Dossan Baigereyev

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.

This paper presents an efficient numerical method for the fractional-order generalization of the stochastic Stokes–Darcy model, which finds application in various engineering, biomedical and environmental problems involving interaction between free fluid flow and flows in porous media. Unlike the classical model, this model allows taking into account the hereditary properties of the process under uncertainty conditions. The proposed numerical method is based on the combined use of the sparse grid stochastic collocation method, finite element/finite difference discretization, a fast numerical algorithm for computing the Caputo fractional derivative, and a cost-effective ensemble strategy. The hydraulic conductivity tensor is assumed to be uncertain in this problem, which is modeled by the reduced Karhunen–Loève expansion. The stability and convergence of the deterministic numerical method have been rigorously proved and validated by numerical tests. Utilizing the ensemble strategy allowed us to solve the deterministic problem once for all samples of the hydraulic conductivity tensor, rather than solving it separately for each sample. The use of the algorithm for computing the fractional derivatives significantly reduced both computational cost and memory usage. This study also analyzes the influence of fractional derivatives on the fluid flow process within the fractional-order Stokes–Darcy model under uncertainty conditions.

This research is devoted to the development of a geographic information system (GIS) for the analysis of geological data. It presents two specialized software modules designed to solve complex geological problems related to potential progress to disturbed masses and magnetotelluric sounding. These modules are integrated into the QGIS environment, offering efficient data processing and analysis capabilities, contributing to a deeper understanding of geological structures. The study presents a mathematical model for the problem of magnetotelluric sounding (MTS) and the continuation of potentials towards the perturbed masses, demonstrating numerical results using the developed algorithm. To confirm the accuracy of the model, a comparative analysis was carried out with empirical data for various chemical elements, which showed high accuracy, especially at shallow depths, with an error rate of less than 2%. In addition, the study highlights the importance of powerful GIS for the analysis and interpretation of geological data, including geochemical, geophysical and remote sensing information. The advanced functionality of QGIS simplifies data processing and visualization, which makes it an invaluable tool for geologists and researchers.

This paper is aimed at efficient numerical implementation of the fractional-order generalization of the stochastic Stokes–Darcy model, which has important scientific, applied, and economic significance in hydrology, the oil industry, and biomedicine. The essence of this generalization of the stochastic model is the introduction of fractional time derivatives in the sense of Caputo’s definition to take into account long-term changes in the properties of media. An efficient numerical method for the implementation of the fractional-order Stokes–Darcy model is proposed, which is based on the use of a higher-order approximation formula for the fractional derivative, higher-order finite difference relations, and a finite element approximation of the problem in the spatial direction. In the paper, a rigorous theoretical analysis of the stability and convergence of the proposed numerical method is carried out, which is confirmed by numerous computational experiments. Further, the proposed method is applied to the implementation of the fractional-order stochastic Stokes–Darcy model using an ensemble technique, in which the approximation is carried out in such a way that the resulting systems of linear equations have the same coefficient matrix for all realizations. Furthermore, evaluation of the discrete fractional derivatives is carried out with the use of parallel threads. The efficiency of applying both approaches has been demonstrated in numerical tests.

In this paper, theoretical estimates for the solution of an approximation problem for a fourth-order partial differential equation with fractional derivatives in the sense of Caputo are obtained. This equation describes a transient fluid flow in a fractured porous medium. To approximate fractional derivatives, a higher-order formula was applied, and a substep scheme technique was employed to retain a global higher-order convergence in the temporal variable. The method of a priori estimates is utilized to prove the unconditional stability of the proposed method with respect to the initial data and the right-hand side of the equation. The convergence of the approximate solution to the weak solution of the original differential problem is rigorously proved.

This article presents an algorithm for the numerical solution of an initial-boundary value problem for a symmetric t-hyperbolic system of partial differential equations. This problem is based on continual filtration model, which describes the propagation of seismic waves in a poroelastic medium saturated with a fluid characterized by such physical parameters as the propagation velocities of longitudinal P- (fast and slow) and transverse S-waves, the density of the medium materials, and porosity. The system of linearized equations of saturated porous media is formulated in terms of physical variables of the velocity–stress tensor of the porous matrix and the velocity–pressure of the saturating fluid in the absence of energy dissipation. The solution is implemented numerically using an explicit finite difference upwind scheme built on a staggered grid to avoid the appearance of oscillations in the solution functions. The program code implementing parallel computing is developed in the high-performance Julia programming language. The possibility of using the approach is demonstrated by the example of solving the problem of propagation of seismic waves from a source located in the formation. Computational experiments based on real data from oil reservoirs have been implemented, and dynamic visualization of solutions consistent with the first waves arrival times has been obtained.

This paper investigates a model of fluid flow in a fractured porous medium under the assumption of a uniform distribution of fractures throughout the volume. This model is based on the use of a fractional differential analogue of Darcy's law, as well as on the assumption that the properties of rock and fluid depend on pressure and its fractional derivative. Unlike previous studies, this study uses a fractional derivative in the Caputo-Fabrizio sense with a non-singular kernel. In this paper, we propose a numerical method for solving this initial boundary value problem and theoretically investigate the order of its convergence. The formulation of a fully discrete scheme is based on application of the finite difference approximation for integer and fractional timederivatives, and the Galerkin method in the spatial variable. A second-order formula is used to approximate both integer derivative and the fractional derivative in the sense of Caputo-Fabrizio. A priori estimates are obtained for both semi-discrete and fully discrete schemes, which imply their second-order convergence in time and space variables. A number of computational experiments were carried out on the example of a model problem to validate the accuracy of the scheme. The results of the numerical tests fully confirm the outcome of the theoretical analysis.

In this paper, we propose an efficient numerical method for solving an initial boundary value problem for a coupled system of equations consisting of a nonlinear parabolic partial integro-differential equation and an elliptic equation with a nonlinear term. This problem has an important applied significance in petroleum engineering and finds application in modeling two-phase nonequilibrium fluid flows in a porous medium with a generalized nonequilibrium law. The construction of the numerical method is based on employing the finite element method in the spatial direction and the finite difference approximation to the time derivative. Newton’s method and the second-order approximation formula are applied for the treatment of nonlinear terms. The stability and convergence of the discrete scheme as well as the convergence of the iterative process is rigorously proven. Numerical tests are conducted to confirm the theoretical analysis. The constructed method is applied to study the two-phase nonequilibrium flow of an incompressible fluid in a porous medium. In addition, we present two examples of models allowing for prediction of the behavior of a fluid flow in a porous medium that are reduced to solving the nonlinear integro-differential equations studied in the paper.