Solid State Ionics

Galaburdin A.V.

Introduction. Many practically significant tasks reduce to nonlinear differential equations. In this study, one of the applications of neural networks for solving specific nonlinear boundary problems for complex-shaped domains is considered. Specifically, the focus is on solving a stationary heat conduction differential equation with a thermal conductivity coefficient dependent on temperature.Materials and Methods. The original nonlinear boundary problem is linearized through Kirchhoff transformation. A neural network is constructed to solve the resulting linear boundary problem. In this context, derivatives of singular solutions to the Laplace equation are used as activation functions, and these singular points are distributed along closed curves encompassing the boundary of the domain. The weights of the network were tuned by minimizing the mean squared error of training. Results. Results for the heat conduction problem are obtained for various complex-shaped domains and different forms of dependence of the thermal conductivity coefficient on temperature. The results are presented in tables that contain the exact solution and the solution obtained using the neural network. Discussion and Conclusion. Based on the computational results, it can be concluded that the proposed method is sufficiently effective for solving the specified type of boundary problems. The use of derivatives of singular solutions to the Laplace equation as activation functions appears to be a promising approach.

Sidoryakina V.V., Solomakha D.A.

Introduction. Detecting oil spills is a critical task in monitoring the marine ecosystem, protecting it, and minimizing the consequences of emergency situations. The development of fast and accurate methods for detecting and mapping oil spills at sea is essential for prompt assessment and response to emergencies. High-resolution aerial photography provides researchers with a tool for remote monitoring of water discoloration. Artificial intelligence technologies contribute to improving and automating the interpretation and analysis of such images. This study aims to develop approaches for identifying oil spilled on water surfaces using neural networks and machine learning techniques.Materials and Methods. Algorithms capable of automatically identifying marine oil spills were developed using computer image analysis and machine learning methods. The U-Net convolutional neural network was employed for image segmentation tasks. The neural network architecture was designed using the PyTorch library implemented in Python. The AdamW optimizer was chosen for training the network. The neural network was trained on a dataset comprising 8,700 images.Results. The performance of oil spill detection on water surfaces was evaluated using metrics such as IoU, Precision, Recall, Accuracy, and F1 score. Calculations based on these metrics demonstrated identification accuracy of approximately 83–88%, confirming the efficiency of the algorithms used.Discussion and Conclusion. The U-Net convolutional network was successfully trained and demonstrated high accuracy in detecting marine oil spills on the given dataset. Future work will focus on developing algorithms using more advanced neural network models and image augmentation methods.

Belova Y.V., Filina A.A., Chistyakov A.E.

Introduction. Mathematical tools integrated with satellite data are typically employed as the primary means for studying aquatic ecosystems and forecasting changes in phytoplankton concentration in shallow water bodies during summer. This approach facilitates accurate monitoring, analysis, and modeling of the spatiotemporal dynamics of biogeochemical processes, considering the combined effects of various physicochemical, biological, and anthropogenic factors impacting the aquatic ecosystem. The authors have developed a mathematical model aligned with satellite data to predict the behavior of summer phytoplankton species in shallow water under accelerated temporal conditions. The model describes oxidative[1]reduction processes, sulfate reduction, and nutrient transformations (phytoplankton mineral nutrition), investigates hypoxia events caused by anthropogenic eutrophication, and forecasts changes in the oxygen and nutrient regimes of the water body.Materials and Methods. To simulate the population dynamics of summer phytoplankton species correlated with satellite data assimilation methods, an operational algorithm for restoring water quality parameters of the Azov Sea was developed based on the Levenberg-Marquardt multidimensional optimization method. The initial distribution of phytoplankton populations was obtained by applying the Local Binary Patterns (LBP) method to satellite images of the Taganrog Bay and was used as input data for the mathematical model.Results. Using integrated hydrodynamic and biological kinetics models combined with satellite data assimilation methods, a software suite was developed. This suite enables short- and medium-term forecasts of the ecological state of shallow water bodies based on diverse input data correlated with satellite information.Discussion and Conclusion. The conducted studies on aquatic systems revealed that improving the accuracy of initial data is one mechanism for enhancing the quality of biogeochemical process forecasting in marine ecosystems. It was established that using satellite data alongside mathematical modeling methods allows for studying the spatiotemporal distribution of pollutants of various origins, plankton populations in the studied water body, and assessing the nature and scale of natural or anthropogenic phenomena to prevent negative economic and social consequences.

Volosova N.К., Volosov K.A., Volosova A.K., Karlov M.I., Pastukhov D.F., Pastukhov Y.F.

Introduction. This study investigates the possibility of increasing the accuracy of numerically solving boundary value problems using the modified Bubnov-Galerkin method with a linear ordinary differential equation, where the coefficients and the right-hand side are continuous functions. The order of the differential equation n must be less than the number of coordinate functions m.Materials and Methods. A modified Petrov-Galerkin method was used to numerically solve the boundary value problem. It employs a system of linearly independent power-type basis functions on the interval [−1,1], each normalized by the unit Chebyshev norm. The system of linear algebraic equations includes only the linearly independent boundary conditions of the original problem.Results. For the first time, an integral quadrature formula with a 22nd order error was developed for a uniform grid. This formula is used to calculate the matrix elements and coefficients in the right-hand side of the system of linear algebraic equations, taking into account the scalar product of two functions based on the new quadrature formula. The study proves a theorem on the existence and uniqueness of a solution for boundary value problems with general non-separated conditions, provided that n linearly independent particular solutions of a homogeneous differential equation of order n are known.Discussion and Conclusion. The hydrodynamic problem in a viscous strong boundary layer with a third-order equation was precisely solved. The analytical solution was compared with its numerical counterpart, and the uniform norm of their difference did not exceed 5·10‒15. The formulas derived using the generalized Bubnov-Galerkin method may be useful for solving boundary value problems with linear ordinary differential equations of the third and higher orders.

Kornilaev N.V., Koledina K.F.

Introduction. Drilling mud losses are among the most common complications encountered during well drilling. Forecasting these losses is a priority as it helps minimize drilling fluid wastage and prevent wellbore incidents. Mud loss events are primarily influenced by the geological properties of the formations being drilled. Understanding the relationship between mud loss occurrences and the geological characteristics of the formations has both fundamental and practical significance. Given the complexity of predicting mud loss probabilities using traditional mathematical models, this study aims to develop a machine-learning-based system to predict the probability of mud losses based on well location and stratigraphic description.Materials and Methods. Experimental data from 735 wells at the Shkapovskoye oil field, including well location coordinates, geological layer indices, and mud loss intensities, were prepared for computational analysis. The dataset was divided into training and testing subsets. The classification problem was addressed using four intensity classes with the following machine learning models: Decision Tree, Random Forest, and Linear Discriminant Analysis.Results. Predictions generated by the three models were compared against the experimental data in the test set. The evaluation metrics included accuracy and recall. All three models achieved an average prediction accuracy of 91%. Linear Discriminant Analysis was identified as the most accurate model.Discussion and Conclusion. High-accuracy predictions enable reliable forecasting of the probability and intensity of mud losses based on the location and stratigraphic description of new wells. The study presents three machine learning methods that demonstrated superior results in solving this problem.

Volosova N.K., Volosov K.A., Volosova A.K., Pastukhov D.F., Pastukhov Y.F.

   Introduction. The paper considers the solution of boundary value problems on an interval for linear ordinary differential equations, in which the coefficients and the right-hand side are continuous functions. The conditions for the orthogonality of the residual equation to the coordinate functions are supplemented by a system of linearly independent boundary conditions. The number of coordinate functions m must exceed the order n of the differential equation.   Materials and Methods. To numerically solve the boundary value problem, a system of linearly independent coordinate functions is proposed on a symmetric interval [−1,1], where each function has a unit Chebyshev’s norm. A modified Petrov-Galerkin method is applied, incorporating linearly independent boundary conditions from the original problem into the system of linear algebraic equations. An integral quadrature formula with twelfth-order error is used to compute the scalar product of two functions.   Results. A criterion for the existence and uniqueness of a solution to the boundary value problem is obtained, provided that n linearly independent solutions of the homogeneous differential equation are known. Formulas are derived for the matrix coefficients and the coefficients of the right-hand side in the system of linear algebraic equations for the vector expansion of the solution in terms of the coordinate function system. These formulas are obtained for second- and third-order linear differential equations. The modified Bubnov-Galerkin method is formulated for differential equations of arbitrary order.   Discussion and Conclusions. The derived formulas for the generalized Bubnov-Galerkin method may be useful for solving boundary value problems involving linear ordinary differential equations. Three boundary value problems with second- and third-order differential equations are numerically solved, with the uniform norm of the residual not exceeding 10–11.

Sidoryakina V.V.

   Introduction. This paper addresses an initial-boundary value problem for the transport of multifractional suspensions applied to coastal marine systems. This problem describes the processes of transport, deposition of suspension particles, and the transitions between its various fractions. To obtain monotonic finite difference schemes for diffusion-convection problems of suspensions, it is advisable to use schemes that satisfy the maximum principle. When constructing a finite difference scheme that adheres to the maximum principle, it is desirable to achieve second-order spatial accuracy for bothinterior and boundary points of the domain under study.   Materials and Methods. This problem presents certain difficulties when considering the boundaries of the geometric domain, where boundary conditions of the second and third kinds are applied. In these cases, to maintain second-order approximation accuracy, an “extended” grid is introduced (a grid supplemented with fictitious nodes). The guidelineis the approximation of the given boundary conditions using the central difference formula, with the exclusion of the concentration function at the fictitious node from the resulting expressions.   Results. Second-order accurate finite difference schemes for the diffusion-convection problem of multifractional suspensions in coastal marine systems are constructed.   Discussion and Conclusion. The proposed schemes are not absolutely stable, and a detailed analysis of stability and convergence, particularly concerning the grid step ratio, remains an important problem that the author plans to address in the future.

Sukhinov A.I., Bugaeva I.A.

   Introduction. This paper addresses a unified spatially inhomogeneous, non-stationary model of interaction between genetically modified crop resources (corn) and the corn borer pest, which is also present on a relatively small section of non-modified corn. The model assumes that insect pests influence both types of crops and are capable of independent movement (taxis) towards the gradient of plant resources. It also considers diffusion processes in the dynamics of all components of the unified model, biomass growth, genetic characteristics of both types of plant resources, processes of crop consumption, phenomena of growth and degradation, diffusion, and mutation of pests. The model allows for predictive calculations aimed at reducing crop losses and increasing the resistance of transgenic crops to pests by slowing down the natural mutation rate of the pest.   Materials and Methods. The mathematical model is an extension of Kostitsin’s model and is formulated as an initial-boundary value problem for a nonlinear system of convection-diffusion equations. These equations describe the spatiotemporal dynamics of biomass density changes in two types of crops — transgenic and non-modified — as well as the specific populations (densities) of three genotypes of pests (the corn borer) resulting from mutations. The authors linearized the convection-diffusion equations by applying a time-lag method on the time grid, with nonlinear terms from eachequation taken from the previous time layer. The terms describing taxis are presented in a symmetric form, ensuring the skew-symmetry of the corresponding continuous operator and, in the case of spatial grid approximation, the finite-difference operator.   Results. A stable monotonic finite-difference scheme is developed, approximating the original problem with second-order accuracy on a uniform 2D spatial grid. Numerical solutions of model problems are provided, qualitatively corresponding to observed processes. Solutions are obtained for various ratios of modified and non-modified sections of the field.   Discussion and Conclusion. The obtained results regarding pest behavior, depending on the type of taxis, could significantly extend the time for pests to acquire Bt resistance. The concentration dynamics of pests moving in the direction of the food gradient differs markedly from the concentration of pests moving towards a mate for reproduction.

Peskova E.E., Snytnikov V.N.

   Introduction. The study is devoted to the numerical investigation of laser radiation’s effect on an oncoming two-phase flow of nanoparticles and multicomponent hydrocarbon gases. Under such exposure, the hydrogen content in the products increases, and methane is bound into more complex hydrocarbons on the surface of catalytic nanoparticles and in the gas phase. The hot walls of the tube serve as the primary source of heat for the reactive two-phase medium containing catalytic nanoparticles.   Materials and Methods. The main method used is mathematical modelling, which includes the numerical solution of a system of equations for a viscous gas-dust two-phase medium, taking into account chemical reactions and laser radiation. The model accounts for the two-phase gas-dust medium’s multicomponent and multi-temperature nature, ordinary differential equations (ODEs) for the temperature of catalytic nanoparticles, ODEs of chemical kinetics, endothermic effects of radical chain reactions, diffusion of light methyl radicals CH3 and hydrogen atoms H, which initiate methaneconversion, as well as absorption of laser radiation by ethylene and particles.   Results. The distributions of parameters characterizing laminar subsonic flows of the gas-dust medium in an axisymmetric tube with chemical reactions have been obtained. It is shown that the absorption of laser radiation by ethylene in the oncoming flow leads to a sharp increase in methane conversion and a predominance of aromatic compounds in the product output.   Discussion and Conclusion. Numerical modelling of the dynamics of reactive two-phase media is of interest for the development of theoretical foundations for the processing of methane into valuable products. The results obtained confirm the need for joint use of mathematical modelling and laboratory experiments in the development of new resource-saving and economically viable technologies for natural gas processing.

Gasilov V.A., Savenko N.O., Sharova Y.S.

Introduction. Currently, frequency modes of operation of electron accelerators based on capillary discharges are actively investigated. Electrons in these systems are accelerated by femtosecond laser pulses passing through the discharge plasma.Materials and Methods. The paper presents results of three-dimensional magnetohydrodynamic modelling of the capillary discharge cycle, including stages of filling a short capillary with working gas (hydrogen), formation of the plasma channel, and restoration of the working medium before the start of the next discharge. Calculations were performed assuming the system is under external cooling, which maintains thermal balance at intermediate stages of the working cycle, and under constant conditions of gas supply and evacuation.Results. The computational experiments demonstrate the capability of generating beams of relativistic electrons with a repetition frequency of approximately one kilohertz.Discussion and Conclusions. The obtained results allow us to speak about the prospects of using LWFA with a short channel length and a high repetition rate of the capillary discharge.

Galaburdin S.A.

Introduction. Many mathematical problems are reduced to solving partial differential equations (PDEs) in domains of complex shapes. Existing analytical and numerical methods do not always provide efficient solutions for such problems. Recently, neural networks have been successfully applied to solve PDEs, typically addressing boundary value problems for domains with simple shapes. This paper attempts to construct a neural network capable of effectively solving boundary value problems for domains of complex shapes.Materials and Methods. A method for constructing a neural network to solve the Dirichlet problem for regions of complex shape is proposed. Derivatives of singular solutions of the Laplace equation are accepted as activation functions. Singular points of these solutions are distributed along closed curves encompassing the boundary of the domain. The adjustment of the network weights is reduced to minimizing the root-mean-square error during training.Results. The results of solving Dirichlet problems for various complex-shaped domains are presented. The results are provided in tables, comparing the exact solution and the solution obtained using the neural network. Figures show the domain shapes and the locations of points where the solutions were determined.Discussion and Conclusion. The presented results indicate a good agreement between the obtained solution and the exact one. It is noted that this method can be easily applied to various boundary value problems. Methods for enhancing the efficiency of such neural networks are suggested.

Vasyukov A.V.

Introduction. The study focuses on modelling the process of ultrasound medical examination in a heterogeneous environment with regions of significantly different sound speeds. Such scenarios typically arise when visualizing brain structures through the skull. The aim of this work is to compare possible approaches to determining the interface between acoustically contrasting media using convolutional neural networks.Materials and Methods. Numerical modelling of the direct problem is performed, obtaining synthetic calculated ultrasonic images based on known geometry and rheology of the area as well as sensor parameters. The calculated images reproduce distortions and artifacts typical for setups involving the skull wall. Convolutional neural networks of 2D and 3D structures following the UNet architecture are used to solve the inverse problem of determining the interface between media based on a sensor signal. The networks are trained on computational datasets and then tested on individual samples not used in training.Results. Numerical B-scans for characteristic setups were obtained. The possibility of localizing the aberrator boundary with good quality for both 2D and 3D convolutional networks was demonstrated. A higher quality result was obtained for the 3D network in the presence of significant noise and artifacts in the input data. It was established that the 3D architecture network can provide the shape of the interface between media in 0.1 seconds.Discussion and Conclusions. The results can be used for the development of transcranial ultrasound technologies. Rapid localization of the skull boundary can be incorporated into imaging algorithms to compensate for distortions caused by differences in sound velocities in bone and soft tissues.

Belyaev K.P., Kuleshov A.A., Novikova A.V., Tuchkova N.P.

Introduction. The study of heat interaction processes and the distribution of heat flaxes in the oceans is important for understanding climate change on Earth. The North Atlantic, which is one of the key components of the global climate system, plays a significant role in regulating the climate of our latitudes. One of the key tools for analyzing heat distribution in the oceans is probabilistic analysis. In this work, using mathematical modelling methods, a statistical analysis of observational data on heat fluxes in the North Atlantic is carried out.Materials and Methods. The used methods include the analysis of random processes specified by the stochastic differential equation (SDE) or the Ito equation, approximation of observational data, and solution of the Fokker-Planck-Kolmogorov (FPK) equation to describe the evolution of the probabilistic distribution of heat in the ocean.Results. Using mathematical modelling methods, a probabilistic analysis of the distribution of heat fluxes in the North Atlantic for the period from 1979 to 2022 has been carried out. The results of the study made it possible to establish patterns of distribution of heat flux in the studied region over the period of time under consideration.Discussion and Conclusions. The results may be useful for further study of climate processes in the North Atlantic, as well as for the development of resource management and environmental protection strategies.

Protsenko E.A., Panasenko N.D., Protsenko S.V.

Introduction. This work is devoted to the mathematical modelling of extreme sea level fluctuations in the Azov Sea using remote sensing data. The aim of the study is to develop and apply a mathematical model that allows more accurate prediction of surge and seiche events caused by extreme wind conditions. The relevance of the work is due to the need to improve the forecasts of hydrodynamic processes in shallow water bodies (such as the Azov Sea), where such phenomena can have significant economic and ecological consequences. The goal of this work is to develop and apply a mathematical model for predicting extreme sea level fluctuations in the Azov Sea caused by wind conditions.Materials and Methods. The study is based on the analysis of remote sensing data and observations of wind speed and direction over the Azov Sea. The primary method used is mathematical modelling, which includes solving the system of shallow water hydrodynamics equations. Wind condition data were collected from November 20 to 25, 2019, during which catastrophic sea level fluctuations were observed. The model considers the components of water flow velocity, water density, hydrodynamic pressure, gravitational acceleration, and turbulence exchange coefficients.Results. The modelling showed that prolonged easterly winds with speeds up to 22 m/s led to significant surge and seiche fluctuations in sea level. The maximum amplitudes of fluctuations were recorded in the central part of the Taganrog Bay, where the wind direction and speed remained almost constant throughout the observation period. Data from various platforms located in different parts of the Azov Sea confirmed a significant decrease in water level in the northeast and an increase in the southwest.Discussion and Conclusions. The study results confirm that using mathematical models in combination with remote sensing data allows more accurate predictions of extreme sea level fluctuations. This is important for developing measures to prevent and mitigate the consequences of surge and seiche events in coastal areas. In the future, it is necessary to improve models by including additional factors such as climate change and anthropogenic impact on the Azov Sea ecosystem.

But I.I., Kiryushina M.A., Elistratov S.A., Elizarova T.G., Tiniakov A.D.

Introduction. We present a novel adaptive mesh refinement (AMR) solver, SWqgdAMR, based on the open software platform AMReX. The new solver is grounded in regularized shallow water equations. This paper details the equations, their discretization, and implementation features within AMReX. The efficacy of SWqgdAMR is demonstrated through two test cases: a two-dimensional circular dam break (collapse of a liquid column) and the collapse of two liquid columns of different heights.Materials and Methods. The SWqgdAMR solver is developed to extend the applicability of regularized equations in problems requiring high computational power and adaptive grids. SWqgdAMR is the first solver based on the quasigas dynamic (QGD) algorithm within the AMReX framework. The implementation and validation of SWqgdAMR represent a crucial step towards the further expansion of the QGD software suite.Results. The AMReX-based shallow water equations solver SWqgdAMR with adaptive mesh refinement is described and tested in detail. Validation of SWqgdAMR involved two-dimensional problems: the breach of a cylindrical dam and the breach of two cylindrical dams of different heights. The presented solver demonstrated high efficiency, with the use of adaptive mesh refinement technology accelerating the computation by 56 times compared to a stationary grid calculation.Discussion and Conclusions. The algorithm can be expanded to include bathymetry, external forces (wind force, bottom friction, Coriolis forces), and the mobility of the shoreline during wetting and drying phases, as has been done in individual codes for regularized shallow water equations (RSWE). The current implementation of the QGD algorithm did not test the potential for parallel computing on graphical cores.