{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T03:20:08Z","timestamp":1776309608030,"version":"3.50.1"},"reference-count":44,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,10,29]],"date-time":"2021-10-29T00:00:00Z","timestamp":1635465600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the present work, an unsteady convection flow of Casson fluid, together with an oscillating vertical plate, is examined. The governing PDEs corresponding to velocity and temperature profile are transformed into linear ODEs with the help of the Laplace transform method. The ordinary derivative model generalized to fractional model is based on a generalized Fourier law. The solutions for energy and velocity equations are obtained after making the equations dimensionless. To check the insight of the physical parameters, especially the symmetric behavior of fractional parameters, it is found that for small and large values of time, fluid properties show dual behavior. Since the fractional derivative exhibits the memory of the function at the chosen value of time, therefore the present fractional model is more suitable in exhibiting memory than the classical model. Such results can be useful in the fitting of real data where needed. In the limiting case when fractional parameters are taken \u03b2=\u03b3 = 0 and \u03b1 = 1 for both velocity and temperature, we get the solutions obtained with ordinary derivatives from the existing literature.<\/jats:p>","DOI":"10.3390\/sym13112039","type":"journal-article","created":{"date-parts":[[2021,11,2]],"date-time":"2021-11-02T22:17:23Z","timestamp":1635891443000},"page":"2039","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["A Prabhakar Fractional Approach for the Convection Flow of Casson Fluid across an Oscillating Surface Based on the Generalized Fourier Law"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0051-1041","authenticated-orcid":false,"given":"Noman","family":"Sarwar","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1484-5114","authenticated-orcid":false,"given":"Muhammad Imran","family":"Asjad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8455-1402","authenticated-orcid":false,"given":"Thanin","family":"Sitthiwirattham","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand"}]},{"given":"Nichaphat","family":"Patanarapeelert","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}]},{"given":"Taseer","family":"Muhammad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences, King Khalid University, Abha 61413, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,29]]},"reference":[{"key":"ref_1","unstructured":"Casson, N. (1959). A flow equation for pigment-oil suspensions of the printing ink type. Rheol. Disperse Syst., Pergamon Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/j.asej.2013.05.003","article-title":"Casson fluid flow and heat transfer past an exponentially porous stretching surface in presence of thermal radiation","volume":"5","author":"Pramanik","year":"2014","journal-title":"Ain Shams Eng. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1145","DOI":"10.1016\/0020-7225(96)00012-2","article-title":"Casson fluid flow in a pipe filled with a homogeneous porous medium","volume":"34","author":"Dash","year":"1996","journal-title":"Int. J. Eng. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2865","DOI":"10.1016\/j.aej.2019.12.023","article-title":"A new model of fractional Casson fluid based on generalized Fick\u2019s and Fourier\u2019s laws together with heat and mass transfer","volume":"59","author":"Sheikh","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_5","first-page":"373","article-title":"A new approach for the solutions of the fractional generalized Casson fluid model described by Caputo fractional operator","volume":"4","author":"Ndolane","year":"2020","journal-title":"Adv. Theory Nonlinear Anal. Its Appl."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Arif, M., Kumam, P., Kumam, W., Khan, I., and Ramzan, M. (2021). A Fractional Model of Casson Fluid with Ramped Wall Temperature: Engineering Applications of Engine Oil. Comput. Math. Methods, e1162.","DOI":"10.1002\/cmm4.1162"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2565","DOI":"10.1007\/s13369-017-2521-3","article-title":"Solutions with Wright function for time fractional free convection flow of Casson fluid","volume":"42","author":"Ali","year":"2017","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"101191","DOI":"10.1016\/j.csite.2021.101191","article-title":"Thermal activity of conventional Casson nanoparticles with ramped temperature due to an infinite vertical plate via fractional derivative approach","volume":"27","author":"Raza","year":"2021","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjp\/i2016-16181-8","article-title":"Unsteady flow of generalized Casson fluid with fractional derivative due to an infinite plate","volume":"131","author":"Khan","year":"2016","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Saqib, M., Shafie, S., Khan, I., Chu, Y.M., and Nisar, K.S. (2020). Symmetric MHD channel flow of nonlocal fractional model of BTF containing hybrid nanoparticles. Symmetry, 12.","DOI":"10.3390\/sym12040663"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"862","DOI":"10.1016\/j.ijheatmasstransfer.2017.05.006","article-title":"Heat transfer enhancement by induced vortices in the vicinity of a rotationally oscillating heated plate","volume":"112","author":"Koffi","year":"2017","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"104528","DOI":"10.1016\/j.icheatmasstransfer.2020.104528","article-title":"Experimental investigation of mixed convection on an oscillating vertical flat plate","volume":"113","author":"Akcay","year":"2020","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"101266","DOI":"10.1016\/j.csite.2021.101266","article-title":"Significances of exponential heating and Darcy\u2019s law for second grade fluid flow over oscillating plate by using Atangana-Baleanu fractional derivatives","volume":"27","author":"Song","year":"2021","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Ali, F., Khan, I., and Shafie, S. (2014). Closed form solutions for unsteady free convection flow of a second grade fluid over an oscillating vertical plate. PLoS ONE, 9.","DOI":"10.1371\/journal.pone.0085099"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Haider, M.I., Asjad, M.I., Ali, R., Ghaemi, F., and Ahmadian, A. (2021). Heat transfer analysis of micropolar hybrid nanofluid over an oscillating vertical plate and Newtonian heating. J. Therm. Anal. Calorim., 144.","DOI":"10.1007\/s10973-021-10698-2"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjc\/s10052-016-4209-3","article-title":"Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo\u2013Fabrizio derivatives","volume":"76","author":"Shah","year":"2016","journal-title":"Eur. Phys. J. C"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.molliq.2018.02.079","article-title":"Effect of thermal diffusion and heat-generation on MHD nanofluid flow past an oscillating vertical plate through porous medium","volume":"257","author":"Sheikholeslami","year":"2018","journal-title":"J. Mol. Liq."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1443","DOI":"10.1016\/j.camwa.2020.07.009","article-title":"A fast method for variable-order Caputo fractional derivative with applications to time-fractional diffusion equations","volume":"80","author":"Fang","year":"2020","journal-title":"Comput. Math. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"884","DOI":"10.1002\/mma.5410","article-title":"Optimal solutions for singular linear systems of Caputo fractional differential equations","volume":"44","author":"Dassios","year":"2021","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s12043-020-01997-8","article-title":"Power law memory of natural convection flow of hybrid nanofluids with constant proportional Caputo fractional derivative due to pressure gradient","volume":"94","author":"Ali","year":"2020","journal-title":"Pramana"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"109705","DOI":"10.1016\/j.chaos.2020.109705","article-title":"A new study on the mathematical modelling of human liver with Caputo\u2013Fabrizio fractional derivative","volume":"134","author":"Baleanu","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1186\/s13662-020-2527-0","article-title":"Analyzing transient response of the parallel RCL circuit by using the Caputo\u2013Fabrizio fractional derivative","volume":"2020","author":"Alizadeh","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2305","DOI":"10.1016\/j.aej.2020.02.022","article-title":"On a nonlinear fractional order model of dengue fever disease under Caputo-Fabrizio derivative","volume":"59","author":"Shah","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2141","DOI":"10.1016\/j.aej.2020.01.030","article-title":"Analysis of some generalized ABC\u2013fractional logistic models","volume":"59","author":"Abdeljawad","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"103507","DOI":"10.1016\/j.rinp.2020.103507","article-title":"Study of transmission dynamics of COVID-19 mathematical model under ABC fractional order derivative","volume":"19","author":"Thabet","year":"2020","journal-title":"Results Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"123516","DOI":"10.1016\/j.physa.2019.123516","article-title":"A new application of fractional Atangana\u2013Baleanu derivatives: Designing ABC-fractional masks in image processing","volume":"542","author":"Ghanbari","year":"2020","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_27","first-page":"576","article-title":"Hilfer\u2013Prabhakar derivatives and some applications","volume":"242","author":"Garra","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Benchohra, M., Bouriah, S., and Nieto, J.J. (2019). Terminal value problem for differential equations with Hilfer\u2013Katugampola fractional derivative. Symmetry, 11.","DOI":"10.3390\/sym11050672"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Asif, M., Ul Haq, S., Islam, S., Abdullah Alkanhal, T., Khan, Z.A., Khan, I., and Nisar, K.S. (2019). Unsteady flow of fractional fluid between two parallel walls with arbitrary wall shear stress using Caputo\u2013Fabrizio derivative. Symmetry, 11.","DOI":"10.3390\/sym11040449"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Polito, F., and Tomovski, Z. (2015). Some properties of Prabhakar-type fractional calculus operators. arXiv.","DOI":"10.7153\/fdc-06-05"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Elnaqeeb, T., Shah, N.A., and Mirza, I.A. (2020). Natural convection flows of carbon nanotubes nanofluids with Prabhakar-like thermal transport. Math. Methods Appl. Sci.","DOI":"10.1002\/mma.6584"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2245","DOI":"10.1007\/s10973-020-09835-0","article-title":"Natural convection flows of Prabhakar-like fractional Maxwell fluids with generalized thermal transport","volume":"143","author":"Shah","year":"2021","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40314-020-01296-3","article-title":"Stability and dynamics of neutral and integro-differential regularized Prabhakar fractional differential systems","volume":"39","author":"Eshaghi","year":"2020","journal-title":"Comput. Appl. Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2327","DOI":"10.1007\/s10973-021-10643-3","article-title":"Thermal analysis of free convection flows of viscous carbon nanotubes nanofluids with generalized thermal transport: A Prabhakar fractional model","volume":"144","author":"Tanveer","year":"2021","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1007\/s12190-019-01277-8","article-title":"Stability region of fractional differential systems with Prabhakar derivative","volume":"62","author":"Alidousti","year":"2020","journal-title":"J. Appl. Math. Comput."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"8817794","DOI":"10.1155\/2021\/8817794","article-title":"New Numerical Algorithm to Solve Variable-Order Fractional Integrodifferential Equations in the Sense of Hilfer-Prabhakar Derivative","volume":"2021","author":"Derakhshan","year":"2021","journal-title":"Abstr. Appl. Anal."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Asjad, M.I., Sarwar, N., Hafeez, M.B., Sumelka, W., and Muhammad, T. (2021). Advancement of Non\u2013Newtonian Fluid with Hybrid Nanoparticles in a Convective Channel and Prabhakar\u2019s Fractional Derivative\u2014Analytical Solution. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5030099"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Basit, A., Asjad, M.I., and Akg\u00fcl, A. (2021). Convective flow of a fractional second grade fluid containing different nanoparticles with Prabhakar fractional derivative subject to non-uniform velocity at the boundary. Math. Methods Appl. Sci.","DOI":"10.1002\/mma.7461"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1292","DOI":"10.1016\/j.jmrt.2021.07.029","article-title":"Unsteady thermal transport flow of Casson nanofluids with generalized Mittag\u2013Leffler kernel of Prabhakar\u2019s type","volume":"14","author":"Wang","year":"2021","journal-title":"J. Mater. Res. Technol."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Sene, N., and Srivastava, G. (2019). Generalized Mittag\u2013Leffler input stability of the fractional differential equations. Symmetry, 11.","DOI":"10.3390\/sym11050608"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Iyiola, O., Oduro, B., Zabilowicz, T., Iyiola, B., and Kenes, D. (2021). System of time fractional models for COVID-19: Modeling, analysis and solutions. Symmetry, 13.","DOI":"10.20944\/preprints202104.0052.v1"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Khalid, A., Khan, I., and Shafie, S. (2015). Exact solutions for unsteady free convection flow of Casson fluid over an oscillating vertical plate with constant wall temperature. Abstract and Applied Analysis, Hindawi.","DOI":"10.1155\/2015\/946350"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"114001","DOI":"10.1088\/1402-4896\/ac102b","article-title":"Advances in transport phenomena with nanoparticles and generalized thermal process for vertical plate","volume":"96","author":"Asjad","year":"2021","journal-title":"Phys. Scr."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1016\/j.cnsns.2017.08.018","article-title":"The Prabhakar or three parameter Mittag\u2013Leffler function: Theory and application","volume":"56","author":"Garra","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2039\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:22:31Z","timestamp":1760167351000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2039"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,29]]},"references-count":44,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["sym13112039"],"URL":"https:\/\/doi.org\/10.3390\/sym13112039","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,10,29]]}}}