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In this paper, we study a nonlinear integro-differential Volterra equation with a fractional Caputo derivative. Based on techniques derived from a study of classical Volterra equations, namely Picard's iterative sequence and the product integration method, we propose a complete analytical and numerical study of this equation. Our study is closed by the development of two numerical examples.

We analyse red refinements of tetrahedral partitions and prove that the measure of degeneracy of some produced tetrahedra may tend to infinity if refinements are constructed in an inappropriate way. The maximum angle condition is shown to be violated in these cases as well.

The work considers problem of reconstruction of differential equations parameters, describing anomalous diffusion processes, on the base of known solutions. As a tool, is used the variational interpolation method elaborated by the authors earlier. The reconstruction time-dependence of diffusivity and determination of fractional time- and space-derivatives order in anomalous diffusion equation is demonstrated. There is shown a possibility of sufficient accuracy with insignificant computational expanses.

In this paper, we consider P ²₀- P ¹ mixed finite element approximations of a class of nonlinear parabolic equations. The backward Euler scheme for temporal discretization is used. Firstly, a new mixed projection is defined and the related a priori error estimates are proved. Secondly, optimal a priori error estimates for pressure variable and velocity variable are derived. Finally, a numerical example is presented to verify the theoretical results.

The paper presents the algorithm of exponential transformation (biasing) and its randomized modification with branching of a Markov chain trajectory for solving the problems of gamma-ray transport in an inhomogeneous medium. These algorithms were applied to a maximum (majorant) cross-section method or the Woodcock tracking which is extremely efficient for the simulation in an inhomogeneous medium. On an example of gamma-ray transport through a thick water slab containing a random amount of air or Al balls, the numerical study of the above algorithms in comparison with the standard simulation algorithm is performed.

In the present work, a numerical scheme is constructed for the approximation of a class of Volterra integral equations of the convolution type with highly oscillatory kernels. The proposed numerical technique transforms the Volterra integral equations of the convolution type into simple algebraic equations. By an inverse transform the problem is converted into an integral representation in the complex plane, and then computed by a suitable quadrature formula. The numerical scheme is applied for a class of linear and nonlinear Volterra integral equations of the convolution type with highly oscillatory kernels, and some of the obtained results are compared with the methods available in the literature. The main advantage of the present scheme is the transformation of a highly oscillatory problem to a non-oscillatory and simple problem. So a large class of a similar type of integral equations having kernels of a highly oscillatory type can be very effectively approximated.

The samples of humic preparations (HP) containing macro- and microelements of plant nutrition were obtained on the basis of humic acids (HA) isolated from brown coal of the Tisul deposit of the Kansk-Achinsk coal basin. The biological activity of humic preparations was tested under laboratory and field conditions on stony and loamy soils using the seeds of high-quality wheat varieties ‘Iren’ and ‘Novosibirskaya 89’. A comparative assessment of the influence of the content of microelements cobalt, manganese, copper and zinc in humic preparations on the growth parameters of wheat was carried out. It is shown that the effectiveness of the use of humic preparations depends on the concentrations of all components, the method of their introduction, as well as on the edaphic properties of soil substrates.

The granulometric and morphological composition of fine coal powders prepared according to a special technique for the size classes -0.2 +0.1, -0.1 +0.063, -0.063 +0.04, -0.04 mm from ten different grades of coal (B, D, G, Zh, K, KS, OS, SS, T, A) of the Kuznetsk coal basin was investigated. Technical and elemental analysis, petrographic studies were carried out with the coal samples using standard methods. The set of values of the reflectivity of vitrinite, the sum of the fusenized components, the thickness of the plastic layer and the yield of volatile substances allowed us to establish the grade identity of the studied samples in accordance with the unified classification of coals on the basis of genetic and technological parameters. According to the results obtained by SEM, changes in the structure of coals and the relief of the fracture surface pattern are observed in the series of metamorphism of the studied coal samples from brown coal to anthracite. The granulometric composition of coal powders was determined by laser diffraction when the samples were dispersed in an aqueous medium with the addition of a surfactant. The particle size distribution in coal powders of the fraction -0.04 mm for all grades of coal is characterized as monomodal, in the fractions -0.063 +0.04 mm - as monomodal asymmetric, in fractions -0.1 +0.063 and -0.2 +0.1 mm - as bimodal. These fractions contain irregularly shaped particles (needle-shaped, oval), as determined by means of SEM. According to the results of the studies of KS grade coal by means of NMR spectroscopy, it is established that the content of aliphatic carbon (CH₃) in the samples increases insignificantly with a decrease in the size of the coal powder fraction - from 3.05 % for the fraction -0.2 +0.1 mm to 3.73 % for the fraction -0.04 mm. The values of the aromaticity index f _a for all fractions remain constant.

Features of phase formation and properties of solid solutions in the nanostructured FePt system synthesized by reducing the aqueous solutions of precursors by hydrazine hydrate are considered. It is established by means of X-ray diffraction studies and elemental analysis involving optical emission spectroscopy that limited solid solutions with the fcc type structure are formed under the conditions of the samples with the component ratio Pt/Fe ≥ 0.5. With iron content ( С _Fe) ≤ 11.6±0.7 at. %, the system is monophase (an fcc-solid solution), with higher iron content it is composed of two phases (an fcc-solid solution with С_Fe = 11.6±0.7 at. % and a diffraction-invisible phase relatively enriched with iron). The size of the coherent scattering region for the solid solution varies from 7 to 9 nm. The presence of Fe³⁺ was revealed in all samples by means of electron paramagnetic resonance. Small amounts of iron hydroxide and oxides were detected by means of XRD in the samples with a high iron content. The temperature regions of O₂ and CO₂ desorption, thermal decomposition of extrinsic FeO(OH) and crystallite agglomeration accompanied by phase transformations were determined in the studies of thermally stimulated processes.

The properties of the coals of Tuva and Mongolia are considered for the purpose of organizing the processing technology for the production of coal smokeless briquettes that reduce the formation of smog and air pollution from private houses with coal heating. Experimental and industrial thermolysis units have been developed and manufactured for conducting research works aimed at the production of smokeless fuel. Fuel briquettes were obtained by pressing the solid products of coal thermolysis (semi-coke) at an elevated temperature in the periodic mode on hydraulic presses in Russia and Mongolia.