The article is dedicated to the memory and scientific legacy of the prominent Ukrainian mathematician Heorhii Mykolajovych Polozhii (1914 1968). The main ideas of the theory of generalized analytic functions, the summary representation method are described and some new application results in mathematical modelling of nonlinear quasi-ideal processes in LEF layers are considered.

Hyperbolic-type differential equations and their iterations are widely used to solve problems related to vibration phenomena and other problems of mechanics and mathematical physics. The methods of solving such equations are the creation of differential and integral operators. In the article, differential operators are constructed that translate arbitrary functions into regular solutions of a hyperbolic equation of the second and higher orders. The Riquier problem for the hyperbolic equation of the fourth order is solved.

The paper proposes an approach for finding the optimal position of sources of known power for the quasi-linear Richards equation in a rectangular area. The Kirchhoff transformation is applied with the introduced scaling of coordinates and powers of submerged sources, which allows formulating a dimensionless problem. The task of this study is to find the position of submerged sources - such that the distribution of moisture at the final moment of time is close to the given values or the given target function.

The authors analyze dynamic processes of filtration in porous media and consider periodic porous media formed by a large number of “blocks” with low permeability, separated by a connected system of “faults” with high permeability. Taking into account the structure of such media in the modeling determines the dependence of filtration equations and their coefficients on small parameters characterizing the microscale of the porous medium and permeability of the blocks. The initial–boundary-value problems for nonstationary equations of filtration in such porous media are considered and the homogenized problems that determine the approximate asymptotic solutions to such problems are given. The homogenized problems are formulated as initial–boundary-value problems for integro-differential equations with convolutions. The estimates for the accuracy of the asymptotics and relevant convergence theorems are proved. Statements about solvability and regularity have been established for the problems that are optimal and do not depend on parameters.

The review describes the main stages of formation and development of the Kyiv School of Mathematical Theory of Filtration. This review focuses on the scientific ideas and results of its leader, a prominent Ukrainian scientist, Academician of the National Academy of Sciences of Ukraine Ivan Ivanovych Lyashko. The journal “Cybernetics and Systems Analysis” systematically publishes the works of I. I. Lyashko’s students, in which his ideas and achievements are further developed.

The authors propose an algorithm for finding the optimal source capacity for the two-dimensional quasi-linear Richards equation in a rectangular domain. The Kirchhoff transformation with scaling of coordinates and capacities of buried sources is used, which allows formulating a dimensionless problem. The existence of a solution to the problem of optimizing moisture transfer in an unsaturated porous medium is substantiated. The task of this study is to find the capacity of sources buried in a porous medium, such that the humidity distribution at the final instant of time is close to the given indicators or the objective function. The numerical solution approximates the optimal values of the sources.

The article is dedicated to several gradient based methods for solving a two-dimensional humidification problem, described by Richards equation. Several assumptions are made: water is assumed incompressible, external pressure and temperature are constant. The initial state and desired function are known, while the optimal source power should be calculated. Kirchhoff transformation is applied to the initial equation to simplify the stated problem. Time and space coordinates are scaled to get linear dimensionless equation, which can be easily discretized over space and time. Numerical methods are applied to rewrite and solve the system. Also gradient methods are applied for cases, where it is possible to define the optimization functional for every allowed source power.

Integral operators that transform arbitrary functions into regular solutions of hyperbolic equations of the second and higher orders are applied to solving boundary value problems. In particular, the Riquet problem for the Euler–Poisson–Darboux equation of the 4th order is posed and solved.

In this paper a one-dimensional nonlinear Richards equation describing fluid flow in porous medium with inserted equalpowered sources is studied. An experimental iterational method is proposed to find source power to minimize the deviation of received humidity values from target values. Modeling was performed using numerical difference approximation of derivatives, resulting into a system of nonlinear equations with dependence from previous time step. The offered method allows to perform modeling for different source power values, and chooses the most suitable one.Iterations stop when they reach average modular difference value less than calculation error of numerical difference scheme. Here explicit scheme was used to save time, equations were tested for unsaturated medium only to avoid flooding the area, so source power is tested with given limitations. Results of simulations and choice for next source power approximations are described and compared until solution is found. This approach is considered as experimental so we plan to perform more analysis in the future.

A Lagrangian formalism is developed, which allows finding the magnetic potential energy of interactions in a system of superconducting inductors (coils with constant magnetic flux) and permanent magnets (coils with direct current). The explicit form of potential energy for the magnetic system with constant magnetic fluxes and direct currents allows the stability analysis in such magnetic systems both at equilibrium and in motion. A number of applications are indicated, which will benefit from this approach such as modeling of cyber-physical or technical systems of magnetic levitation.

The Marchuk–Petrov mathematical model of antiviral immune response is generalized for complex consideration of diffusion perturbations, concentrated influences, body temperature response, and logistic population dynamics of viral elements and antibodies to the development of infectious disease. A step-by-step procedure for numerically asymptotic solution to the corresponding singularly perturbed problems with delays is developed. The authors present the results of computer simulation that illustrate the “model” reduction of the maximum level of antigens in the epicenter of infection due to their diffusion “scattering,” body temperature response, and logistic population dynamics of viruses on the nature of infectious disease, including the presence of concentrated sources of antigens. It is emphasized that such a systemic effect of these factors can reduce the initial supercritical concentration of antigens to a level after which their neutralization and excretion are provided by the existing level of immune protection, which is important in deciding whether to use external “therapeutic” effects.

The paper deals with the Korteweg-de Vries equation with variable coefficients and a small parameter at the highest derivative. The asymptotic step-like solution to the equation is obtained by the non-linear WKB technique. An algorithm of constructing the higher terms of the asymptotic step-like solutions is presented. The theorem on the accuracy of the higher asymptotic approximations is proven. The proposed technique is demonstrated by example of the equation with given variable coefficients. The main term and the first asymptotic approximation of the given example are found, their analysis is done and statement of the approximate solutions accuracy is presented.

Paper provides a research of the mathematical model of a superconducting magnetic suspension in zero gravity. The model consists of a special configuration of the superconducting inductors and uniform magnetic field. The stabilization of suspension in dependence of angle between the magnetic field induction vector and the axis of the inductance coil that is a suspended «free» rigid body is researched. Analysis of the model on stability of equilibrium is carried out and the conditions of spatial magnetic well existence providing the contactless confinement of a rigid body in zero gravity are found.

Integral operators based on the Riemann function, which transform arbitrary analytical functions into regular solutions of equations of elliptic, parabolic, and hyperbolic types of second order, are constructed. The Riemann operator method is generalized for the biaxisymmetric Helmholtz equation. A method for finding solutions to the above equations in analytical form is developed. In some cases, formulas for inverting integral representations of solutions are constructed. The conditions for solving the Cauchy problem for the axisymmetric Helmholtz equation are formulated.

The authors propose an analog of the Galerkin method for the initial–boundary-value problem that describes drug delivery in the artery wall using a drug-coated stent. The method of numerical solution of the initial–boundary-value problem is constructed and the theorems on its convergence to the solution are proved.

The Korteweg–de Vries equation with variable coefficients and a small parameter of the first level at the higher-order derivative is considered. The concept of a stepwise asymptotic solution is introduced. On the basis of the nonlinear WKB method, an algorithm for creating such solutions is developed and substantiated. The order on small parameter of the asymptotic accuracy of the constructed approximate solution satisfying the original equation is established.

Optimization and controllability problems for systems described by partial differential equations, where coefficients and the right-hand sides belong to different functional spaces, are considered. In particular, pharmacokinetic problems lead to such models. A model described by a general differential equation with zero initial and boundary conditions is analyzed. Coefficients are assumed positive in this area, concentrated sources are modeled by the Dirac delta function. The search of feasible control that minimizes the quality functional is performed. Based on the space of measurable and square-integrable functions, adjunction for functions smooth in the research area according to the norm, and conjugate problems are constructed. Negative spaces are introduced for the conjugate problem, and generalized solution to the problem is investigated.

Integral operators that translate arbitrary functions into regular solutions of a hyperbolic equation of second and higher orders are constructed. The Cauchy problem for the fourth-order hyperbolic equation is solved. The use of the theory of special functions made it possible to present solutions of partial derivative equations in a form convenient for the analysis. Along the way, integral convolution equations with special functions in the kernel are solved.

In this article we consider using modern fast alternation direction optimization methods for cancer diagnostics. During last years the number of people that suffer from cancer significantly increased. The aim of this work was to make research and to determine the most important factors that influence durability of life after surgery and to predict this durability. We collect data from a group of people with melanoma, define 36 characteristics that can influence durability of life and with help of LASSO problem and multiblock ADMM with Nesterov acceleration we determine the most important characteristics, f in d life expectancy and calculate the difference between real life durability and predicted values. Multiblock ADMM with Nesterov acceleration is an optimization method, based on combination of two approaches – splitting of original optimiza problem into N subproblems and finding Nesterov acceleration step on each iteration.