{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,22]],"date-time":"2025-10-22T03:06:13Z","timestamp":1761102373798,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,4,3]],"date-time":"2019-04-03T00:00:00Z","timestamp":1554249600000},"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 context of the development of the research project (Boulaaras et al.) which was supported by the Deanship of Scientific Research of Qassim University under the Project No. 1140 during the academic year 2016\/1437, this research will introduce a two dimensional mathematical model based on the finite elements spatial method combined with a finite differences time scheme applied to the diffusion equations. This model will be applied in the hypothetical example where the obtained results will be compared with the real experimental data. Such a comparison will allow to predict the soil temperature for different depths and at different time periods according to certain conditions on the weather and take into account the stability conditions of the used numerical method. It\u2019s worthy of note that some of the concreted mathematical problems will be addressed by the experimental scientific theories of functional analysis and then the numerical simulation of the theoretical study will be provided thanks to sophisticated methods with more convergence and stability. This mathematical model would be reduced by means of physical measurement and also in charge of the physical cost.<\/jats:p>","DOI":"10.3390\/sym11040478","type":"journal-article","created":{"date-parts":[[2019,4,4]],"date-time":"2019-04-04T03:13:42Z","timestamp":1554347622000},"page":"478","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Two-Dimensional Mathematical Model of Heat Propagation Equations and Their Significance for Soil Temperature"],"prefix":"10.3390","volume":"11","author":[{"given":"Anis Ben","family":"Dhahbi","sequence":"first","affiliation":[{"name":"Physics Department, College of Science and Arts at ArRass, Qassim University, P.O. Box 53, ArRass 51921, Saudi Arabia"},{"name":"Unit of Research in Nuclear and High Energy Physics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunis 2092, Tunisia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3458-2403","authenticated-orcid":false,"given":"Salah","family":"Boulaaras","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Science and Arts at ArRass, Qassim University, P.O. Box 53, ArRass 51921, Saudi Arabia"},{"name":"Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1 Ahmed Benbella, Oran 31000, Algeria"}]},{"given":"Taha","family":"Radwan","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Science and Arts at ArRass, Qassim University, P.O. Box 53, ArRass 51921, Saudi Arabia"},{"name":"Department of Mathematics and Statistics, Port Said University, Port Said 42511, Egypt"}]},{"given":"Nadia","family":"Mezouar","sequence":"additional","affiliation":[{"name":"Faculty of Economics, University Mustapha Stambouli of Mascara, Mascara 29000, Algeria"}]},{"given":"Khaled","family":"Zennir","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Science and Arts at ArRass, Qassim University, P.O. Box 53, ArRass 51921, Saudi Arabia"}]},{"given":"Mohamed","family":"Haiour","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Annaba University, P.O. Box 12, Annaba 23000, Algeria"}]},{"given":"Ali","family":"Allahem","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University, P.O. Box 53, Buraydah 51921, Saudi Arabia"}]},{"given":"Sewelem","family":"Ghanem","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Science and Arts at ArRass, Qassim University, P.O. Box 53, ArRass 51921, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2019,4,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.jcp.2004.01.004","article-title":"Dispersive and dissipative behavior of high order Discontinuous Galerkin finite element methods","volume":"198","author":"Ainsworth","year":"2004","journal-title":"J. Comput. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"5203","DOI":"10.1002\/mma.5071","article-title":"Existence of positive weak solutions for a class of Kirrchoff elliptic systems with multiple parameters","volume":"41","author":"Boulaaras","year":"2018","journal-title":"Math. Meth. Appl. 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