{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T04:49:15Z","timestamp":1773722955716,"version":"3.50.1"},"reference-count":51,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2021,9,29]],"date-time":"2021-09-29T00:00:00Z","timestamp":1632873600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The aim of this work is to solve the case study singular model involving the Neumann\u2013Robin, Dirichlet, and Neumann boundary conditions using a novel computing framework that is based on the artificial neural network (ANN), global search genetic algorithm (GA), and local search sequential quadratic programming method (SQPM), i.e., ANN-GA-SQPM. The inspiration to present this numerical framework comes through the objective of introducing a reliable structure that associates the operative ANNs features using the optimization procedures of soft computing to deal with such stimulating systems. Four different problems that are based on the singular equations involving Neumann\u2013Robin, Dirichlet, and Neumann boundary conditions have been occupied to scrutinize the robustness, stability, and proficiency of the designed ANN-GA-SQPM. The proposed results through ANN-GA-SQPM have been compared with the exact results to check the efficiency of the scheme through the statistical performances for taking fifty independent trials. Moreover, the study of the neuron analysis based on three and 15 neurons is also performed to check the authenticity of the proposed ANN-GA-SQPM.<\/jats:p>","DOI":"10.3390\/s21196498","type":"journal-article","created":{"date-parts":[[2021,10,8]],"date-time":"2021-10-08T21:26:20Z","timestamp":1633728380000},"page":"6498","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Artificial Neural Networks to Solve the Singular Model with Neumann\u2013Robin, Dirichlet and Neumann Boundary Conditions"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8092-4665","authenticated-orcid":false,"given":"Kashif","family":"Nisar","sequence":"first","affiliation":[{"name":"Faculty of Computing and Informatics, Universiti Malaysia Sabah, Jalan UMS, Kota Kinabalu Sabah 88400, Malaysia"}]},{"given":"Zulqurnain","family":"Sabir","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Hazara University, Mansehra 21120, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9953-822X","authenticated-orcid":false,"given":"Muhammad","family":"Asif Zahoor Raja","sequence":"additional","affiliation":[{"name":"Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, Douliou, Yunlin 64002, Taiwan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1793-5905","authenticated-orcid":false,"given":"Ag Asri","family":"Ag Ibrahim","sequence":"additional","affiliation":[{"name":"Faculty of Computing and Informatics, Universiti Malaysia Sabah, Jalan UMS, Kota Kinabalu Sabah 88400, Malaysia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8657-3800","authenticated-orcid":false,"given":"Joel","family":"J. P. C. Rodrigues","sequence":"additional","affiliation":[{"name":"PPGEE, Federal University of Piau\u00ed (UFPI), Teresina 64049-550, Brazil"},{"name":"Covilh\u00e3 Delegation, Instituto de Telecomunica\u00e7\u00f5es, 6201-001 Covilh\u00e3, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7008-1366","authenticated-orcid":false,"given":"Samy","family":"Refahy Mahmoud","sequence":"additional","affiliation":[{"name":"GRC Department, Faculty of Applied Studies, Jeddah, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4340-9602","authenticated-orcid":false,"given":"Bhawani Shankar","family":"Chowdhry","sequence":"additional","affiliation":[{"name":"NCRA Condition Monitoring Systems Lab, Mehran University of Engineering and Technology, Jamshoro 76020, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4274-4927","authenticated-orcid":false,"given":"Manoj","family":"Gupta","sequence":"additional","affiliation":[{"name":"Department of Electronics and Communication Engineering, JECRC University Jaipur, Rajasthan 303905, India"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1137\/1017036","article-title":"On the generalized Emden\u2013Fowler equation","volume":"17","author":"Wong","year":"1975","journal-title":"Siam Rev."},{"key":"ref_2","first-page":"543","article-title":"Adomian decomposition method for a reliable treatment of the Emden\u2013Fowler equation","volume":"161","author":"Wazwaz","year":"2005","journal-title":"Appl. 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