{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T02:55:45Z","timestamp":1768445745753,"version":"3.49.0"},"reference-count":61,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T00:00:00Z","timestamp":1627603200000},"content-version":"vor","delay-in-days":210,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100007270","name":"University of Michigan","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100007270","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>The main aim of this study is to introduce a 2\u2010layered artificial neural network (ANN) for solving the Black\u2013Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed to change the model into a sequence of ordinary differential equations (ODE). Subsequently, each of these ODEs is solved with the aid of an ANN. Adam optimization is employed as the learning paradigm since it can add the foreknowledge of slowing down the process of optimization when getting close to the actual optimum solution. The model also takes advantage of fine\u2010tuning for speeding up the process and domain mapping to confront the infinite domain issue. Finally, the accuracy, speed, and convergence of the method for solving several types of the Black\u2013Scholes model are reported.<\/jats:p>","DOI":"10.1155\/2021\/5511396","type":"journal-article","created":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T19:38:29Z","timestamp":1627673909000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Novel ANN Method for Solving Ordinary and Time\u2010Fractional Black\u2013Scholes Equation"],"prefix":"10.1155","volume":"2021","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6172-2766","authenticated-orcid":false,"given":"Saeed","family":"Bajalan","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7380-2491","authenticated-orcid":false,"given":"Nastaran","family":"Bajalan","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2021,7,30]]},"reference":[{"key":"e_1_2_9_1_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2015.08.016"},{"key":"e_1_2_9_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2015.08.016"},{"key":"e_1_2_9_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2015.08.016"},{"key":"e_1_2_9_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2015.08.016"},{"key":"e_1_2_9_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2014.11.016"},{"key":"e_1_2_9_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2015.08.016"},{"key":"e_1_2_9_7_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10614-015-9506-7"},{"key":"e_1_2_9_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2017.01.019"},{"key":"e_1_2_9_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2017.03.019"},{"key":"e_1_2_9_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10614-016-9561-8"},{"key":"e_1_2_9_11_2","first-page":"89","article-title":"Collocation method based on modified cubic b-spline for option pricing models","volume":"22","author":"Rashidinia J.","year":"2017","journal-title":"Mathematical Communications"},{"key":"e_1_2_9_12_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00009-017-0913-y"},{"key":"e_1_2_9_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2017.07.033"},{"key":"e_1_2_9_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2006.08.071"},{"key":"e_1_2_9_15_2","first-page":"51","article-title":"The fractional black\u2013scholes equation","volume":"3","author":"Wyss W.","year":"2000","journal-title":"Fractional Calculus and Applied Analysis"},{"key":"e_1_2_9_16_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00780-004-0144-5"},{"key":"e_1_2_9_17_2","volume-title":"Stochastic Models for Fractional Calculus","author":"Meerschaert A.","year":"2012"},{"key":"e_1_2_9_18_2","doi-asserted-by":"publisher","DOI":"10.1007\/s40314-019-0957-7"},{"key":"e_1_2_9_19_2","doi-asserted-by":"publisher","DOI":"10.1090\/S0033-569X-2014-01373-2"},{"key":"e_1_2_9_20_2","first-page":"1","article-title":"Analytical solution of fractional black\u2013scholes european option pricing equation by using laplace transform","volume":"2","author":"Kumar S.","year":"2012","journal-title":"Computational and Applied Mathematics"},{"key":"e_1_2_9_21_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2009.05.015"},{"key":"e_1_2_9_22_2","unstructured":"LiH. 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