{"id":"https://openalex.org/W2010465708","doi":"https://doi.org/10.1080/00207160212108","title":"On some 4-Point Spline Collocation Methods for Solving Ordinary Initial Value Problems","display_name":"On some 4-Point Spline Collocation Methods for Solving Ordinary Initial Value Problems","publication_year":2002,"publication_date":"2002-01-01","ids":{"openalex":"https://openalex.org/W2010465708","doi":"https://doi.org/10.1080/00207160212108","mag":"2010465708"},"language":"en","primary_location":{"id":"doi:10.1080/00207160212108","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160212108","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":null,"display_name":"H. M. El-Hawary","orcid":null},"institutions":[{"id":"https://openalex.org/I91041137","display_name":"Assiut University","ror":"https://ror.org/01jaj8n65","country_code":"EG","type":"education","lineage":["https://openalex.org/I91041137"]}],"countries":["EG"],"is_corresponding":true,"raw_author_name":"H. M. El-Hawary","raw_affiliation_strings":["a   Dept. of Math., Faculty of Science , Assuit University ,  Assuit ,  Egypt  E-mail:","Dept. of Math., Faculty of Science , Assuit University , Assuit, Egypt E-mail: elhawary@aun.eun.eg"],"affiliations":[{"raw_affiliation_string":"a   Dept. of Math., Faculty of Science , Assuit University ,  Assuit ,  Egypt  E-mail:","institution_ids":["https://openalex.org/I91041137"]},{"raw_affiliation_string":"Dept. of Math., Faculty of Science , Assuit University , Assuit, Egypt E-mail: elhawary@aun.eun.eg","institution_ids":["https://openalex.org/I91041137"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5110247571","display_name":"S. A. Mahmoud","orcid":"https://orcid.org/0009-0007-1762-8637"},"institutions":[{"id":"https://openalex.org/I91041137","display_name":"Assiut University","ror":"https://ror.org/01jaj8n65","country_code":"EG","type":"education","lineage":["https://openalex.org/I91041137"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"S. M. Mahmoud","raw_affiliation_strings":["b   Dept. of Math., Faculty of Science , Assuit University ,  Assuit ,  Egypt","Dept. of Math., Faculty of Science , Assuit University , Assuit , Egypt"],"affiliations":[{"raw_affiliation_string":"b   Dept. of Math., Faculty of Science , Assuit University ,  Assuit ,  Egypt","institution_ids":["https://openalex.org/I91041137"]},{"raw_affiliation_string":"Dept. of Math., Faculty of Science , Assuit University , Assuit , Egypt","institution_ids":["https://openalex.org/I91041137"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I91041137"],"apc_list":null,"apc_paid":null,"fwci":1.5656,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.80801491,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"79","issue":"8","first_page":"931","last_page":"944"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","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/T11416","display_name":"Numerical methods for differential equations","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/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9984999895095825,"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":0.9984999895095825,"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":0.727280855178833},{"id":"https://openalex.org/keywords/collocation","display_name":"Collocation (remote sensing)","score":0.597001850605011},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.5542457103729248},{"id":"https://openalex.org/keywords/collocation-method","display_name":"Collocation method","score":0.5412435531616211},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4965851902961731},{"id":"https://openalex.org/keywords/initial-value-problem","display_name":"Initial value problem","score":0.46572041511535645},{"id":"https://openalex.org/keywords/beta","display_name":"BETA (programming language)","score":0.4556993246078491},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4345967471599579},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.37494850158691406}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.727280855178833},{"id":"https://openalex.org/C80023036","wikidata":"https://www.wikidata.org/wiki/Q5147531","display_name":"Collocation (remote sensing)","level":2,"score":0.597001850605011},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.5542457103729248},{"id":"https://openalex.org/C2126413","wikidata":"https://www.wikidata.org/wiki/Q1779479","display_name":"Collocation method","level":4,"score":0.5412435531616211},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4965851902961731},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.46572041511535645},{"id":"https://openalex.org/C2776174256","wikidata":"https://www.wikidata.org/wiki/Q830842","display_name":"BETA (programming language)","level":2,"score":0.4556993246078491},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4345967471599579},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.37494850158691406},{"id":"https://openalex.org/C127313418","wikidata":"https://www.wikidata.org/wiki/Q1069","display_name":"Geology","level":0,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C62649853","wikidata":"https://www.wikidata.org/wiki/Q199687","display_name":"Remote sensing","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/00207160212108","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160212108","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":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W284380720","https://openalex.org/W301056823","https://openalex.org/W1971092129","https://openalex.org/W1972574198","https://openalex.org/W1977449329","https://openalex.org/W1979146918","https://openalex.org/W2019230842","https://openalex.org/W2027748182","https://openalex.org/W2042388195","https://openalex.org/W2050572505","https://openalex.org/W2051998357","https://openalex.org/W2053673060","https://openalex.org/W2055896346","https://openalex.org/W2064946172","https://openalex.org/W2085562953","https://openalex.org/W2131785782","https://openalex.org/W2989404328","https://openalex.org/W3021591933","https://openalex.org/W4255552441"],"related_works":["https://openalex.org/W2078318892","https://openalex.org/W2791210196","https://openalex.org/W1987365920","https://openalex.org/W322101810","https://openalex.org/W2384941625","https://openalex.org/W2889856026","https://openalex.org/W2009058751","https://openalex.org/W1989122168","https://openalex.org/W2131278000","https://openalex.org/W2953063785"],"abstract_inverted_index":{"This":[0],"paper":[1],"discusses":[2],"collocation":[3,13],"schemes":[4],"based":[5],"on":[6,144],"seventh":[7],"C":[8],"3":[9],"-splines":[10],"with":[11,122],"four":[12],"points":[14],"s_{i-1+\\alpha}=x_{i-1}+\\alpha":[15],"h":[16,19,22,124,142],",":[17,20,33,103,105,157,159],"x_{i-1+\\beta}=x_{i-1}+\\beta":[18],"x_{i-1+\\theta}=x_{i-1}+\\theta":[21],"and":[23,74,97],"x_i=x_{i-1}+h":[24],"in":[25,46],"each":[26],"subinterval":[27],"[":[28],"x":[29,34],"i":[30,35,37],"@":[31],"1":[32,62,131,162],"],":[36],"=":[38],"1(1)":[39],"N":[40],"for":[41],"solving":[42],"initial":[43],"value":[44],"problems":[45],"ordinary":[47],"differential":[48],"equations":[49],"including":[50],"stiff":[51],"equations.":[52],"Here":[53],"0":[54,96],"<":[55,57,59,61,107,126,128,130],"f":[56,102,125,141,156],"g":[58,104,127,158],"\u00e8":[60,106,129,160],"are":[63,72,99,117,134,138],"arbitrarily":[64],"given.":[65],"It":[66,119],"is":[67,79],"shown":[68],"that":[69,121],"the":[70,75,110,115,132,147],"methods":[71,116,133],"convergent":[73],"order":[76],"of":[77,114,149,151],"convergence":[78],"seven":[80],"if:":[81],"\\eqalign":[82],"{&\\beta(\\theta":[83],"-\\theta^2)+\\beta^2(\\theta^2-\\theta)+\\alpha^2[\\theta^2-\\theta":[84],"+\\beta^2(1+4\\theta":[85],"-5\\theta^2)+\\theta(4\\theta^2-4\\theta":[86],"-1)]\\cr":[87],"&\\qquad":[88],"\\qquad":[89,90],"+\\alpha[\\theta":[91],"-\\theta^2+\\beta":[92],"(1+4\\theta":[93],"-4\\theta^2)+\\beta^2(4\\theta^2+4\\theta":[94],"-1)]\\le":[95],"they":[98,137],"unstable":[100],"if":[101,140],"0.7279115.":[108],"Moreover,":[109],"absolute":[111,152],"stability":[112,153],"properties":[113],"considered.":[118],"shows":[120],"0.888035":[123],"A-stable":[135],"while":[136],"not":[139],"0.5;":[143],"other":[145],"hand,":[146],"sizes":[148],"regions":[150],"increase":[154],"when":[155],"M":[161],"m.":[163],"Keywords:":[164],"Initial":[165],"Value":[166],"ProblemsSpline":[167],"FunctionsAbsolute":[168],"StabilityConsistencyStiff":[169],"Differential":[170],"Equation":[171]},"counts_by_year":[],"updated_date":"2026-04-16T08:26:57.006410","created_date":"2025-10-10T00:00:00"}
