{"id":"https://openalex.org/W2113061500","doi":"https://doi.org/10.1155/2012/349618","title":"A Note on Fourth Order Method for Doubly Singular Boundary Value Problems","display_name":"A Note on Fourth Order Method for Doubly Singular Boundary Value Problems","publication_year":2012,"publication_date":"2012-12-05","ids":{"openalex":"https://openalex.org/W2113061500","doi":"https://doi.org/10.1155/2012/349618","mag":"2113061500"},"language":"lv","primary_location":{"id":"doi:10.1155/2012/349618","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2012/349618","pdf_url":"https://downloads.hindawi.com/archive/2012/349618.pdf","source":{"id":"https://openalex.org/S120697665","display_name":"Advances in Numerical Analysis","issn_l":"1687-9562","issn":["1687-9562","1687-9570"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Numerical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://downloads.hindawi.com/archive/2012/349618.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101630479","display_name":"R.K. Pandey","orcid":"https://orcid.org/0000-0003-1216-0282"},"institutions":[{"id":"https://openalex.org/I145894827","display_name":"Indian Institute of Technology Kharagpur","ror":"https://ror.org/03w5sq511","country_code":"IN","type":"education","lineage":["https://openalex.org/I145894827"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"R. K. Pandey","raw_affiliation_strings":["Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India","institution_ids":["https://openalex.org/I145894827"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5067377863","display_name":"Gopal Gupta","orcid":"https://orcid.org/0000-0001-9727-0362"},"institutions":[{"id":"https://openalex.org/I145894827","display_name":"Indian Institute of Technology Kharagpur","ror":"https://ror.org/03w5sq511","country_code":"IN","type":"education","lineage":["https://openalex.org/I145894827"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"G. K. Gupta","raw_affiliation_strings":["Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India","institution_ids":["https://openalex.org/I145894827"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.17056426,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"2012","issue":null,"first_page":"1","last_page":"21"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9975000023841858,"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":0.9975000023841858,"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":0.9771000146865845,"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":0.9675999879837036,"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/materials-science","display_name":"Materials science","score":0.3896338641643524},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3628159761428833},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.15141937136650085}],"concepts":[{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.3896338641643524},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3628159761428833},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.15141937136650085}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1155/2012/349618","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2012/349618","pdf_url":"https://downloads.hindawi.com/archive/2012/349618.pdf","source":{"id":"https://openalex.org/S120697665","display_name":"Advances in Numerical Analysis","issn_l":"1687-9562","issn":["1687-9562","1687-9570"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Numerical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:93be8adef7af41eb904f19d61090254c","is_oa":false,"landing_page_url":"https://doaj.org/article/93be8adef7af41eb904f19d61090254c","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Advances in Numerical Analysis, Vol 2012 (2012)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/2012/349618","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2012/349618","pdf_url":"https://downloads.hindawi.com/archive/2012/349618.pdf","source":{"id":"https://openalex.org/S120697665","display_name":"Advances in Numerical Analysis","issn_l":"1687-9562","issn":["1687-9562","1687-9570"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Numerical Analysis","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2113061500.pdf","grobid_xml":"https://content.openalex.org/works/W2113061500.grobid-xml"},"referenced_works_count":28,"referenced_works":["https://openalex.org/W42040310","https://openalex.org/W624851556","https://openalex.org/W1976203147","https://openalex.org/W1982929698","https://openalex.org/W1983073767","https://openalex.org/W1996740660","https://openalex.org/W1996986887","https://openalex.org/W2001558750","https://openalex.org/W2009240935","https://openalex.org/W2013137589","https://openalex.org/W2014943752","https://openalex.org/W2023128601","https://openalex.org/W2031925666","https://openalex.org/W2032225945","https://openalex.org/W2033853386","https://openalex.org/W2038973894","https://openalex.org/W2045693757","https://openalex.org/W2048923395","https://openalex.org/W2075201176","https://openalex.org/W2077221255","https://openalex.org/W2085989646","https://openalex.org/W2100864874","https://openalex.org/W2103238585","https://openalex.org/W2106007809","https://openalex.org/W2146747898","https://openalex.org/W2494779131","https://openalex.org/W4244602987","https://openalex.org/W4248164709"],"related_works":["https://openalex.org/W2899084033","https://openalex.org/W2898370298","https://openalex.org/W4292492973","https://openalex.org/W2744391499","https://openalex.org/W3120461830","https://openalex.org/W4230250635","https://openalex.org/W3041790586","https://openalex.org/W2018879842","https://openalex.org/W3144504424","https://openalex.org/W2325423348"],"abstract_inverted_index":{"We":[0],"present":[1],"a":[2],"fourth":[3],"order":[4],"finite":[5,79],"difference":[6],"method":[7,106,116],"for":[8],"doubly":[9],"singular":[10,100],"boundary":[11,30],"value":[12],"problem":[13],"<mml:math":[14,32,50,59,64,70,75,82,88,102],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[15,33,51,60,65,71,76,83,89,103],"id=\"M1\"><mml:mrow><mml:mo":[16],"stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo":[17],"stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo":[18,20,23,44,46,91],"stretchy=\"false\">)</mml:mo><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>\u2032</mml:mi></mml:mrow><mml:mo":[19],"stretchy=\"false\">)</mml:mo><mml:mo":[21],"stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>\u2032</mml:mi><mml:mo>=</mml:mo><mml:mi>q</mml:mi><mml:mo":[22],"stretchy=\"false\">)</mml:mo><mml:mi>f</mml:mi><mml:mo":[24],"stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo":[25],"stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mi>":[26],"</mml:mi><mml:mn>0</mml:mn><mml:mo>&lt;</mml:mo><mml:mi>x</mml:mi><mml:mi>":[27],"</mml:mi><mml:mo>\u2264</mml:mo><mml:mn>1</mml:mn></mml:math>":[28],"with":[29],"conditions":[31,122],"id=\"M2\"><mml:mi>y</mml:mi><mml:mo":[34],"stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo":[35,40,85],"stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>A</mml:mi><mml:mi>":[36],"</mml:mi><mml:mo":[37],"stretchy=\"false\">(</mml:mo><mml:mtext>or</mml:mtext><mml:mi>":[38],"</mml:mi><mml:mi>y</mml:mi><mml:mi>'</mml:mi><mml:mo":[39],"stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mi>":[41],"</mml:mi><mml:mtext>or</mml:mtext><mml:mi>":[42],"</mml:mi><mml:msub><mml:mrow><mml:mi>lim</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>\u2192</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mi>p</mml:mi><mml:mo":[43],"stretchy=\"false\">)</mml:mo><mml:mi>y</mml:mi><mml:mi>'</mml:mi><mml:mo":[45],"stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo":[47],"stretchy=\"false\">)</mml:mo></mml:math>":[48,92],"and":[49,74,87,123],"id=\"M3\"><mml:mi>\u03b1</mml:mi><mml:mi>y</mml:mi><mml:mo":[52],"stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo":[53,56],"stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:mi>":[54],"</mml:mi><mml:mi>\u03b2</mml:mi><mml:mi>y</mml:mi><mml:mi>'</mml:mi><mml:mo":[55],"stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>\u03b3</mml:mi></mml:math>,":[57],"where":[58],"id=\"M4\"><mml:mi>\u03b1</mml:mi><mml:mo":[61],"stretchy=\"false\">(</mml:mo><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn><mml:mo":[62],"stretchy=\"false\">)</mml:mo></mml:math>,":[63,69],"id=\"M5\"><mml:mi>":[66],"</mml:mi><mml:mi>\u03b2</mml:mi><mml:mo":[67],"stretchy=\"false\">(</mml:mo><mml:mo>\u2265</mml:mo><mml:mn>0</mml:mn><mml:mo":[68],"id=\"M6\"><mml:mi>":[72],"</mml:mi><mml:mi>\u03b3</mml:mi></mml:math>":[73],"id=\"M7\"><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:math>":[77],"are":[78],"constants.":[80],"Here":[81],"id=\"M8\"><mml:mi>p</mml:mi><mml:mo":[84],"stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math>":[86],"id=\"M9\"><mml:mi>q</mml:mi><mml:mo":[90],"is":[93,107,117],"allowed":[94],"to":[95],"be":[96],"discontinuous":[97],"at":[98],"the":[99,115],"point":[101],"id=\"M10\"><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math>.":[104],"The":[105,112],"based":[108],"on":[109],"uniform":[110],"mesh.":[111],"accuracy":[113],"of":[114],"established":[118],"under":[119],"quite":[120],"general":[121],"also":[124],"corroborated":[125],"through":[126],"one":[127],"numerical":[128],"example.":[129]},"counts_by_year":[{"year":2020,"cited_by_count":2},{"year":2017,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
