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-03-31T00:00:00"}
