{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T20:50:41Z","timestamp":1780087841027,"version":"3.54.0"},"reference-count":75,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,16]],"date-time":"2021-08-16T00:00:00Z","timestamp":1629072000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this study, a novel application of neurocomputing technique is presented for solving nonlinear heat transfer and natural convection porous fin problems arising in almost all areas of engineering and technology, especially in mechanical engineering. The mathematical models of the problems are exploited by the intelligent strength of Euler polynomials based Euler neural networks (ENN\u2019s), optimized with a generalized normal distribution optimization (GNDO) algorithm and Interior point algorithm (IPA). In this scheme, ENN\u2019s based differential equation models are constructed in an unsupervised manner, in which the neurons are trained by GNDO as an effective global search technique and IPA, which enhances the local search convergence. Moreover, a temperature distribution of heat transfer and natural convection porous fin are investigated by using an ENN-GNDO-IPA algorithm under the influence of variations in specific heat, thermal conductivity, internal heat generation, and heat transfer rate, respectively. A large number of executions are performed on the proposed technique for different cases to determine the reliability and effectiveness through various performance indicators including Nash\u2013Sutcliffe efficiency (NSE), error in Nash\u2013Sutcliffe efficiency (ENSE), mean absolute error (MAE), and Thiel\u2019s inequality coefficient (TIC). Extensive graphical and statistical analysis shows the dominance of the proposed algorithm with state-of-the-art algorithms and numerical solver RK-4.<\/jats:p>","DOI":"10.3390\/e23081053","type":"journal-article","created":{"date-parts":[[2021,8,16]],"date-time":"2021-08-16T21:27:45Z","timestamp":1629149265000},"page":"1053","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":54,"title":["Application of Euler Neural Networks with Soft Computing Paradigm to Solve Nonlinear Problems Arising in Heat Transfer"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8680-723X","authenticated-orcid":false,"given":"Naveed Ahmad","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, KP, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Osamah Ibrahim","family":"Khalaf","sequence":"additional","affiliation":[{"name":"Al-Nahrain Nanorenewable Energy Research Center Baghdad, Al-Nahrain University, Baghdad 10001, Iraq"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4606-7222","authenticated-orcid":false,"given":"Carlos Andr\u00e9s Tavera","family":"Romero","sequence":"additional","affiliation":[{"name":"COMBA R&D Laboratory, Faculty of Engineering, Universidad Santiago de Cali, Cali 76001, Colombia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4040-6211","authenticated-orcid":false,"given":"Muhammad","family":"Sulaiman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, KP, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9512-7191","authenticated-orcid":false,"given":"Maharani A.","family":"Bakar","sequence":"additional","affiliation":[{"name":"Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, Kuala Nerus 21300, Terengganu, Malaysia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1016\/j.icheatmasstransfer.2011.03.025","article-title":"The application of differential transformation method to nonlinear equations arising in heat transfer","volume":"38","author":"Yaghoobi","year":"2011","journal-title":"Int. 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