Modelo comparativo de la ecuación de calor unidimensional: perspectivas deterministas y estocásticas
Article Sidebar
Main Article Content
Abstract
Introduction: the incorporation of uncertainty in heat conduction models allows us to represent random perturbations present in various physical processes. Objectives: the objective of this research is to compare the deterministic and stochastic formulations of the one-dimensional heat equation under additive Gaussian white noise and to analyze the influence of thermal diffusivity on the propagation of uncertainty. Methodology: a validated and reproducible computational simulation for stochastic partial differential equations was implemented in the MATLAB software, considering a deterministic model using the Crank-Nicolson scheme and a stochastic formulation using the Euler-Maruyama method. Uncertainty was modeled with additive Gaussian white noise and evaluated using 200 Monte Carlo simulations in aluminum, stainless steel, and wood. RMSE, maximum variance, and average thermal energy were used as metrics. Results: the means of the Monte Carlo assembly showed a high agreement with the deterministic solution. The increase in noise intensity increased the statistical dispersion, while the average thermal energy remained practically constant. Conclusions: in conclusion, thermal diffusivity acts as a natural uncertainty dampening mechanism, mainly affecting the variability of the solution without significantly modifying the average dynamics of the diffusive process. General area of study: Applied Mathematics. Specific area of study: Computational simulation. Type of study: Original articles.
Downloads
Article Details
dssfdsf
dsfdsf
References
Adamowicz, A. (2022). Determination of thermal diffusivity values based on the inverse problem of heat conduction – numerical analysis. Acta Mechanica et Automatica, 16(4), 399-407. https://doi.org/10.2478/ama-2022-0048
Anton, R., Cohen, D., & Quer-Sardanyons, L. (2017). A fully discrete approximation of the one-dimensional stochastic heat equation. arXiv [math.NA]. https://doi.org/https://doi.org/10.48550/arXiv.1711.08340
Azorin Penalva, A., & Yaulema Castañeda, J. L. (2021). Case analysis numerical solution of differential equations with uncertainty and applications. Conciencia Digital, 4(3.1), 253-272. https://doi.org/10.33262/concienciadigital.v4i3.1.1828
Bayer, C., Ben Hammouda, C., & Tempone, R. (2024). Multilevel Monte Carlo with numerical smoothing for robust and efficient computation of probabilities and densities. SIAM Journal on Scientific Computing, 46(3), A1514–A1548. https://doi.org/10.1137/22M1495718
Benítez Guarnizo, G. M., Toledo Toledo, J. F., & Martínez Ochoa, L. L. (2025). Implementación de un prototipo de calentador solar para evaluar las variaciones de temperatura producidas en el interior de un cuarto de estudio en la ciudad de Loja. Alfa Publicaciones, 7(2.1), 122–149. https://doi.org/10.33262/ap.v7i2.1.613
Bergman, T. L., Lavine, A. S., Incropera, F. P., & DeWitt, D. P. (2019). Fundamentals of heat and mass transfer. John Wiley & Sons, Inc. https://books.google.com.ec/books/about/Fundamentals_of_Heat_and_Mass_Transfer.html?id=0kAxEAAAQBAJ&redir_esc=y
Bogoi, A., Dan, C.-I., Strătilă, S., Cican, G., & Crunteanu, D.-E. (2023). Assessment of stochastic numerical schemes for stochastic differential equations with “white noise” using itô’s integral. Symmetry, 15(11), 2038. https://doi.org/10.3390/sym15112038
Cîmpean, I., Nachit, Y., & Tudor, C. A. (2026). Quartic variation of the solution to the semilinear stochastic heat equation: Limit behavior and asymptotic independence with respect to the data. Journal of Mathematical Analysis and Applications, 557(1), 130258. https://doi.org/10.1016/j.jmaa.2025.130258
Crank, J., & Nicolson, P. (1947). A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Mathematical Proceedings of the Cambridge Philosophical Society, 43(1), 50–67. https://doi.org/10.1017/S0305004100023197
Dong, J., Zhao, W., & Li, H. (2025). A structure-preserving reduced-order finite difference approach for a class of semilinear stochastic partial differential equations driven by white noise. Journal of Mathematical Analysis and Applications, 552(2), 129807. https://doi.org/10.1016/j.jmaa.2025.129807
Elizalde Pin, R. A., Macao Ortega, J. A., & Marrero Ramírez, S. (2024). Análisis comparativo de consumo de energía en una estación de bombeo de agua con diferentes algoritmos de control voltaje/frecuencia. Alfa Publicaciones, 6(2.2), 48–67. https://doi.org/10.33262/ap.v6i2.2.488
Ermakov, S. & Smilovitsky, M. G. (2021). Monte-Carlo for solving large linear systems of ordinary differential equations. Vestnik of Saint Petersburg University Mathematics Mechanics Astronomy, 8 (1), 37–48. https://doi.org/10.21638/spbu01.2021.104
Esmaeilbeigi, M., Chatrabgoun, O., & Shafa, M. (2019). Numerical solution of time-dependent stochastic partial differential equations using RBF partition of unity collocation method based on finite difference. Engineering Analysis with Boundary Elements, 104, 120–134. https://doi.org/10.1016/j.enganabound.2019.03.013
Evans, L. (2022). Partial differential equations (Vol. 19). American Mathematical Society. https://books.google.com.ec/books/about/Partial_Differential_Equations.html?id=Ott1EAAAQBAJ&redir_esc=y
Ghajar, A. J., & Çengel, Y. A. (2025). Heat and mass transfer: fundamentals & applications. McGraw Hill LLC. https://books.google.com.ec/books/about/Heat_and_Mass_Transfer_Fundamentals_and.html?id=B89MnwEACAAJ&redir_esc=y
Guzmán Clavijo, C. R., & Alvear Calle, D. A. (2021). Application of the thermal balance equation to determine hygrothermal comfort in a single-family house in canton Giron. Ciencia Digital, 5(2), 149-164. https://doi.org/10.33262/cienciadigital.v5i2.1722
Higham, D. J. (2001). An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review, 43(3), 525–546. https://doi.org/10.1137/S0036144500378302
Khattab, A. G., Semary, M. S., Hammad, D. A., & Fareed, A. F. (2024). Exploring stochastic heat equations: a numerical analysis with fast discrete fourier transform techniques. Axioms, 13(12), 886. https://doi.org/10.3390/axioms13120886
Kloeden, P. E., & Platen, E. (1992). Numerical solution of stochastic differential equations. Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-12616-5
Kroese, D. P., Taimre, T., & Botev, Z. I. (2011). Handbook of Monte Carlo methods. Wiley. https://www.wiley.com/en-us/Handbook+of+Monte+Carlo+Methods-p-9781118014943
LeVeque, R. J. (2007). Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems. Society for Industrial and Applied Mathematics (SIAM). https://books.google.com.ec/books/about/Finite_Difference_Methods_for_Ordinary_a.html?id=lZysESfSWwgC
Li, Y., & Su, C. (2022). Stochastic heat transfer analysis and reliability assessment under non-stationary random thermal load using the explicit time-domain method. International Journal of Heat and Mass Transfer, 194, 123011. https://doi.org/10.1016/j.ijheatmasstransfer.2022.123011
Lord, G. J., & Tambue, A. (2018). A modified semi–implicit Euler–Maruyama scheme for finite element discretization of SPDEs with additive noise. Applied Mathematics and Computation, 332, 105–122. https://doi.org/10.1016/j.amc.2018.03.014
Navas Muñoz, M. J., Matovelle Bustos, C., Vélez Arcentales , A., & Córdova, F. (2022). Evaluación de modelos hidráulicos unidimensionales y bidimensionales para la generación de mapas de inundaciones en un río de montaña. Alfa Publicaciones, 4(1), 163–182. https://doi.org/10.33262/ap.v4i1.181
Ogethakpo, A. & Nkonyeasua, I. (2025). Analyzing Computational Approaches for Differential Equations: A Study of MATLAB, Mathematica, and Maple. arXiv [cs.MS]. https://doi.org/10.48550/arXiv.2510.02346
Rice, J. A. (2007). Mathematical statistics and data analysis (3rd ed.). Cengage Learning. https://korivernon.com/documents/MathematicalStatisticsandDataAnalysis3ed.pdf
Seyer, L., Enguehard, F., & Rochais, D. (2024). Deterministic and stochastic approaches for the modeling of conduction-radiation coupling within non-Beerian semi-transparent media. Journal of Quantitative Spectroscopy and Radiative Transfer, 325 (109086), 109086. https://doi.org/10.1016/j.jqsrt.2024.109086
Sterr, B., Mahravan, E., & Kim, D. (2021). Uncertainty quantification of heat transfer in a microchannel heat sink with random surface roughness. International Journal of Heat and Mass Transfer, 174(121307), 121307. https://doi.org/10.1016/j.ijheatmasstransfer.2021.121307
Villasante, A., Fernández-Serrano, Á., Osuna-Sequera, C., & Hermoso, E. (2025). Methodology for stiffness prediction in structural timber using cross-validation RMSE analysis. Journal of Building Engineering, 107(112767), 112767. https://doi.org/10.1016/j.jobe.2025.112767
Welford, B. P. (1962). Note on a method for calculating corrected sums of squares and products semantic scholar. Technometrics, 4(3), 419-420. https://doi.org/10.1080/00401706.1962.10490022
Zhang, A., & Li, Y. (2023). Thermal conductivity of aluminum alloys—a review. Materials, 16(8), 2972. https://doi.org/10.3390/ma16082972