{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,12]],"date-time":"2026-03-12T00:46:05Z","timestamp":1773276365616,"version":"3.50.1"},"reference-count":28,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,9,26]],"date-time":"2021-09-26T00:00:00Z","timestamp":1632614400000},"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>The transient temperature distribution through a convective-radiative moving rod with temperature-dependent internal heat generation and non-linearly varying temperature-dependent thermal conductivity is elaborated in this investigation. Symmetries are intrinsic and fundamental features of the differential equations of mathematical physics. The governing energy equation subjected to corresponding initial and boundary conditions is non-dimensionalized into a non-linear partial differential equation (PDE) with the assistance of relevant non-dimensional terms. Then the resultant non-dimensionalized PDE is solved analytically using the two-dimensional differential transform method (2D DTM) and multivariate Pade approximant. The consequential impact of non-dimensional parameters such as heat generation, radiative, temperature ratio, and conductive parameters on dimensionless transient temperature profiles has been scrutinized through graphical elucidation. Furthermore, these graphs indicate the deviations in transient thermal profile for both finite difference method (FDM) and 2D DTM-multivariate Pade approximant by considering the forced convective and nucleate boiling heat transfer mode. The results reveal that the transient temperature profile of the moving rod upsurges with the change in time, and it improves for heat generation parameter. It enriches for the rise in the magnitude of Peclet number but drops significantly for greater values of the convective-radiative and convective-conductive parameters.<\/jats:p>","DOI":"10.3390\/sym13101793","type":"journal-article","created":{"date-parts":[[2021,9,27]],"date-time":"2021-09-27T23:08:31Z","timestamp":1632784111000},"page":"1793","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":41,"title":["Analysis of Transient Thermal Distribution in a Convective\u2013Radiative Moving Rod Using Two-Dimensional Differential Transform Method with Multivariate Pade Approximant"],"prefix":"10.3390","volume":"13","author":[{"given":"Ganeshappa","family":"Sowmya","sequence":"first","affiliation":[{"name":"Department of PG Mathematics, The National College (Auto), Bangalore 560070, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6542-0490","authenticated-orcid":false,"given":"Ioannis E.","family":"Sarris","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, University of West Attica, 12244 Athens, Greece"}]},{"given":"Chandra Sen","family":"Vishalakshi","sequence":"additional","affiliation":[{"name":"Department of PG Mathematics, Government Science College, Chitradurga 577501, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1107-7743","authenticated-orcid":false,"given":"Ravikumar Shashikala Varun","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Studies and Research in Mathematics, Davangere University, Davangere 577002, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1950-4666","authenticated-orcid":false,"given":"Ballajja Chandrappa","family":"Prasannakumara","sequence":"additional","affiliation":[{"name":"Department of Studies and Research in Mathematics, Davangere University, Davangere 577002, India"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Khan, N.S., Usman, A.H., Sohail, A., Hussanan, A., Shah, Q., Ullah, N., Kumam, P., Thounthong, P., and Humphries, U.W. (2021). A Framework for the Magnetic Dipole Effect on the Thixotropic Nanofluid Flow Past a Continuous Curved Stretched Surface. Crystals, 11.","DOI":"10.3390\/cryst11060645"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-020-77419-x","article-title":"Impact of Cattaneo-Christov heat flux model on MHD hybrid nano-micropolar fluid flow and heat transfer with viscous and joule dissipation effects","volume":"11","author":"Tassaddiq","year":"2021","journal-title":"Sci. Rep."},{"key":"ref_3","unstructured":"Kumar, R.S.V., Dhananjaya, P.G., Kumar, R.N., Gowda, R.J.P., and Prasannakumara, B.C. (2021). Modeling and theoretical investigation on Casson nanofluid flow over a curved stretching surface with the influence of magnetic field and chemical reaction. Int. J. Comput. Methods Eng. Sci. Mech., 1\u20138."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"100982","DOI":"10.1016\/j.csite.2021.100982","article-title":"Flow and heat transfer of hybrid nanofluid induced by an exponentially stretching\/shrinking curved surface","volume":"25","author":"Wahid","year":"2021","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Yusuf, T., Mabood, F., Prasannakumara, B., and Sarris, I. (2021). Magneto-Bioconvection Flow of Williamson Nanofluid over an Inclined Plate with Gyrotactic Microorganisms and Entropy Generation. Fluids, 6.","DOI":"10.3390\/fluids6030109"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1007\/s10973-020-09720-w","article-title":"Cu\u2013Al2O3\u2013H2O hybrid nanofluid flow with melting heat transfer, irreversibility analysis and nonlinear thermal radiation","volume":"143","author":"Mabood","year":"2021","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"705","DOI":"10.1016\/j.applthermaleng.2016.04.121","article-title":"Convection\u2013radiation heat transfer study of moving fin with temperature-dependent thermal conductivity, heat transfer coefficient and heat generation","volume":"103","author":"Dogonchi","year":"2016","journal-title":"Appl. Therm. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"5002","DOI":"10.1002\/htj.21864","article-title":"Thermal performance of straight porous fin with variable thermal conductivity under magnetic field and radiation effects","volume":"49","author":"Madhura","year":"2020","journal-title":"Heat Transf."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1139\/cjp-2019-0435","article-title":"Exact solution of a non-linear fin problem of temperature-dependent thermal conductivity and heat transfer coefficient","volume":"98","author":"Sun","year":"2020","journal-title":"Can. J. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Das, R., and Kundu, B. (2021). Prediction of Heat-Generation and Electromagnetic Parameters from Temperature Response in Porous Fins. J. Thermophys. Heat Transf., 1\u20139.","DOI":"10.2514\/1.T6224"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1193","DOI":"10.1016\/0017-9310(94)90205-4","article-title":"Analytical solution for the transient temperature distribution in a moving rod or plate of finite length with surface heat transfer","volume":"37","author":"Choudhury","year":"1994","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1523","DOI":"10.1016\/j.ijthermalsci.2011.03.014","article-title":"Convection-radiation from a continuously moving, variable thermal conductivity sheet or rod undergoing thermal processing","volume":"50","author":"Aziz","year":"2011","journal-title":"Int. J. Therm. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/j.ijthermalsci.2014.12.019","article-title":"Spectral collocation method for convective\u2013radiative transfer of a moving rod with variable thermal conductivity","volume":"90","author":"Sun","year":"2015","journal-title":"Int. J. Therm. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Sarwe, D.U., Shanker, B., Mishra, R., Kumar, R.S.V., and Shekar, M.N.R. (2021). Simultaneous impact of magnetic and Arrhenius activation energy on the flow of Casson hybrid nanofluid over a vertically moving plate. Int. J. Thermofluid Sci. Technol., 8.","DOI":"10.36963\/IJTST.2021080202"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1574","DOI":"10.1007\/s10891-020-02262-w","article-title":"Application of an Integral Numerical Technique for a Temperature-Dependent Thermal Conductivity Fin with Internal Heat Generation","volume":"93","author":"Onyejekwe","year":"2020","journal-title":"J. Eng. Phys. Thermophys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjp\/s13360-020-00206-0","article-title":"A new analytical solution of longitudinal fin with variable heat generation and thermal conductivity using DRA","volume":"135","author":"Kezzar","year":"2020","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Biswal, B., Sarkar, B., and Mahanta, P. (2020). New Approach for Determining Fin Performances of an Annular Disc Fin with Internal Heat Generation. Advances in Mechanical Engineering, Springer. Lecture Notes in Mechanical Engineering.","DOI":"10.1007\/978-981-15-0124-1"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Venkitesh, V., and Mallick, A. (2020). Thermal analysis of a convective\u2013conductive\u2013radiative annular porous fin with variable thermal parameters and internal heat generation. J. Therm. Anal. Calorim., 1\u201315.","DOI":"10.1007\/s10973-020-10384-9"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Sowmya, G., and Gireesha, B.J. (2021). Thermal stresses and efficiency analysis of a radial porous fin with radiation and variable thermal conductivity and internal heat generation. J. Therm. Anal. Calorim., 1\u201312.","DOI":"10.1007\/s10973-021-10801-7"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/j.jppr.2014.01.005","article-title":"Approximate solution of the nonlinear heat transfer equation of a fin with the power-law temperature-dependent thermal conductivity and heat transfer coefficient","volume":"3","author":"Mosayebidorcheh","year":"2014","journal-title":"Propuls. Power Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1513","DOI":"10.1002\/mma.3585","article-title":"General exact solution of the fin problem with the power law temperature-dependent thermal conductivity","volume":"39","author":"Kader","year":"2016","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"403","DOI":"10.4028\/www.scientific.net\/DDF.387.403","article-title":"Predicting the Temperature Distribution in Longitudinal Fins of Various Profiles with Power Law Thermal Properties Using the Variational Iteration Method","volume":"387","author":"Ndlovu","year":"2018","journal-title":"Defect Diffus. Forum"},{"key":"ref_23","unstructured":"Zhou, J.K. (1986). Differential Transformation and Its Applications for Electrical Circuits, Huazhong University Press."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2010\/491319","article-title":"Using Differential Transform Method and Pad\u00e9 Approximant for Solving MHD Flow in a Laminar Liquid Film from a Horizontal Stretching Surface","volume":"2010","author":"Rashidi","year":"2010","journal-title":"Math. Probl. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.molliq.2013.12.034","article-title":"Squeezing Cu\u2013water nanofluid flow analysis between parallel plates by DTM-Pad\u00e9 Method","volume":"193","author":"Domairry","year":"2014","journal-title":"J. Mol. Liq."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"105213","DOI":"10.1088\/1402-4896\/ac0c94","article-title":"Thermal behaviour of annular hyperbolic fin with temperature dependent thermal conductivity by differential transformation method and Pade approximant","volume":"96","author":"Sarwe","year":"2021","journal-title":"Phys. Scr."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1002\/fld.2305","article-title":"Multivariate pad\u00e9 approximation for solving partial differential equations (PDE)","volume":"66","author":"Turut","year":"2011","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2013\/746401","article-title":"Multivariate Pad\u00e9 Approximation for Solving Nonlinear Partial Differential Equations of Fractional Order","volume":"2013","author":"Turut","year":"2013","journal-title":"Abstr. Appl. Anal."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/10\/1793\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:05:18Z","timestamp":1760166318000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/10\/1793"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,26]]},"references-count":28,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2021,10]]}},"alternative-id":["sym13101793"],"URL":"https:\/\/doi.org\/10.3390\/sym13101793","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9,26]]}}}