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Coffee berry disease (CBD), resulting from the Colletotrichum kahawae fungal pathogen, poses a severe risk to coffee crops worldwide. Focused on coffee berries, it triggers substantial economic losses in regions relying heavily on coffee cultivation. The devastating impact extends beyond agricultural losses, affecting livelihoods and trade economies. Experimental insights into coffee berry disease provide crucial information on its pathogenesis, progression, and potential mitigation strategies for control, offering valuable knowledge to safeguard the global coffee industry. In this paper, we investigated the mathematical model of coffee berry disease, with a focus on the dynamics of the coffee plant and Colletotrichum kahawae pathogen populations, categorized as susceptible, exposed, infected, pathogenic, and recovered (SEIPR) individuals. To address the system of nonlinear differential equations and obtain semi-analytical solution for the coffee berry disease model, a novel analytical approach combining the Shehu transformation, Akbari – Ganji, and Pade approximation method (SAGPM) was utilized. A comparison of analytical results with numerical simulations demonstrates that the novel SAGPM is excellent efficiency and accuracy. Furthermore, the sensitivity analysis of the coffee berry disease model examines the effects of all parameters on the basic reproduction number $R_0$. Moreover, in order to examine the behavior of the model individuals, we varied some parameters in CBD. Through this analysis, we obtained valuable insights into the responses of the coffee berry disease model under various conditions and scenarios. This research offers valuable insights into the utilization of SAGPM and sensitivity analysis for analyzing epidemiological models, providing significant utility for researchers in the field.

X-ray diffraction experiment allows determining of magnitudes of complex coefficients in the decomposition of the studied electron density distribution into Fourier series. The determination of the lost in the experiment phase values poses the central problem of the method, namely the phase problem. Some methods for solving of the phase problem result in a set of feasible solutions. Cluster analysis method may be used to investigate the composition of this set and to extract one or several typical solutions. An essential feature of the approach is the estimation of the closeness of two solutions by the map correlation between two aligned Fourier syntheses calculated with the use of phase sets under comparison. An interactive computer program ClanGR was designed to perform this analysis.

Comparative analysis of two models of porous medium (Dacry and Brinkman) on an example of mathematical simulation of transient natural convection in a porous vertical cylindrical cavity with heat-conducting shell of finite thickness in conditions of convective cooling from an environment has been carried out. The boundary-value problem of mathematical physics formulated in dimensionless variables such as stream function, vorticity and temperature has been solved by implicit finite difference method. The presented verification results validate used numerical approach and also confirm that the solution is not dependent on the mesh size. Features of the conjugate heat transfer problems with considered models of porous medium have been determined.

Problems of multiple covering ($k$-covering) of a bounded set $G$ with equal circles of a given radius are well known. They are thoroughly studied under the assumption that $G$ is a finite set. There are several papers concerned with studying this problem in the case where $G$ is a connected set. In this paper, we study the problem of minimizing the number of circles that form a $k$-covering, $k \geqslant 1$, provided that $G$ is a bounded convex plane domain.

For the above-mentioned problem, we state a 0-1 linear model, a general integer linear model, and a nonlinear model, imposing a constraint on the minimum distance between the centers of covering circles. The latter constraint is due to the fact that in practice one can place at most one device at each point. We establish necessary and sufficient solvability conditions for the linear models and describe one (easily realizable) variant of these conditions in the case where the covered set $G$ is a rectangle.

We propose some methods for finding an approximate number of circles of a given radius that provide the desired $k$-covering of the set $G$, both with and without constraints on distances between the circles’ centers. We treat the calculated values as approximate upper bounds for the number of circles. We also propose a technique that allows one to get approximate lower bounds for the number of circles that is necessary for providing a $k$-covering of the set $G$. In the general linear model, as distinct from the 0-1 linear model, we require no additional constraint. The difference between the upper and lower bounds for the number of circles characterizes the quality (acceptability) of the constructed $k$-covering.

We state a nonlinear mathematical model for the $k$-covering problem with the above-mentioned constraints imposed on distances between the centers of covering circles. For this model, we propose an algorithm which (in certain cases) allows one to find more exact solutions to covering problems than those calculated from linear models.

For implementing the proposed approach, we have developed computer programs and performed numerical experiments. Results of numerical experiments demonstrate the effectiveness of the method.

The mathematical model based on the linear integro-differential Boltzmann equation is considered in this article. The model describes the radiation transfer in the scattering medium irradiated by a point source. The inverse problem for the transfer equation is defined. This problem consists of determining the scattering coefficient from the time-angular distribution of the radiation flux density at a given point in space. The Neumann series representation for solving the radiation transfer equation is analyzed in the study of the inverse problem. The zero member of the series describes the unscattered radiation, the first member of the series describes a single-scattered field, the remaining members of the series describe a multiple-scattered field. When calculating the approximate solution of the radiation transfer equation, the single scattering approximation is widespread to calculated an approximate solution of the equation for regions with a small optical thickness and a low level of scattering. An analytical formula is obtained for finding the scattering coefficient by using this approximation for problem with additional restrictions on the initial data. To verify the adequacy of the obtained formula the Monte Carlo weighted method for solving the transfer equation is constructed and software implemented taking into account multiple scattering in the medium and the space-time singularity of the radiation source. As applied to the problems of high-frequency acoustic sensing in the ocean, computational experiments were carried out. The application of the single scattering approximation is justified, at least, at a sensing range of about one hundred meters and the double and triple scattered fields make the main impact on the formula error. For larger regions, the single scattering approximation gives at the best only a qualitative evaluation of the medium structure, sometimes it even does not allow to determine the order of the parameters quantitative characteristics of the interaction of radiation with matter.

In this work, the problem of constructing an electronic analogue of heterogeneous DNA is solved with the help of the methods of mathematical modeling. Electronic analogs of that type, along with other physical models of living systems, are widely used as a tool for studying the dynamic and functional properties of these systems. The solution to the problem is based on an algorithm previously developed for homogeneous (synthetic) DNA and modified in such a way that it can be used for the case of inhomogeneous (native) DNA. The algorithm includes the following steps: selection of a model that simulates the internal mobility of DNA; construction of a transformation that allows you to move from the DNA model to its electronic analogue; search for conditions that provide an analogy of DNA equations and electronic analogue equations; calculation of the parameters of the equivalent electrical circuit. To describe inhomogeneous DNA, the model was chosen that is a system of discrete nonlinear differential equations simulating the angular deviations of nitrogenous bases, and Hamiltonian corresponding to these equations. The values of the coefficients in the model equations are completely determined by the dynamic parameters of the DNA molecule, including the moments of inertia of nitrous bases, the rigidity of the sugar-phosphate chain, and the constants characterizing the interactions between complementary bases in pairs. The inhomogeneous Josephson line was used as a basis for constructing an electronic model, the equivalent circuit of which contains four types of cells: A-, T-, G-, and C-cells. Each cell, in turn, consists of three elements: capacitance, inductance, and Josephson junction. It is important that the A-, T-, G- and C-cells of the Josephson line are arranged in a specific order, which is similar to the order of the nitrogenous bases (A, T, G and C) in the DNA sequence. The transition from DNA to an electronic analog was carried out with the help of the A-transformation which made it possible to calculate the values of the capacitance, inductance, and Josephson junction in the A-cells. The parameter values for the T-, G-, and C-cells of the equivalent electrical circuit were obtained from the conditions imposed on the coefficients of the model equations and providing an analogy between DNA and the electronic model.

In this paper we propose high-order (tensor) methods for two types of saddle point problems. Firstly, we consider the classic min-max saddle point problem. Secondly, we consider the search for a stationary point of the saddle point problem objective by its gradient norm minimization. Obviously, the stationary point does not always coincide with the optimal point. However, if we have a linear optimization problem with linear constraints, the algorithm for gradient norm minimization becomes useful. In this case we can reconstruct the solution of the optimization problem of a primal function from the solution of gradient norm minimization of dual function. In this paper we consider both types of problems with no constraints. Additionally, we assume that the objective function is $\mu$-strongly convex by the first argument, $\mu$-strongly concave by the second argument, and that the $p$-th derivative of the objective is Lipschitz-continous.

For min-max problems we propose two algorithms. Since we consider strongly convex a strongly concave problem, the first algorithm uses the existing tensor method for regular convex concave saddle point problems and accelerates it with the restarts technique. The complexity of such an algorithm is linear. If we additionally assume that our objective is first and second order Lipschitz, we can improve its performance even more. To do this, we can switch to another existing algorithm in its area of quadratic convergence. Thus, we get the second algorithm, which has a global linear convergence rate and a local quadratic convergence rate.

Finally, in convex optimization there exists a special methodology to solve gradient norm minimization problems by tensor methods. Its main idea is to use existing (near-)optimal algorithms inside a special framework. I want to emphasize that inside this framework we do not necessarily need the assumptions of strong convexity, because we can regularize the convex objective in a special way to make it strongly convex. In our article we transfer this framework on convex-concave objective functions and use it with our aforementioned algorithm with a global linear convergence and a local quadratic convergence rate.

Since the saddle point problem is a particular case of the monotone variation inequality problem, the proposed methods will also work in solving strongly monotone variational inequality problems.

Stochastic optimization is a current area of research due to significant advances in machine learning and their applications to everyday problems. In this paper, we consider two fundamentally different methods for solving the problem of stochastic optimization — online and offline algorithms. The corresponding algorithms have their qualitative advantages over each other. So, for offline algorithms, it is required to solve an auxiliary problem with high accuracy. However, this can be done in a distributed manner, and this opens up fundamental possibilities such as, for example, the construction of a dual problem. Despite this, both online and offline algorithms pursue a common goal — solving the stochastic optimization problem with a given accuracy. This is reflected in the comparison of the computational complexity of the described algorithms, which is demonstrated in this paper.

The comparison of the described methods is carried out for two types of stochastic problems — convex optimization and saddles. For problems of stochastic convex optimization, the existing solutions make it possible to compare online and offline algorithms in some detail. In particular, for strongly convex problems, the computational complexity of the algorithms is the same, and the condition of strong convexity can be weakened to the condition of $\gamma$-growth of the objective function. From this point of view, saddle point problems are much less studied. Nevertheless, existing solutions allow us to outline the main directions of research. Thus, significant progress has been made for bilinear saddle point problems using online algorithms. Offline algorithms are represented by just one study. In this paper, this example demonstrates the similarity of both algorithms with convex optimization. The issue of the accuracy of solving the auxiliary problem for saddles was also worked out. On the other hand, the saddle point problem of stochastic optimization generalizes the convex one, that is, it is its logical continuation. This is manifested in the fact that existing results from convex optimization can be transferred to saddles. In this paper, such a transfer is carried out for the results of the online algorithm in the convex case, when the objective function satisfies the $\gamma$-growth condition.

Instrument of numerical-analytical modeling of characteristics of propagation of electromagnetic waves in chaotic space plasma with taking into account effects of gravitation is developed for interpretation of data of measurements of astrophysical precision instruments of new education. The task of propagation of waves in curved (Riemann’s) space is solved in Euclid’s space by introducing of the effective index of refraction of vacuum. The gravitational potential can be calculated for various model of distribution of mass of astrophysical objects and at solution of Poisson’s equation. As a result the effective index of refraction of vacuum can be evaluated. Approximate model of the effective index of refraction is suggested with condition that various objects additively contribute in total gravitational field. Calculation of the characteristics of electromagnetic waves in the gravitational field of astrophysical objects is performed by the approximation of geometrical optics with condition that spatial scales of index of refraction a lot more wavelength. Light differential equations in Euler’s form are formed the basis of numerical-analytical instrument of modeling of trajectory characteristic of waves. Chaotic inhomogeneities of space plasma are introduced by model of spatial correlation function of index of refraction. Calculations of refraction scattering of waves are performed by the approximation of geometrical optics. Integral equations for statistic moments of lateral deviations of beams in picture plane of observer are obtained. Integrals for moments are reduced to system of ordinary differential equations the firsts order with using analytical transformations for cooperative numerical calculation of arrange and meansquare deviations of light. Results of numerical-analytical modeling of trajectory picture of propagation of electromagnetic waves in interstellar space with taking into account impact of gravitational fields of space objects and refractive scattering of waves on inhomogeneities of index of refraction of surrounding plasma are shown. Based on the results of modeling quantitative estimation of conditions of stochastic blurring of the effect of gravitational lensing of electromagnetic waves at various frequency ranges is performed. It’s shown that operating frequencies of meter range of wavelengths represent conditional low-frequency limit for observational of the effect of gravitational lensing in stochastic space plasma. The offered instrument of numerical-analytical modeling can be used for analyze of structure of electromagnetic radiation of quasar propagating through group of galactic.

The article covered results of three-dimensional modeling of ecologic situation of shallow water on the example of the Azov Sea with using schemes of increased order of accuracy on multiprocessor computer system of Southern Federal University. Discrete analogs of convective and diffusive transfer operators of the fourth order of accuracy in the case of partial occupancy of cells were constructed and studied. The developed scheme of the high (fourth) order of accuracy were used for solving problems of aquatic ecology and modeling spatial distribution of polluting nutrients, which caused growth of phytoplankton, many species of which are toxic and harmful. The use of schemes of the high order of accuracy are improved the quality of input data and decreased the error in solutions of model tasks of aquatic ecology. Numerical experiments were conducted for the problem of transportation of substances on the basis of the schemes of the second and fourth orders of accuracy. They’re showed that the accuracy was increased in 48.7 times for diffusion-convection problem. The mathematical algorithm was proposed and numerically implemented, which designed to restore the bottom topography of shallow water on the basis of hydrographic data (water depth at individual points or contour level). The map of bottom relief of the Azov Sea was generated with using this algorithm. It’s used to build fields of currents calculated on the basis of hydrodynamic model. The fields of water flow currents were used as input data of the aquatic ecology models. The library of double-layered iterative methods was developed for solving of nine-diagonal difference equations. It occurs in discretization of model tasks of challenges of pollutants concentration, plankton and fish on multiprocessor computer system. It improved the precision of the calculated data and gave the possibility to obtain operational forecasts of changes in ecologic situation of shallow water in short time intervals.