Purpose. We have found exact solutions to boundary-value problems for the inhomogeneous Helmholtz equation with the polynomial right-hand side in a multidimensional infinite layer bounded by two hyperplanes. Methodology and Approach. The paper considers Dirichlet and Dirichlet-Neumann boundary-value problems with polynomials in the right-hand sides of the boundary conditions. The Fourier transform of generalized functions of slow growth is applied. Results. It is shown that the Dirichlet and Dirichlet-Neumann boundary-value problems with polynomials in the right-hand sides of the boundary conditions for the inhomogeneous Helmholtz equation with the polynomial right-hand side have a solution that is a quasi-polynomial containing, in addition to power functions, hyperbolic or trigonometric functions. This solution is unique in the class of functions of slow growth if the parameter of the equation is not an eigenvalue. An algorithm for constructing this solution is presented and examples are considered. Theoretical and Practical Implications. Exact solutions to boundary-value problems for one of the well-known equations of mathematical physics have been obtained.

Purpose. We investigate the methods for reducing first-order partial differential equations with power nonlinearity to loaded equations using the example of the Hopf equation. The solution to the reduced equation is applied to a sequential approximation of the solution to a nonlinear equation by solutions to a linearized equation. Methodology and Approach. Two methods of reduction are proposed. In the first of them, the desired function in the nonlinear term is replaced by its average value for the spatial variable. To solve an auxiliary ordinary differential equation, a second reduction is possible, namely, to an algebraic equation. In the second method, an integral transition is made to the loaded equation. The resulting auxiliary equation is solved using a partial solution to the corresponding differential inequality. Results. The proposed methods of reduction after some additional transformations allow one to obtain initial approximations for starting the iterative process of searching for approximate solutions to a nonlinear problem. The possibility of using partial solutions associated with the differential inequality equation is shown. Theoretical and Practical Implications. We have demonstrated the possibility of applying reduction to loaded equations to find approximate solutions to first-order partial differential equations with power nonlinearity.

Purpose. We have identified the impact of virtual reality (distorted or false information) on the human nervous system. Methodology and Approach. We use a mathematical model of an artificial neural network (ANN), built in the image and likeness of the human nervous system. To conduct a correct computer experiment, we selected: (1) the impact of the game system on the ANN, which provides a large amount of virtual information, and (2) the ANN was configured only for visual perception. Distorted (virtual) information is sent to the input of the pre-trained ANN, and the ANS is retrained taking into account the distorted information. After several sessions in the virtual reality environment of the ANN, the response of the ANN to the original actual reality is studied. A Python program has been developed to manage the ANS. Results. It is shown that the conducted model experiment on the effect of virtual reality on the ANN, previously trained on traditional objects of the surrounding reality, makes these objects either difficult to recognize or, in general, unrecognizable. Theoretical and Practical implications. A suitable computer model was built for the first time that allows one to study the effect of virtual reality on the human nervous system; it is shown for the first time quantitatively and qualitatively how virtual reality affects the ANN and, consequently, the human nervous system.

Purpose. We have developed a model of mirror-diffuse boundary conditions that satisfy the Fuchs and Soffer models. Methodology and Approach. The model is based on mirror-diffuse boundary conditions. In this case, the possible dependence of the specularity coefficient on the angle of incidence of the electron on the surface is taken into account. Results. The proposed boundary conditions satisfy the Andreev reflection coefficient with an almost tangent incidence of the electrons on the surface. In the limiting case they also reproduce the known mirror-diffuse Fuchs boundary conditions. Theoretical and Practical Implications. The proposed model satisfies all known models. This model can be used to describe kinetic processes near the surface of a metal, in thin films, wires, in small metal particles, and when describing the size effect in a metal.

Purpose. We study of the dynamics of atomic-molecular conversion in a Bose-Einstein condensate. Methodology and Approach. The paper explores the behavior of the potential energy of some nonlinear oscillator as a function of the magnitude of the photon density of the second pulse. Results. An equation is found for the density of photons of the second pulse in the system of Bose-condensed atoms and molecules. Theoretical and Practical Implications. We have obtained the solutions and found critical values of parameters at which evolution modes change.

Purpose. The aim of the paper is to study the effect of the magnetic field on a jet of a stationary plasma thruster. Methodology and Approach. Considering the collisionless movement of ions, it is possible to derive an expression for the distribution function of ions coming out of a ring hole. Then, a scheme is constructed to calculate the density of ions as a corresponding integral of the distribution function. Results. We have obtained the patterns of the ion density distribution in a three-dimensional space, which show the possibility of controlling the thrust vector by the magnetic field. Theoretical and Practical Implications. The obtained results indicate a further development of the direction in which the methods of kinetic theory in plasma are used. The results of this work are of practical importance for specialists involved in the design and development of new types of electric propulsion engines.

Purpose. We have developed a method for calculating the surface tension of molten non-transition metals. Methodology and Approach. The proposed method is based on the model assuming that the change in surface energy of a drop of molten metal spreading on the substrate corresponds to the work performed by the drop weight and electrochemical interaction. For the calculated values of surface tension of molten metals, statistical corrections and electrochemical correction are obtained. Statistical corrections are based on the calculation of deviation of the molten metals’ surface tension from the values approximated by regression curves. The electrochemical correction takes into account the change in the surface tension of molten metals due to accumulation of atoms near the surface of the drop. Results. The calculated values of the surface tension of molten metals are found, in agreement with the experimental data. Practical relevance. The obtained results make it possible to use this method for calculating the surface tension of non-transition metals, the values of which are difficult to measure experimentally.

Purpose. We examine experimentally the permittivity in the microwave range and approximate numerically the frequency dependences of the permittivity of a nematic liquid crystal. Methodology and Approach. At a frequency of 39 GHz, measurements of the permittivity and dielectric loss are performed using the waveguide method, and at frequencies up to 10MHz, using the capacitive method. Results. Experimental values of the permittivity of a nematic liquid crystal LC-1289 at a frequency of 39 GHz are obtained. Numerical approximation of dielectric spectra is performed. The dipole relaxation times associated with the rotation of molecules around the short and long axes and the corresponding activation energies are calculated. Theoretical and Practical Implications. The values of the permittivity and dielectric loss are obtained for evaluating the performance characteristics of the working bodies of information display devices.