The fractional advection–dispersion equation is used in groundwater hydrology for modeling the movements of contaminants/solute particles along with flowing groundwater at the seepage velocity in porous media. This model is used for the prediction of the transport of nonreactive dissolved contaminants in groundwater. This paper establishes the existence and the uniqueness of solutions represented as fractional bi-variate power series of some initial-value problems and boundary-value problems for the fractional advection–dispersion equation. Moreover, a method to approximate the solutions using fractional polynomials in two variables and to evaluate the errors in a suitable rectangle is designed. Illustrative examples showing the applicability of the theoretical results are presented.

The study of the combination of chemical kinetics with transport phenomena is the main step for reactor design. It is possible to deviate from the model behaviour, the cause of which may be fluid channelling, fluid recirculation, or creation of stagnant regions in the vessel, by using a dispersion model. In this paper, the known general solution of the dispersion model for closed vessels is given in a new, straightforward form. In order to improve the classical theoretical solution, a hybrid of analytical and numerical methods is used. It is based on the general analytic solution and the least-squares method by fitting the results of a tracer test carried out on the vessel with the values of the analytic solution. Thus, the accuracy of the estimation for the vessel dispersion number is increased. The presented method can be used to similar problems modelled by a partial differential equation and some boundary conditions which are not sufficient to ensure the uniqueness of the solution.

Atmospheric pollution greatly impacts public health, since accumulating scientific data found credible evidence between pollutant exposure and various causes of morbimortality. In the 42 counties of Romania some pollutant substances and particles are monitored trough 152 stations. The study focuses on particulate matter (PM ₁₀ and PM _2.5 ) data analysis in correlation with public health statistics. Spearman tests were used to correlate contemporaneous yearly average PM values to morbidity and mortality and showed a statistically significant weak positive correlation between PM ₁₀ and PM _2.5 average yearly concentrations (μg/m3) and morbidities for lung cancer (LC) ( r = 0.211, p < 0.001; r = 0.166, p = 0.022, respectively) and asthma ( r = 0.264, p < 0.001; r = 0.31, p < 0.001, respectively). For mortalities, PM _2.5 showed a positive weak correlation with congenital and chromosomal anomalies (CCA) ( r = 0.21, p = 0.004) while PM ₁₀ showed weaker positive correlations with multiple mortality causes (tuberculosis, CCA, early childhood mortalities (< 27 days, < 1 year and < 5 years). Other correlation results involving morbidities or mortalities from other causes were not reliable due to the high impact of SARS-Cov2 pandemic on hospital admissions and deaths but also on data reporting. The Welch 2-sample T-test performed for the comparison of the most (Gorj, 30.03 μg/m ³ ) and least (Harghita, 16.80 μg/m ³ ) PM ₁₀ polluted counties showed an increase in LC ( p = 0.002) and asthma ( p = 0.279) admissions for the polluted county. The 3-year exposure window emerged as a predictor for morbidity for lung cancer and asthma using single and dual linear regression models. For asthma, the fine fraction (PM _2.5 ) remained the primary independent driver, adding 5.28 admissions per 100,000 inhabitants for every increase with 1 μg/m ³ . For lung cancer, the coarse particle fraction exhibited the strongest independent effect on hospitalizations, adding 2.85 admissions per 100,000 inhabitants for every unit increase in concentration. Based on the results of the linear regression models, avoidable burden maps for morbidity from lung cancer and asthma were built, which can be useful tools for adapting public health and environmental policies on the dynamic of the PM concentrations in time and space.