{"id":"https://openalex.org/W7134188048","doi":"https://doi.org/10.1155/jama/9383545","title":"An Efficient Numerical Model for the Black\u2013Scholes Equations","display_name":"An Efficient Numerical Model for the Black\u2013Scholes Equations","publication_year":2026,"publication_date":"2026-01-01","ids":{"openalex":"https://openalex.org/W7134188048","doi":"https://doi.org/10.1155/jama/9383545"},"language":"en","primary_location":{"id":"doi:10.1155/jama/9383545","is_oa":true,"landing_page_url":"https://doi.org/10.1155/jama/9383545","pdf_url":null,"source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"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":"Journal of Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://doi.org/10.1155/jama/9383545","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5128363183","display_name":"Yan Zhou","orcid":null},"institutions":[{"id":"https://openalex.org/I83776822","display_name":"Shandong Institute of Business and Technology","ror":"https://ror.org/03rrkrc24","country_code":"CN","type":"education","lineage":["https://openalex.org/I83776822"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yan Zhou","raw_affiliation_strings":["School of Management ,  Shandong Technology and Business University ,  Yantai ,  Shandong ,  China ,  sdibt.edu.cn"],"raw_orcid":"https://orcid.org/0000-0003-4986-139X","affiliations":[{"raw_affiliation_string":"School of Management ,  Shandong Technology and Business University ,  Yantai ,  Shandong ,  China ,  sdibt.edu.cn","institution_ids":["https://openalex.org/I83776822"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5128369337","display_name":"Yunxing Zhang","orcid":null},"institutions":[{"id":"https://openalex.org/I182707071","display_name":"Ludong University","ror":"https://ror.org/028h95t32","country_code":"CN","type":"education","lineage":["https://openalex.org/I182707071"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yunxing Zhang","raw_affiliation_strings":["School of Hydraulic and Civil Engineering ,  Ludong University ,  Yantai ,  Shandong ,  China ,  ytnc.edu.cn"],"raw_orcid":"https://orcid.org/0000-0001-6993-0802","affiliations":[{"raw_affiliation_string":"School of Hydraulic and Civil Engineering ,  Ludong University ,  Yantai ,  Shandong ,  China ,  ytnc.edu.cn","institution_ids":["https://openalex.org/I182707071"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5128363183"],"corresponding_institution_ids":["https://openalex.org/I83776822"],"apc_list":{"value":1025,"currency":"USD","value_usd":1025},"apc_paid":{"value":1025,"currency":"USD","value_usd":1025},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.35395454,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"2026","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.15219999849796295,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.15219999849796295,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.1290999948978424,"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/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.0940999984741211,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.8995000123977661},{"id":"https://openalex.org/keywords/convection\u2013diffusion-equation","display_name":"Convection\u2013diffusion equation","score":0.6463000178337097},{"id":"https://openalex.org/keywords/diagonal","display_name":"Diagonal","score":0.5217999815940857},{"id":"https://openalex.org/keywords/term","display_name":"Term (time)","score":0.5091999769210815},{"id":"https://openalex.org/keywords/benchmark","display_name":"Benchmark (surveying)","score":0.5065000057220459},{"id":"https://openalex.org/keywords/temporal-discretization","display_name":"Temporal discretization","score":0.4474000036716461},{"id":"https://openalex.org/keywords/diagonally-dominant-matrix","display_name":"Diagonally dominant matrix","score":0.39989998936653137},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.3889000117778778},{"id":"https://openalex.org/keywords/convection","display_name":"Convection","score":0.38850000500679016}],"concepts":[{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.8995000123977661},{"id":"https://openalex.org/C17456955","wikidata":"https://www.wikidata.org/wiki/Q343279","display_name":"Convection\u2013diffusion equation","level":2,"score":0.6463000178337097},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6266000270843506},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.6265000104904175},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.5217999815940857},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.5091999769210815},{"id":"https://openalex.org/C185798385","wikidata":"https://www.wikidata.org/wiki/Q1161707","display_name":"Benchmark (surveying)","level":2,"score":0.5065000057220459},{"id":"https://openalex.org/C2778327290","wikidata":"https://www.wikidata.org/wiki/Q17146496","display_name":"Temporal discretization","level":3,"score":0.4474000036716461},{"id":"https://openalex.org/C84507854","wikidata":"https://www.wikidata.org/wiki/Q952222","display_name":"Diagonally dominant matrix","level":3,"score":0.39989998936653137},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.3889000117778778},{"id":"https://openalex.org/C10899652","wikidata":"https://www.wikidata.org/wiki/Q160329","display_name":"Convection","level":2,"score":0.38850000500679016},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.38420000672340393},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3686999976634979},{"id":"https://openalex.org/C205951836","wikidata":"https://www.wikidata.org/wiki/Q1147751","display_name":"Finite difference method","level":2,"score":0.3596999943256378},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.3325999975204468},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.3262999951839447},{"id":"https://openalex.org/C571446","wikidata":"https://www.wikidata.org/wiki/Q6510183","display_name":"Diffusion equation","level":3,"score":0.3181000053882599},{"id":"https://openalex.org/C69357855","wikidata":"https://www.wikidata.org/wiki/Q163214","display_name":"Diffusion","level":2,"score":0.2989000082015991},{"id":"https://openalex.org/C101303905","wikidata":"https://www.wikidata.org/wiki/Q17104838","display_name":"Numerical solution of the convection\u2013diffusion equation","level":4,"score":0.2962999939918518},{"id":"https://openalex.org/C181330731","wikidata":"https://www.wikidata.org/wiki/Q2068418","display_name":"Finite difference","level":2,"score":0.28679999709129333},{"id":"https://openalex.org/C60866291","wikidata":"https://www.wikidata.org/wiki/Q5140577","display_name":"Coefficient matrix","level":3,"score":0.27230000495910645},{"id":"https://openalex.org/C176321772","wikidata":"https://www.wikidata.org/wiki/Q1430640","display_name":"Numerical stability","level":3,"score":0.2721000015735626},{"id":"https://openalex.org/C113313756","wikidata":"https://www.wikidata.org/wiki/Q332791","display_name":"Diagonal matrix","level":3,"score":0.2720000147819519},{"id":"https://openalex.org/C10184394","wikidata":"https://www.wikidata.org/wiki/Q5165491","display_name":"Continuous modelling","level":2,"score":0.26919999718666077},{"id":"https://openalex.org/C105427703","wikidata":"https://www.wikidata.org/wiki/Q665193","display_name":"Discretization of continuous features","level":4,"score":0.26829999685287476},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.267300009727478},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.257099986076355},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2565000057220459},{"id":"https://openalex.org/C500300565","wikidata":"https://www.wikidata.org/wiki/Q925667","display_name":"Computer simulation","level":2,"score":0.25130000710487366}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1155/jama/9383545","is_oa":true,"landing_page_url":"https://doi.org/10.1155/jama/9383545","pdf_url":null,"source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"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":"Journal of Applied Mathematics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1155/jama/9383545","is_oa":true,"landing_page_url":"https://doi.org/10.1155/jama/9383545","pdf_url":null,"source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"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":"Journal of Applied Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W1506342804","https://openalex.org/W1971954987","https://openalex.org/W2073537687","https://openalex.org/W2077791698","https://openalex.org/W2085051954","https://openalex.org/W2580657778","https://openalex.org/W2604409950","https://openalex.org/W3013876872","https://openalex.org/W3133468063","https://openalex.org/W4206947073","https://openalex.org/W4213061062","https://openalex.org/W4230835142","https://openalex.org/W4231057775","https://openalex.org/W4281791347","https://openalex.org/W4285751563","https://openalex.org/W4286470391","https://openalex.org/W4316126906","https://openalex.org/W4365141810","https://openalex.org/W4379982810","https://openalex.org/W4399007270","https://openalex.org/W4400179546","https://openalex.org/W4402925826","https://openalex.org/W4403337325","https://openalex.org/W4406833470","https://openalex.org/W4408573724","https://openalex.org/W4409716292"],"related_works":[],"abstract_inverted_index":{"In":[0],"this":[1,23],"paper,":[2],"a":[3,36,107,129],"novel":[4],"numerical":[5,161],"model":[6,149,203],"for":[7,66,103,113,143,178],"the":[8,26,29,46,54,61,67,73,78,82,91,95,104,115,121,125,137,144,147,160,172,175,179,186,194,201],"Black\u2013Scholes":[9,31,116],"equations":[10],"is":[11,33,42,57,141,150,169],"developed.":[12],"To":[13],"address":[14],"some":[15],"potential":[16],"issues":[17],"that":[18,120,135,171,200],"may":[19],"arise":[20],"when":[21],"solving":[22,114],"equation":[24,32],"using":[25,60],"conventional":[27,138,208],"model,":[28,139],"original":[30],"reformulated":[34],"as":[35],"convection\u2013diffusion":[37],"equation.":[38],"The":[39],"Crank\u2013Nicolson":[40],"scheme":[41,177],"utilized":[43],"to":[44,90],"discretize":[45],"diffusion":[47,68],"and":[48,53,94,155,159,196],"source":[49],"terms":[50],"in":[51,211],"time,":[52],"convection":[55,83,105,180],"term":[56,69,84,181],"dealt":[58],"explicitly":[59],"Adams\u2013Bashforth":[62],"method.":[63],"Spatial":[64],"derivatives":[65],"are":[70,111,163],"approximated":[71],"with":[72,124,152,165],"central":[74],"difference":[75],"scheme.":[76],"However,":[77],"spatial":[79],"discretization":[80,101,176],"of":[81,109,136,174],"can":[85,190],"be":[86,191],"chosen":[87],"adaptively":[88],"according":[89],"problem":[92],"setting":[93],"parameter":[96],"values.":[97],"By":[98],"employing":[99],"various":[100],"schemes":[102],"term,":[106],"series":[108],"models":[110,127],"constructed":[112],"equations.":[117],"We":[118,198],"demonstrated":[119],"discrete":[122],"matrix":[123],"new":[126,148,202],"has":[128,182],"better":[130],"diagonal":[131],"dominance":[132],"property":[133],"than":[134],"which":[140],"beneficial":[142],"solution.":[145],"Subsequently,":[146],"validated":[151],"several":[153],"single\u2010asset":[154],"multiasset":[156],"benchmark":[157],"problems,":[158],"results":[162],"compared":[164],"analytic":[166],"solutions.":[167],"It":[168],"observed":[170],"choice":[173],"little":[183],"impact":[184],"on":[185,193],"computational":[187],"efficiency":[188],"but":[189],"significant":[192],"accuracy":[195],"robustness.":[197],"find":[199],"shows":[204],"clear":[205],"advantages":[206],"over":[207],"methods,":[209],"particularly":[210],"stable":[212],"market":[213],"conditions.":[214]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2026-03-09T00:00:00"}