{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,6]],"date-time":"2026-07-06T21:16:28Z","timestamp":1783372588257,"version":"3.54.6"},"reference-count":20,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2017,8,18]],"date-time":"2017-08-18T00:00:00Z","timestamp":1503014400000},"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>This communication addresses a comparison of newly presented non-integer order derivatives with and without singular kernel, namely Michele Caputo\u2013Mauro Fabrizio (CF)         C F    (     \u2202 \u03b2   \/  \u2202  t \u03b2     )       and Atangana\u2013Baleanu (AB)         A B    (     \u2202 \u03b1   \/  \u2202  t \u03b1     )       fractional derivatives. For this purpose, second grade fluids flow with combined gradients of mass concentration and temperature distribution over a vertical flat plate is considered. The problem is first written in non-dimensional form and then based on AB and CF fractional derivatives, it is developed in fractional form, and then using the Laplace transform technique, exact solutions are established for both cases of AB and CF derivatives. They are then expressed in terms of newly defined M-function       M  q p   ( z )      and generalized Hyper-geometric function       p    \u03a8  q    ( z )      . The obtained exact solutions are plotted graphically for several pertinent parameters and an interesting comparison is made between AB and CF derivatives results with various similarities and differences.<\/jats:p>","DOI":"10.3390\/e19080279","type":"journal-article","created":{"date-parts":[[2017,8,21]],"date-time":"2017-08-21T04:12:17Z","timestamp":1503288737000},"page":"279","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":76,"title":["Atangana\u2013Baleanu and Caputo Fabrizio Analysis of Fractional Derivatives for Heat and Mass Transfer of Second Grade Fluids over a Vertical Plate: A Comparative Study"],"prefix":"10.3390","volume":"19","author":[{"given":"Arshad","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Computer Science and IT, Sarhad University of Science and IT Peshawar, Peshawar 25000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kashif","family":"Ali Abro","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences and Related Studies, Mehran University of Engineering Technology, Jamshoro 76062, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6165-8055","authenticated-orcid":false,"given":"Asifa","family":"Tassaddiq","sequence":"additional","affiliation":[{"name":"Computer Sciences and Information Technology College, Majmaah University Majmaah, Majmaah 11952, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2056-9371","authenticated-orcid":false,"given":"Ilyas","family":"Khan","sequence":"additional","affiliation":[{"name":"Basic Engineering Sciences Department, College of Engineering Majmaah University, Majmaah 11952, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,8,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1615\/HeatTransRes.2014007241","article-title":"Free convection flow of a second-grade fluid with ramped wall temperature","volume":"45","author":"Samiulhaq","year":"2014","journal-title":"Heat Transf. 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