Aim. The purpose is to find exact solutions of boundary value problems and the Cauchy problem for the Poisson equation in a half-space with polynomial data. Methodology. The paper considers the Dirichlet and Neumann boundary value problems in a half-space and the Cauchy problem with polynomial data for the Poisson equation. These problems are solved using the Fourier transform of generalized functions of slow growth. Results. It is shown that the Cauchy problem with polynomial data for the Poisson equation has a solution that is a polynomial. This solution is the only one in the class of functions of slow growth in hyperplanes parallel to the hyperplane on which the initial conditions are specified. The polynomial solution is obtained explicitly. Each solution from an infinite set of solutions to the Dirichlet or Neumann problem is a solution to some Cauchy problem. Research implications. We have obtained exact solutions to boundary value problems and the Cauchy problem with polynomial data for the Poisson equation.

Aim. The purpose is to find an approximate solution to the first initial boundary value problem for a parabolic equation with a power-law nonlinearity. The problem is solved using an approximate analytical method based on the application of an a priori estimation of the solution to the problem for the linearization of the original equation. Methodology. The first step in applying the method is to reduce the nonlinear equation to the loaded equation, by replacing the nonlinear member with its integral in the spatial variable. Following this, an a priori estimate of the obtained problem is established in a suitable functional space. By integrating the loaded equation with respect to the spatial variable, a transition is made to the nonlinear ordinary differential equation associated with it. The latter is linearized using the a priori estimate of the loaded problem, in which the upper bound of inequality is chosen. Results. A formula is obtained that expresses the solution to the loaded equation in terms of its norm and the solution to the associated ordinary differential equation. Approximation of the solution to a nonlinear equation is proposed to be performed by using an iterative process for solving a sequence of linear problems. An example illustrating the application of the method to a model problem is presented. Research implications. The applied procedure makes it possible to obtain an analytical expression for an approximate solution to a nonlinear problem. The described method can be applied to partial differential equations of any type and order, containing the natural degree of the desired function or its derivative.

Aim. The paper presents an experimental study of thermoelectric properties of colloidal solutions and the effect of dialysis purification on these properties, using the example of colloidal solutions of silver iodide. Methodology. The paper uses standard methods for measuring the coefficient of thermoelectric EMF and the coefficient of electrical conductivity used for electrolyte and colloidal solutions. To purify colloidal solutions from the ions present in them, the method of dialysis purification using semipermeable membranes is used. Results. It is shown that during the removal of ions from colloidal solutions, their thermoelectric EMF increases in absolute value, while the coefficient of electrical conductivity decreases. The observed increase cannot be explained only by the effect of an increase in the thermoelectric strength of the ionic electrolyte solution with a decrease in its concentration. The results obtained can be explained in the framework of the thermodynamics of irreversible processes as a consequence of an increase in the transfer numbers of large colloidal particles, which, unlike ions, have initially high values of the transfer heat. Research implications. The results of the study contribute to the theory of transport phenomena in dispersed colloidal systems.

Aim. We have found asymptotically exact values for the distribution function of pairs of molecules in a shock-compressed, binary mixture of gases with a strong difference in the concentrations and molecular weights of its components. Methodology. The research relies on the mathematical methods of theoretical physics related to the calculation of the threshold frequency of collisions based on the kinetic Boltzmann equation. Results. Asymptotically exact analytical expressions are found for the distribution functions of pairs of molecules by the absolute values of their relative velocities. The maxima of these functions are also determined in pairs: light-light, light-heavy and heavy-heavy components. These maxima correspond to the greatest intensities of the effects of high-speed overshoot in the components of the shock-compressed gas mixture. Research implication. The effect of high-speed overshoot in the components of a shock-compressed gas mixture is realized by experimentally modeling it in shock pipes (for example, in the processes of pyrolysis of carbon and carbon-hydrogen compounds). The results obtained in this work are essential for the optimal performance of such experiments.

Aim. The purpose is to determine the relationship between the material structure parameters and the optical properties of its constituent substances on the one hand and its scattering and absorption characteristics on the other hand. Methodology. Use is made of terahertz spectral-selective transmission measurements of packets of thin dielectric webs with periodically modulated thickness (fabrics) depending on the number of layers in the packet. Results. A method for measuring the scattering depth and the penetration depth of radiation into the substance of the scattering and absorbing dielectric structure is developed. Research implications. When describing the scattering of electromagnetic radiation in a dielectric structure, the macroscopic wave approximation is valid.

Aim of the article is to show that under the action of raying solar radiation on the surface of the metal plate will occur its resonance heating at the frequency where ω0 - is the optical frequency of radiation, εF - is an Fermi’s energy, - is Planck constant. Methodology. Thanks to the method of the kinetic equation, the dependence of the thermal conductivity coefficient on the coordinate counted deep into the plate is calculated x. Results. The analytical value of the surface temperature heated by a resonance optical heat source for various types of materials, in particular, for metal, sand and dielectric, is given. Research implications. A theoretical substantiation of the possibility of resonance heating of surfaces by a raying flow of energy is proposed.