{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T00:58:48Z","timestamp":1781571528099,"version":"3.54.5"},"reference-count":32,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,5,28]],"date-time":"2024-05-28T00:00:00Z","timestamp":1716854400000},"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>This paper presents the numerical solution of the heat conduction model with a fractional derivative of the Riemann\u2013Liouville type with respect to the spatial variable. The considered mathematical model assumes the dependence on temperature of the material parameters (such as specific heat, density, and thermal conductivity) of the model. In the paper, the boundary conditions of the first and second types are considered. If the heat flux equal to zero is assumed on the left boundary, then the thermal symmetry is obtained, which results in a simplification of the problem and the possibility of considering only half the area. The numerical examples presented in the paper illustrate the effectiveness and convergence of the discussed computational method.<\/jats:p>","DOI":"10.3390\/sym16060667","type":"journal-article","created":{"date-parts":[[2024,5,28]],"date-time":"2024-05-28T13:32:55Z","timestamp":1716903175000},"page":"667","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Numerical Solution for the Heat Conduction Model with a Fractional Derivative and Temperature-Dependent Parameters"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7255-6951","authenticated-orcid":false,"given":"Rafa\u0142","family":"Brociek","sequence":"first","affiliation":[{"name":"Department of Mathematics Applications and Methods for Artificial Intelligence, Faculty of Applied Mathematics, Silesian University of Technology, 44-100 Gliwice, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9291-4891","authenticated-orcid":false,"given":"Edyta","family":"Hetmaniok","sequence":"additional","affiliation":[{"name":"Department of Mathematics Applications and Methods for Artificial Intelligence, Faculty of Applied Mathematics, Silesian University of Technology, 44-100 Gliwice, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9265-5711","authenticated-orcid":false,"given":"Damian","family":"S\u0142ota","sequence":"additional","affiliation":[{"name":"Department of Mathematics Applications and Methods for Artificial Intelligence, Faculty of Applied Mathematics, Silesian University of Technology, 44-100 Gliwice, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"7835","DOI":"10.1002\/mma.7229","article-title":"Fractional viscoelastic models with Caputo generalized fractional derivative","volume":"46","author":"Bhangale","year":"2023","journal-title":"Math. 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Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"624","DOI":"10.1016\/j.apm.2022.11.036","article-title":"New description of the mechanical creep response of rocks by fractional derivative theory","volume":"116","author":"Kamdem","year":"2023","journal-title":"Appl. Math. Model."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"118440","DOI":"10.1016\/j.ijheatmasstransfer.2019.118440","article-title":"Comparison of mathematical models with fractional derivative for the heat conduction inverse problem based on the measurements of temperature in porous aluminum","volume":"143","author":"Brociek","year":"2019","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S096249292000001X","article-title":"Numerical methods for nonlocal and fractional models","volume":"29","author":"Du","year":"2020","journal-title":"Acta Numer."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.apnum.2020.04.015","article-title":"Numerical simulation of initial value problems with generalized Caputo-type fractional derivatives","volume":"156","author":"Odibat","year":"2020","journal-title":"Appl. Numer. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1378","DOI":"10.1515\/math-2021-0093","article-title":"On multi-step methods for singular fractional q-integro-differential equations","volume":"19","author":"Hajiseyedazizi","year":"2021","journal-title":"Open Math."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"B\u0142asik, M. (2023). The Implicit Numerical Method for the Radial Anomalous Subdiffusion Equation. Symmetry, 15.","DOI":"10.20944\/preprints202307.2132.v1"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Hou, J., Meng, X., Wang, J., Han, Y., and Yu, Y. (2023). Local Error Estimate of an L1-Finite Difference Scheme for the Multiterm Two-Dimensional Time-Fractional Reaction\u2013Diffusion Equation with Robin Boundary Conditions. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7060453"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1515\/fca-2019-0003","article-title":"A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications","volume":"22","author":"Sun","year":"2019","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"102093","DOI":"10.1016\/j.asej.2022.102093","article-title":"Application of fractional derivatives in a Darcy medium natural convection flow of MHD nanofluid","volume":"14","author":"Khan","year":"2023","journal-title":"Ain Shams Eng. J."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Baleanu, D., and Lopes, A.M. (2019). Volume 7 Applications in Engineering, Life and Social Sciences, Part A, De Gruyter.","DOI":"10.1515\/9783110571929"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1767","DOI":"10.1007\/s10915-019-01062-6","article-title":"Numerical Schemes for Solving the Time-Fractional Dual-Phase-Lagging Heat Conduction Model in a Double-Layered Nanoscale Thin Film","volume":"81","author":"Ji","year":"2019","journal-title":"J. Sci. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1865","DOI":"10.1134\/S0965542522110033","article-title":"Numerical Solution of Two and Three-Dimensional Fractional Heat Conduction Equations via Bernstein Polynomials","volume":"62","author":"Gholizadeh","year":"2022","journal-title":"Comput. Math. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Kukla, S., Siedlecka, U., and Ciesielski, M. (2022). Fractional Order Dual-Phase-Lag Model of Heat Conduction in a Composite Spherical Medium. Materials, 15.","DOI":"10.3390\/ma15207251"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"119405","DOI":"10.1016\/j.applthermaleng.2022.119405","article-title":"Reconstruction of aerothermal heating for the thermal protection system of a reusable launch vehicle","volume":"219","author":"Brociek","year":"2023","journal-title":"Appl. Therm. Eng."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"108229","DOI":"10.1016\/j.ijthermalsci.2023.108229","article-title":"Estimation of aerothermal heating for a thermal protection system with temperature dependent material properties","volume":"188","author":"Brociek","year":"2023","journal-title":"Int. J. Therm. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"124772","DOI":"10.1016\/j.ijheatmasstransfer.2023.124772","article-title":"Identification of aerothermal heating for thermal protection systems taking into account the thermal resistance between layers","volume":"218","author":"Brociek","year":"2024","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Zingales, M., and Alaimo, G. (2014, January 23\u201325). A physical description of fractional-order Fourier diffusion. Proceedings of the ICFDA\u201914 International Conference on Fractional Differentiation and Its Applications 2014, Catania, Italy.","DOI":"10.1109\/ICFDA.2014.6967411"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"418","DOI":"10.2478\/s13540-011-0026-4","article-title":"Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder","volume":"14","author":"Povstenko","year":"2011","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Povstenko, Y. (2015). Fractional Thermoelasticity, Springer.","DOI":"10.1007\/978-3-319-15335-3"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1140\/epjst\/e2011-01394-2","article-title":"Approximate solutions to fractional subdiffusion equations","volume":"193","author":"Hristov","year":"2011","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1515\/fca-2017-0021","article-title":"Determination of two unknown thermal coefficients through an inverse one-phase fractional Stefan problem","volume":"20","author":"Ceretani","year":"2017","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_26","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_27","unstructured":"\u00d6zi\u015fik, M. (1980). Heat Conduction, Wiley & Sons."},{"key":"ref_28","unstructured":"Jaluria, Y., and Torrance, K. (2003). Computational Heat Transfer, Taylor & Francis."},{"key":"ref_29","unstructured":"Mochnacki, B., and Suchy, J. (1995). Numerical Methods in Computations of Foundry Processes, PFTA."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.cam.2004.01.033","article-title":"Finite difference approximations for fractional advection-dispersion flow equations","volume":"172","author":"Meerschaert","year":"2006","journal-title":"J. Comput. Appl. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1140\/epjp\/i2018-11951-x","article-title":"Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator","volume":"133","author":"Owolabi","year":"2018","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Cai, M., and Li, C. (2020). Numerical Approaches to Fractional Integrals and Derivatives: A Review. Mathematics, 8.","DOI":"10.3390\/math8010043"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/667\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:49:39Z","timestamp":1760107779000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/667"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,28]]},"references-count":32,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["sym16060667"],"URL":"https:\/\/doi.org\/10.3390\/sym16060667","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,28]]}}}