This study presents an integrated approach that combines the Levenberg–Marquardt (LM) and Luus–Jaakola (LJ) algorithms to enhance parameter estimation for various applications. The LM algorithm, known for its precision in solving non-linear least squares problems, is effectively paired with the LJ algorithm, a robust stochastic optimization method, to improve accuracy and computational efficiency. This hybrid LM-LJ methodology is demonstrated through several case studies, including the optimization of MESH equations in distillation processes, modeling controlled diffusion in biopolymer films, and analyzing heat and mass transfer during the drying of cylindrical quince slices. By overcoming the convergence issues typical of gradient-based methods and performing global searches without initial parameter bounds, this approach effectively handles complex models and closely aligns simulation results with experimental data. These capabilities highlight the versatility of this approach in engineering and environmental modeling, significantly enhancing parameter estimation in systems governed by differential equations.

En este proyecto se caracterizaron diferentes formulaciones de hidromiel artesanal, evaluando la producción de alcohol y la actividad antioxidante. Para ello, se prepararon tres variantes de hidromiel: natural, con infusión de jamaica-manzanilla y con infusión de jengibre-menta. El proceso de fermentación se llevó a cabo utilizando miel de mezquite y la cepa de levadura (Saccharomyces bayanus) durante tres semanas. Además, se compararon las formulaciones con una variante con infusión de jamaica-manzanilla madurada durante 18 meses. Se evaluaron propiedades como color, acidez y la capacidad antioxidante mediante la técnica de 1,1-difenil-2-pricrilhidrazilo (DPPH). Además, se emplearon técnicas cromatográficas para un análisis preliminar de la producción de alcohol y se realizó la modelación del consumo de biomasa, sustrato y generación de producto a partir de la medición del consumo de azúcar con los grados Brix, estimando estos parámetros mediante la solución de Ecuaciones Diferenciales Ordinarias (EDOs) en MATLAB. Los resultados mostraron variaciones significativas en las propiedades organolépticas y antioxidantes según la formulación y el proceso de maduración.

In the field of applied sciences, systems are frequently modeled using mathematical frameworks that include systems of nonlinear algebraic equations. Identifying their roots, whether real or complex, is of critical importance. The widespread use of complex numbers in science and engineering highlights the importance of accurately determining the complex roots of equations. This article presents a study in which the complex roots of a system of equations are identified through an approach that utilizes homotopy continuation, with the curve being traced using a hyperspherical path tracking technique. Furthermore, this article details five case studies on electrical networks where this method is applied to solve systems of equations containing imaginary coefficients to find mesh currents. The path tracking shows the behavior of system equation in each case study. Finally, an analysis of the precision of the solutions obtained in these case studies is provided, demonstrating an accuracy of up to 15 SDs in a single iteration during the refinement stage.

This study presents a numerical comparison of dimensional and non-dimensional implementations of the classical benchmark problem of steady natural convection in a two-dimensional differentially heated square cavity over the Rayleigh-number range 10³ ≤ Ra ≤ 10¹². The novelty of this work lies in the systematic comparison of both formulations under identical numerical conditions, providing an implementation-oriented assessment of their benchmark accuracy, mesh sensitivity, continuation strategy, and computational efficiency. The governing equations of mass, momentum, and energy conservation were solved under the Boussinesq approximation using primitive variables and the finite-element method. Since both formulations are theoretically equivalent descriptions of the same physical problem, the purpose of this work is not to reassess their physical validity, but to examine their numerical behavior under identical benchmark conditions in terms of mesh sensitivity, continuation strategy, benchmark accuracy, and computational efficiency. In the dimensional implementation, temperature differences of 1, 10, 25, and 50 K were considered, and the corresponding cavity lengths were determined from the Rayleigh-number definition. The main calculations were performed with ΔT = 25 K, while ΔT = 50 K was retained only as an exploratory sensitivity case. Boundary-layer refinement was applied along the vertical walls over the range 10³ ≤ Ra ≤ 10¹². The results show that, once the near-wall gradients are properly resolved, both implementations predict essentially identical average Nusselt numbers, with a maximum relative difference of 0.022%. Temperature contours, stream-function distributions, and centerline profiles also exhibited the same structural behavior in both cases. For the reference mesh, the dimensional implementation exhibited a lower computational cost than the non-dimensional implementation, resulting in an approximately 31% reduction in computation time. These results demonstrate that solving the governing equations directly in dimensional variables provides a numerically efficient and physically interpretable alternative while preserving the benchmark accuracy of the classical non-dimensional formulation.

This study presents a numerical investigation of natural convection in a two-dimensional inclined square cavity filled with air and subjected to differential heating. The effects of the Rayleigh number and cavity inclination on heat transfer and flow behavior were investigated for 10³ ≤ Ra ≤ 10⁹ and inclination angles of 0°, 15°, 30°, and 45°. The dimensionless Navier–Stokes and energy equations were solved using the finite-element method under the Boussinesq approximation with a steady laminar formulation. A structured quadrilateral mesh with boundary-layer refinement was employed near the differentially heated walls. Mesh-refinement tests and comparisons with benchmark data for the classical square cavity were used to assess the numerical accuracy of the model. The results show that the average Nusselt number increases with the Rayleigh number, reflecting the progressive intensification of buoyancy-driven heat transfer. The effect of inclination is non-monotonic and depends on the Rayleigh number. At Ra = 10⁴, the highest average Nusselt number is obtained at 45°, whereas for Ra ≥ 10⁵, the maximum is consistently observed at 15°. At Ra = 10⁹, the average Nusselt number is 54.475, 55.617, 53.224, and 49.408 for inclination angles of 0°, 15°, 30°, and 45°, respectively. The results indicate that moderate cavity inclination can enhance heat transfer by favorably modifying the interaction between buoyancy and the imposed thermal gradient, whereas larger inclinations progressively reduce the heat-transfer rate. The present results provide a systematic characterization of the coupled effects of Rayleigh number and cavity inclination within the scope of the steady two-dimensional formulation considered.