Purpose The purpose of this paper is to solve Navier–Stokes equations including the effects of temperature and inner pipe rotation for fully developed turbulent flow in eccentric annuli by using finite difference scheme with fixing non-linear terms. Design/methodology/approach A mathematical model is proposed for fully developed turbulent flow including the effects of temperature and inner pipe rotation in eccentric annuli. Obtained equation is solved numerically via central difference approximation. In this process, the non-linear term is frozen. In so doing, the non-linear equation can be considered as a linear one. Findings The convergence analysis is studied before using the method to the proposed momentum equation. It reflects that the method approaches to the exact solution of the equation. The numerical solution of the mathematical model shows that pressure gradient can be predicted with a good accuracy when it is compared with experimental data collected from experiments conducted at Izmir Katip Celebi University Flow Loop. Originality/value The originality of this work is that Navier–Stokes equations including temperature and inner pipe rotation effects for fully developed turbulent flow in eccentric annuli are solved numerically by a finite difference method with frozen non-linear terms.

The iterative splitting methods have been extensively applied to solve complicated systems of differential equations. In this process, we split the complex problem into several sub-problems, each of which can be solved sequentially. In this paper, we construct a new symmetric iterative splitting scheme based on the Magnus expansion for solving non-autonomous problems. We also study its convergence properties by using the concepts of stability, consistency, and order. Several numerical examples are illustrated to confirm the theoretical results by comparing frequently used methods.

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Background The aim of this paper was to evaluate the change in 25-hidroxyvitamin D (25(OH)D) levels before and during the COVID-19 pandemic. Methods In this retrospective, cross-sectional and methodological study included 86,772 patients (18–75 years) samples who were admitted to the Izmir Dokuz Eylul University Hospital (latitude and longitude (Turkey): 27 E 09; 38 N 25, respectively) for various reasons and whose 25(OH)D levels were measured in the biochemistry unit between 2019–2020 and 2020–2021 (before and during the COVID-19 outbreak). A time series analysis of monthly averages for 25(OH)D was performed. For the purpose of seasonal study, the mean levels of 25(OH)D are grouped by years. Data were modeled in terms of 25(OH)D levels using the MATLAB Curve Fitting Toolbox. Results There was no significant difference between the sexes according to 25(OH)D levels (p>0.05). 25(OH)D levels were significantly higher in the summer months and lower in the winter months (p<0.001). When comparing the spring months, 25(OH)D levels in 2020 (18 ± 10) were found to be significantly lower than in 2019 (22 ± 12) (p<0.001); on the contrary, when examined based on the summer, autumn, and winter months, it was determined that 25(OH)D levels increased in 2020 (summer: 25 ± 13, autumn: 25 ± 14, and winter: 19 ± 10) compared to 2019 (summer: 23 ± 11, autumn: 22 ± 10, and winter: 19 ± 11) (p<0.001). In the estimates curve obtained with an error margin of 11% in the time series analysis, it was estimated that the 25(OH)D averages after the pandemic would be similar to those before the pandemic. Conclusions Restrictions, partial or complete closures, and curfews can significantly affect individuals’ 25(OH)D levels during the COVID-19 outbreak. There is a need for multicenter studies with larger populations covering different regions to strengthen and support our results.

Nonlinear oscillation problems are extensively used in engineering and applied sciences. Due to non-availability of the analytic solutions, numerical approaches have been used for these equations. In this study, a numerical method which is based on Newton-Raphson linearization and Fréchet derivative is suggested. The convergence analysis is also studied locally. The present method is tested on three examples: damped oscillator, Van-der Pol equation and Schrödinger equation. It is shown that the obtained solutions via the present method are more accurate than those of the well-known second order Runge-Kutta method. When examining the present method, preservation of characteristic properties of these equations is also considered. The obtained results show that the present method is applicable with respect to the efficiency and the physical compatibility.