The present paper is devoted to the construction and study of numerical methods for solving an initial boundary value problem for a differential equation containing several terms with fractional time derivatives in the sense of Caputo. This equation is suitable for describing the process of fluid flow in fractured porous media under some physical assumptions, and has an important applied significance in petroleum engineering. Two different approaches to constructing numerical schemes depending on orders of the fractional derivatives are proposed. The semi-discrete and fully discrete numerical schemes for solving the problem are analyzed. The construction of a fully discrete scheme is based on applying the finite difference approximation to time derivatives and the finite element method in the spatial direction. The approximation of the fractional derivatives in the sense of Caputo is carried out using the L1-method. The convergence of both numerical schemes is rigorously proved. The results of numerical tests conducted for model problems are provided to confirm the theoretical analysis. In addition, the proposed computational method is applied to study the flow of oil in a fractured porous medium within the framework of the considered model. Based on the results of the numerical tests, it was concluded that the model reproduces the characteristic features of the fluid flow process in the medium under consideration.

This paper presents a numerical method for solving the convection-diffusion equation with a fractional-order Caputo derivative to model air pollution in urban environments. The developed finite element scheme accounts for memory effects, offering a more accurate representation of pollutant transport compared to classical models. Stability and convergence of the method are theoretically proven and supported by numerical experiments. The model effectively identifies pollutant accumulation zones and can forecast air quality under various weather conditions. The results have practical value for improving environmental monitoring systems and planning measures to reduce pollution levels.

In this article, a numerical method is proposed and investigated for an initial boundary value problem governed by a fractional differential generalization of the nonlinear transient filtration law which describes fluid motion in a porous medium. This type of equation is widely used to describe complex filtration processes such as fluid movement in horizontal wells in fractured geological formations. To construct the numerical method, a high-order approximation formula for the fractional derivative in the sense of Caputo is applied, and a combination of the finite difference method with the finite element method is used. The article proves the uniqueness and continuous dependence of the solution on the input data in differential form, as well as the stability and convergence of the proposed numerical scheme. The linearization of nonlinear terms is carried out by the Newton method, which allows for achieving high accuracy in solving complex problems. The research results are confirmed by a series of numerical tests that demonstrate the applicability of the developed method in real engineering problems. The practical significance of the presented approach lies in its ability to accurately and effectively model filtration processes in shale formations, which allows engineers and geologists to make more informed decisions when designing and operating oil fields.

This paper proposes a new approach to predicting the distribution of harmful substances in the atmosphere based on the combined use of the parameter estimation technique and machine learning algorithms. The essence of the proposed approach is based on the assumption that the concentration values predicted by machine learning algorithms at observation points can be used to refine the pollutant concentration field when solving a differential equation of the convection-diffusion-reaction type. This approach reduces to minimizing an objective functional on some admissible set by choosing the atmospheric turbulence coefficient. We consider two atmospheric turbulence models and restore its unknown parameters by using the limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm. Three ensemble machine learning algorithms are analyzed for the prediction of concentration values at observation points, and comparison of the predicted values with the measurement results is presented. The proposed approach has been tested on an example of two cities in the Republic of Kazakhstan. In addition, due to the lack of data on pollution sources and their intensities, an approach for identifying this information is presented.

This paper studies a differential equation representing the differences of two operators. One of the operators is generated by linear differential expressions that depend on time. The second operator is the Ionkin operator with respect to the spatial variable. In this paper, the differential operator with respect to time is generated by two-point Birkhoff regular boundary conditions. At the same time, the elliptic operator with respect to the spatial variable does not satisfy the so-called Agmon conditions. Moreover, the operator with respect to the spatial variable is not self-adjoint. In the beginning, the solvability of the problem is proved. In the final part, the uniqueness of the solution is proved. Direct application of the methods of the authors’ previous works to prove the uniqueness of the solution to the problem is quite problematic. However, the authors managed to modify the reasoning of previous works to prove the uniqueness of the solution to the problem.

This paper investigates the dispersion of atmospheric pollutants in urban environments using a fractional-order convection–diffusion-reaction model with dynamic line sources associated with vehicle traffic. The model includes Caputo fractional time derivatives and Riesz fractional space derivatives to account for memory effects and non-local transport phenomena characteristic of complex urban air flows. Vehicle trajectories are generated stochastically on the road network graph using Dijkstra’s algorithm, and each moving vehicle acts as a mobile line source of pollutant emissions. To reflect the daily variability of emissions, a time-dependent modulation function determined by unknown parameters is included in the source composition. These parameters are inferred by solving an inverse problem using synthetic concentration measurements from several fixed observation points throughout the area. The study presents two main contributions. Firstly, a detailed numerical analysis of how fractional derivatives affect pollutant dispersion under realistic time-varying mobile source conditions, and secondly, an evaluation of the performance of the proposed parameter estimation method for reconstructing time-varying emission rates. The results show that fractional-order models provide increased flexibility for representing anomalous transport and retention effects, and the proposed method allows for reliable recovery of emission dynamics from sparse measurements.