Transcendental functions are a fundamental building block of science and engineering. Among them, a relatively new function denominated as Lambert W is highlighted. The importance of such function relies on the fact that it can perform novel isolation of variables. In this work, we propose two accurate piece-wise approximate solutions, one for the lower branch and another one for the upper branch, respectively. The proposed analytic approximations are obtained by using the power series extender method (PSEM) in combination with asymptotic solutions. In addition, we will compare some published approximations with our proposal, highlighting our advantages in terms of significant digits and speed of evaluation. Furthermore, the approximations are validated by the successful simulation of a problem of economy and other acoustic waves of nonlinear ions.

This article introduces two approximations that allow the evaluation of Fresnel integrals without the need for using numerical algorithms. These equations accomplish the characteristic of being continuous in the same interval as Fresnel. Both expressions have been determined applying the least squares method to suitable expressions. Accuracy of equations improves as x increases; as for small values of x, it is possible to achieve an absolute error less than 8×10-5. To probe the efficiency of the equations, two case studies are presented, both applied in the optics field. The first case is related to the semi-infinite opaque screen for Fresnel diffraction. In this case study Fresnel integrals are evaluated with the proposed equations to calculate the irradiance distribution and the Cornu spiral for diffraction computations of the Fresnel diffraction; obtained results show a good accuracy. The second case is related to the double aperture problem for Fresnel diffraction.

This study analyzes the dynamics of temperature and moisture in a cylindrical silo with a conical roof and floor used for storing corn in the Bajío region of Mexico, considering conditions both with and without aeration. The model incorporates external temperature fluctuations, solar radiation, grain moisture equilibrium with air humidity through the sorption isotherm (water activity), and grain respiration to simulate real storage conditions. The model is based on continuity, momentum, energy, and moisture conservation equations in porous media. This model was solved using the finite element method (FEM) to evaluate temperature and interstitial humidity variations during January and May, representing cold and warm environmental conditions, respectively. The simulations show that, without aeration, grain temperature progressively accumulates in the center and bottom region of the silo, reaching critical values for safe storage. In January, the low ambient temperature favors the natural dissipation of heat. In contrast, in May, the combination of high ambient temperatures and solar radiation intensifies thermal accumulation, increasing the risk of grain deterioration. However, implementing aeration periods allowed for a reduction in the silo’s internal temperature, achieving more homogeneous cooling and reducing the threats of mold and insect proliferation. For January, an airflow rate of 0.15 m³/(min·ton) was optimal for maintaining the temperature within the safe storage range (≤17 °C). In contrast, in May, neither this airflow rate nor the accumulation of 120 h of aeration was sufficient to achieve optimal storage temperatures. This indicates that, under warm conditions, the aeration strategy needs to be reconsidered, assessing whether a higher airflow rate, longer periods, or a combination of both could improve heat dissipation. The results also show that interstitial relative humidity remains stable with nocturnal aeration, minimizing moisture absorption in January and preventing excessive drying in May. However, it was identified that aeration period management must be adaptive, taking environmental conditions into account to avoid issues such as re-wetting or excessive grain drying.

This work presents the application of homotopy perturbation method (HPM) and Optimized Direct Padé (ODP) to obtain a handy and easily computable approximate solution of the nonlinear differential equation to model the oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics. On one hand, the obtained HPM solution is fully symbolic in terms of the coefficients of the equation, allowing us to use the same solution for different values of the maximum reaction rate, the Michaelis constant, and the permeability of the cell membrane. On the other hand, the numerical experiments show the high accuracy of the proposed ODP solution, resulting in 3.58×10-4 as the lowest absolute relative error (A.R.E.) for a set of coefficients. In addition, a novel technique is proposed to reduce the number of algebraic operations during the process of application of ODP method through the use of the Taylor series, which help to simplify the algebraic expressions used. The powerful process to obtain the solution shows that the Optimized Direct Padé and homotopy perturbation method are suitable methods to use.

This work introduces a novel modification of classical perturbation method (PM), denominated Optimized Distribution of Boundary Conditions Perturbation Method (ODBCPM) with the purpose to improve the performance of PM in the solution of ordinary differential equations (ODES). We will see that the main proposal of ODBCPM rests above all in the redistribution and optimization of the boundary conditions of the problem to be solved among the iterations of the proposed method. The solution of a couple of heat relevant problems indicates the potentiality of ODBCPM even for the case of large values of the perturbative parameter.

This paper introduces a novel methodology for vehicle routing services called Route Optimization with Multiple Delivery Points (ROMP), which works by modeling urban street networks as analog electrical circuits. This methodology translates road networks into a linear electrical circuit where the resistances of circuit branches represent parameters like vehicular flow and street length, derived from geographic positions between intersections. By applying Modified Nodal Analysis (MNA) to this circuit, ROMP identifies high-current paths that closely approximate minimal travel distances. The practical performance of ROMP is demonstrated through three case studies, showing its potential to yield shorter routes and faster route-finding compared to OpenRouteService (ORS). The resultant improvements can lead to fuel savings, reduced labor costs, and enhanced logistics operations, particularly in applications involving a single origin and multiple delivery points, such as goods delivery and patient transport. In addition, this proposal supports sustainability by optimizing routes, which helps reduce the environmental impact of transportation and lower greenhouse gas emissions. Furthermore, shorter travel distances and improved efficiency promote better energy use, enhancing air quality and urban sustainability. Future work aims to integrate new street models and real-time traffic data to expand ROMP’s applicability in vehicle routing research.