{"id":"https://openalex.org/W4378229123","doi":"https://doi.org/10.3390/sym15061144","title":"New Numerical Results on Existence of Volterra\u2013Fredholm Integral Equation of Nonlinear Boundary Integro-Differential Type","display_name":"New Numerical Results on Existence of Volterra\u2013Fredholm Integral Equation of Nonlinear Boundary Integro-Differential Type","publication_year":2023,"publication_date":"2023-05-24","ids":{"openalex":"https://openalex.org/W4378229123","doi":"https://doi.org/10.3390/sym15061144"},"language":"en","primary_location":{"id":"doi:10.3390/sym15061144","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15061144","pdf_url":"https://www.mdpi.com/2073-8994/15/6/1144/pdf?version=1684994931","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/15/6/1144/pdf?version=1684994931","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5026883056","display_name":"Hawsar HamaRashid","orcid":null},"institutions":[{"id":"https://openalex.org/I64952554","display_name":"University of Sulaimani","ror":"https://ror.org/00saanr69","country_code":"IQ","type":"education","lineage":["https://openalex.org/I64952554"]}],"countries":["IQ"],"is_corresponding":false,"raw_author_name":"Hawsar HamaRashid","raw_affiliation_strings":["Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq"],"raw_orcid":"https://orcid.org/0000-0002-1493-6518","affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq","institution_ids":["https://openalex.org/I64952554"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5079512627","display_name":"H. M. Srivastava","orcid":"https://orcid.org/0000-0002-9277-8092"},"institutions":[{"id":"https://openalex.org/I184693016","display_name":"China Medical University","ror":"https://ror.org/00v408z34","country_code":"TW","type":"education","lineage":["https://openalex.org/I184693016"]},{"id":"https://openalex.org/I212119943","display_name":"University of Victoria","ror":"https://ror.org/04s5mat29","country_code":"CA","type":"education","lineage":["https://openalex.org/I212119943"]},{"id":"https://openalex.org/I35928602","display_name":"Kyung Hee University","ror":"https://ror.org/01zqcg218","country_code":"KR","type":"education","lineage":["https://openalex.org/I35928602"]},{"id":"https://openalex.org/I4210126829","display_name":"China Medical University Hospital","ror":"https://ror.org/0368s4g32","country_code":"TW","type":"healthcare","lineage":["https://openalex.org/I4210126829"]},{"id":"https://openalex.org/I87490745","display_name":"Azerbaijan University","ror":"https://ror.org/00g1psz15","country_code":"AZ","type":"education","lineage":["https://openalex.org/I87490745"]}],"countries":["AZ","CA","KR","TW"],"is_corresponding":true,"raw_author_name":"Hari Mohan Srivastava","raw_affiliation_strings":["Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea","Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku 1007, Azerbaijan","Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada","Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"],"raw_orcid":"https://orcid.org/0000-0002-9277-8092","affiliations":[{"raw_affiliation_string":"Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea","institution_ids":["https://openalex.org/I35928602"]},{"raw_affiliation_string":"Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku 1007, Azerbaijan","institution_ids":["https://openalex.org/I87490745"]},{"raw_affiliation_string":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada","institution_ids":["https://openalex.org/I212119943"]},{"raw_affiliation_string":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan","institution_ids":["https://openalex.org/I4210126829","https://openalex.org/I184693016"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5076519011","display_name":"Mudhafar Hama","orcid":"https://orcid.org/0000-0001-9012-0943"},"institutions":[{"id":"https://openalex.org/I64952554","display_name":"University of Sulaimani","ror":"https://ror.org/00saanr69","country_code":"IQ","type":"education","lineage":["https://openalex.org/I64952554"]}],"countries":["IQ"],"is_corresponding":false,"raw_author_name":"Mudhafar Hama","raw_affiliation_strings":["Department of Mathematics, College of Science, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq"],"raw_orcid":"https://orcid.org/0000-0001-9012-0943","affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq","institution_ids":["https://openalex.org/I64952554"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5059025015","display_name":"Pshtiwan Othman Mohammed","orcid":"https://orcid.org/0000-0001-6837-8075"},"institutions":[{"id":"https://openalex.org/I64952554","display_name":"University of Sulaimani","ror":"https://ror.org/00saanr69","country_code":"IQ","type":"education","lineage":["https://openalex.org/I64952554"]}],"countries":["IQ"],"is_corresponding":false,"raw_author_name":"Pshtiwan Othman Mohammed","raw_affiliation_strings":["Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq"],"raw_orcid":"https://orcid.org/0000-0001-6837-8075","affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq","institution_ids":["https://openalex.org/I64952554"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034642808","display_name":"Eman Al-Sarairah","orcid":"https://orcid.org/0000-0002-0223-4711"},"institutions":[{"id":"https://openalex.org/I100086675","display_name":"Al-Hussein Bin Talal University","ror":"https://ror.org/019dkd780","country_code":"JO","type":"education","lineage":["https://openalex.org/I100086675"]},{"id":"https://openalex.org/I176601375","display_name":"Khalifa University of Science and Technology","ror":"https://ror.org/05hffr360","country_code":"AE","type":"education","lineage":["https://openalex.org/I176601375"]}],"countries":["AE","JO"],"is_corresponding":false,"raw_author_name":"Eman Al-Sarairah","raw_affiliation_strings":["Department of Mathematics, Al-Hussein Bin Talal University, P.O. Box 20, Ma\u2019an 71111, Jordan","Department of Mathematics, Khalifa University, Abu Dhabi P.O. Box 127788, United Arab Emirates","Department of Mathematics, Al-Hussein Bin Talal University, P.O. Box 20, Ma'an 71111, Jordan"],"raw_orcid":"https://orcid.org/0000-0002-0223-4711","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Al-Hussein Bin Talal University, P.O. Box 20, Ma\u2019an 71111, Jordan","institution_ids":["https://openalex.org/I100086675"]},{"raw_affiliation_string":"Department of Mathematics, Khalifa University, Abu Dhabi P.O. Box 127788, United Arab Emirates","institution_ids":["https://openalex.org/I176601375"]},{"raw_affiliation_string":"Department of Mathematics, Al-Hussein Bin Talal University, P.O. Box 20, Ma'an 71111, Jordan","institution_ids":["https://openalex.org/I100086675"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5034381260","display_name":"Musawa Yahya Almusawa","orcid":"https://orcid.org/0000-0001-8485-7113"},"institutions":[{"id":"https://openalex.org/I91450128","display_name":"Jazan University","ror":"https://ror.org/02bjnq803","country_code":"SA","type":"education","lineage":["https://openalex.org/I91450128"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Musawa Yahya Almusawa","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Jazan University, Jazan 45142, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0001-8485-7113","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Jazan University, Jazan 45142, Saudi Arabia","institution_ids":["https://openalex.org/I91450128"]}]}],"institutions":[],"countries_distinct_count":8,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5079512627"],"corresponding_institution_ids":["https://openalex.org/I184693016","https://openalex.org/I212119943","https://openalex.org/I35928602","https://openalex.org/I4210126829","https://openalex.org/I87490745"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":3.1916,"has_fulltext":true,"cited_by_count":15,"citation_normalized_percentile":{"value":0.91809567,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":95,"max":99},"biblio":{"volume":"15","issue":"6","first_page":"1144","last_page":"1144"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":1.0,"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":1.0,"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/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.9977999925613403,"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":0.9926000237464905,"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":0.7475864291191101},{"id":"https://openalex.org/keywords/integral-equation","display_name":"Integral equation","score":0.657492458820343},{"id":"https://openalex.org/keywords/volterra-integral-equation","display_name":"Volterra integral equation","score":0.6079280376434326},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5837815999984741},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.5103489756584167},{"id":"https://openalex.org/keywords/fredholm-integral-equation","display_name":"Fredholm integral equation","score":0.5092031359672546},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.4885631501674652},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.48568591475486755},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4707750976085663},{"id":"https://openalex.org/keywords/integro-differential-equation","display_name":"Integro-differential equation","score":0.4341011643409729},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3834154009819031},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.3616943061351776}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7475864291191101},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.657492458820343},{"id":"https://openalex.org/C21965488","wikidata":"https://www.wikidata.org/wiki/Q2632613","display_name":"Volterra integral equation","level":3,"score":0.6079280376434326},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5837815999984741},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.5103489756584167},{"id":"https://openalex.org/C89285879","wikidata":"https://www.wikidata.org/wiki/Q2991898","display_name":"Fredholm integral equation","level":3,"score":0.5092031359672546},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.4885631501674652},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.48568591475486755},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4707750976085663},{"id":"https://openalex.org/C70985411","wikidata":"https://www.wikidata.org/wiki/Q2380676","display_name":"Integro-differential equation","level":4,"score":0.4341011643409729},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3834154009819031},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.3616943061351776},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","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/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","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/C45473103","wikidata":"https://www.wikidata.org/wiki/Q851503","display_name":"Riccati equation","level":3,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/sym15061144","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15061144","pdf_url":"https://www.mdpi.com/2073-8994/15/6/1144/pdf?version=1684994931","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:9e78b9942f754d3e997f16f3e35ae2ee","is_oa":false,"landing_page_url":"https://doaj.org/article/9e78b9942f754d3e997f16f3e35ae2ee","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":"Symmetry, Vol 15, Iss 6, p 1144 (2023)","raw_type":"article"},{"id":"pmh:oai:dspace.library.uvic.ca:1828/16005","is_oa":false,"landing_page_url":"http://hdl.handle.net/1828/16005","pdf_url":null,"source":{"id":"https://openalex.org/S4306401509","display_name":"UVic\u2019s Research and Learning Repository (University of Victoria)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I212119943","host_organization_name":"University of Victoria","host_organization_lineage":["https://openalex.org/I212119943"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:mdpi.com:/2073-8994/15/6/1144/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym15061144","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym15061144","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15061144","pdf_url":"https://www.mdpi.com/2073-8994/15/6/1144/pdf?version=1684994931","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4378229123.pdf","grobid_xml":"https://content.openalex.org/works/W4378229123.grobid-xml"},"referenced_works_count":40,"referenced_works":["https://openalex.org/W634523187","https://openalex.org/W1558573846","https://openalex.org/W1559244587","https://openalex.org/W1979966692","https://openalex.org/W1985585033","https://openalex.org/W2002138473","https://openalex.org/W2003806194","https://openalex.org/W2005839730","https://openalex.org/W2006367435","https://openalex.org/W2030974514","https://openalex.org/W2050210825","https://openalex.org/W2055356295","https://openalex.org/W2073947030","https://openalex.org/W2140804801","https://openalex.org/W2164931015","https://openalex.org/W2324290783","https://openalex.org/W2936676491","https://openalex.org/W2964206098","https://openalex.org/W2968751023","https://openalex.org/W3016683849","https://openalex.org/W3027634808","https://openalex.org/W3041144680","https://openalex.org/W3132001005","https://openalex.org/W3133855275","https://openalex.org/W3162085095","https://openalex.org/W3196948123","https://openalex.org/W3203989271","https://openalex.org/W3206623638","https://openalex.org/W3217578944","https://openalex.org/W4303005599","https://openalex.org/W4309686613","https://openalex.org/W4313460522","https://openalex.org/W4366493163","https://openalex.org/W6646846542","https://openalex.org/W6651231001","https://openalex.org/W6651859850","https://openalex.org/W6658054297","https://openalex.org/W6669058023","https://openalex.org/W6680828364","https://openalex.org/W6780381690"],"related_works":["https://openalex.org/W2017640945","https://openalex.org/W2401195586","https://openalex.org/W2184420462","https://openalex.org/W2945803575","https://openalex.org/W2003647985","https://openalex.org/W4367557662","https://openalex.org/W2118560105","https://openalex.org/W2946528495","https://openalex.org/W2517687681","https://openalex.org/W4240464390"],"abstract_inverted_index":{"Symmetry":[0],"is":[1,14],"presented":[2],"in":[3,16],"many":[4],"works":[5],"involving":[6],"differential":[7],"and":[8,39,47,79,105,127,141,165,169],"integral":[9,21,74],"equations.":[10],"Whenever":[11],"a":[12,62,70,113],"human":[13],"involved":[15],"the":[17,36,51,56,59,81,98,106,132,135,139,144,167],"design":[18],"of":[19,53,83,143,149,158,171],"an":[20],"equation,":[22],"they":[23],"naturally":[24],"tend":[25],"to":[26,111,160],"opt":[27],"for":[28,69,146],"symmetric":[29],"features.":[30],"The":[31,90,152],"most":[32,125],"common":[33,95],"examples":[34,157],"are":[35,43,124],"Green":[37],"functions":[38],"linguistic":[40],"kernels":[41],"that":[42,121,137],"often":[44],"designed":[45],"symmetrically":[46],"regularly":[48],"distributed":[49],"over":[50],"universe":[52],"discourse.":[54],"In":[55,129],"current":[57],"study,":[58],"authors":[60,91,133,153],"report":[61],"study":[63],"on":[64],"boundary":[65],"value":[66],"problem":[67],"(BVP)":[68],"nonlinear":[71],"integro":[72],"Volterra\u2013Fredholm":[73],"equation":[75],"with":[76],"variable":[77],"coefficients":[78],"show":[80,155],"existence":[82,140],"solution":[84,145],"by":[85,103],"applying":[86],"some":[87],"fixed-point":[88],"theorems.":[89],"employ":[92],"various":[93,147],"numerical":[94,114,156],"approaches":[96],"as":[97],"homotopy":[99],"analysis":[100],"methodology":[101],"established":[102],"Liao":[104],"modified":[107],"Adomain":[108],"decomposition":[109],"technique":[110],"produce":[112],"approximate":[115],"solution,":[116],"then":[117],"graphical":[118],"depiction":[119],"reveals":[120],"both":[122],"methods":[123],"effective":[126],"convenient.":[128],"this":[130],"regard,":[131],"address":[134],"requirements":[136],"ensure":[138],"uniqueness":[142],"variations":[148],"nonlinearity":[150],"power.":[151],"also":[154],"how":[159],"apply":[161],"our":[162,172],"primary":[163],"theorems":[164],"test":[166],"convergence":[168],"validity":[170],"suggested":[173],"approach.":[174]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":8},{"year":2023,"cited_by_count":3}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}