Modeling and visualizing of the formation of a snub dodecahedron in the AutoCAD system

The article is devoted to modeling and visualization of the formation of flat-nosed (snub-nosed) dodecahedron (snub dodecahedron). The purpose of the research is to model the snub dodecahedron (flat-nosed dodecahedron) and visualize the process of its formation. The formation of the faces of the flat-nosed dodecahedron consists in the truncation of the edges and vertices of the Platonic dodecahedron with the subsequent rotation of the new faces around their centers. The values of the truncation of the dodecahedron edges, the angle of rotation of the faces and the length of the edge of the flat-nosed dodecahedron are the parameters of three equations composed as the distances between the vertices of triangles located between the faces of the snub dodecahedron. The solution of these equations was carried out by the method of successive approximations. The results of the calculations were used to create an electronic model of the flat-nosed dodecahedron and visualize its formation. The task was generally achieved in the AutoCAD system using programs in the AutoLISP language. Software has been created for calculating the parameters of modeling a snub dodecahedron and visualizing its formation.

MODELING OF STRESS-DEFORMED STATE OF BENDED REINFORCED CONCRETE ELEMENTS IN ZONES OF CLEAN AND TRANSVERSE BEND

The article deals with the analysis of the stress-strain state (SSS) of a bent reinforced concrete element in zones of pure and transverse bending. It is assumed that a bent element in the process of loading (after the formation of normal and oblique cracks) is divided into blocks, united by uncracked concrete and reinforcement that has adhesion to concrete. SSS was formed using the results of experimental studies of special prototypes in the PC “Lira-SAPR”. A fi nite element model of a prototype has been developed in the form of a reinforced concrete rectangular beam loaded with two identical concentrated forces in the span. By the method of successive approximations, the process of formation and formation of a system of cracks is realized, with which the beam is divided into blocks during loading. The results of calculating the fi nite element model and their comparison with experimental data are presented.

Calculation of the Shaping of a Space Umbrella Antenna During Strong Bending of Radial Rods Connected Along Parallels by Tensile Cables

Abstract— A cyclically symmetric umbrella antenna is considered, the frame of which consists of flexible inextensible radial rods connected in nodes along parallels by tensile cables. In the initial transport position, the multilink rods are packed in packages oriented in the direction of the system axis. After the packing ties are removed, the rods are deployed in radial planes under the action of elastic springs connecting the links, and are fixed in rectilinear positions at a given angle with respect to the axis, at which all cables connecting the same type of rod nodes take the form of regular polygons, while remaining loose. Further, under the action of the force of a damping hydraulic cylinder with pre-compressed springs, the root parts of all rods are slowly turned to the stops. In the final position, the radial rods, connected at the nodes by tensioned cables, take a curved shape. The tensile stiffnesses of the cables are determined so that the radial and axial coordinates of the nodes of the curved rods coincide with the coordinates of the points of the given surface of revolution. A model of strong bending of a flexible inextensible rod is constructed taking into account the unknown radial reactions of tensioned cables acting on it at the nodes. The links of the rod are considered as “cantilever” elements connected in series at the nodes in local coordinate systems, which can make large displacements and turns. The bending of each element is described by two specified functions, the shrinkage of the element due to bending is taken into account in a quadratic approximation. The obtained nonlinear deformation equations of the system, taking into account the geometric connections at the nodes, are solved by the method of successive approximations with respect to the unknown reactions of the cables. The obtained values of the reactions are then used to determine the required tensile stiffness of the cables at the given coordinates of the nodes. As an example of the calculation, a parabolic antenna is considered for various numbers of radial rods and components of links. The estimates of the accuracy of the proposed computational model of the antenna shaping are carried out.

Inverse Problem for the Sobolev Type Equation of Higher Order

The article investigates the inverse problem for a complete, inhomogeneous, higher-order Sobolev type equation, together with the Cauchy and overdetermination conditions. This problem was reduced to two equivalent problems in the aggregate: regular and singular. For these problems, the theory of polynomially bounded operator pencils is used. The unknown coefficient of the original equation is restored using the method of successive approximations. The main result of this work is a theorem on the unique solvability of the original problem. This study continues and generalizes the authors’ previous research in this area. All the obtained results can be applied to the mathematical modeling of various processes and phenomena that fit the problem under study.

On one class of solutions of the quasilinear matrix differential equations

In the mathematical description of various phenomena and processes that arise in mathematical physics, electrical engineering, economics, one has to deal with matrix differential equations. Therefore, these equations are relevant both for mathematicians and for specialists in other areas of natural science. Many studies are devoted to them, in which the solvability of matrix equations in various function spaces, boundary value problems for matrix differential equations, and other problems were investigated. In this article, a quasilinear matrix equation is considered, the coefficients of which can be represented in the form of absolutely and uniformly converging Fourier series with coefficients and frequency slowly varying in a certain sense. The problem is posed of obtaining sufficient conditions for the existence of particular solutions of a similar structure for the equation under consideration. For this purpose, the corresponding linear equation is considered first. It is written down in component-wise form, and, based on the assumptions made, the existence of the only particular solution of the specified structure is proved. Then, using the method of successive approximations and the principle of contracting mappings, the existence of a unique particular solution of the indicated structure for the original quasilinear equation are proved.

On the singular Volterra integral equation of the boundary value problem for heat conduction in a degenerating domain

In this paper, we consider a singular Volterra type integral equation of the second kind, to which some boundary value problems of heat conduction in domains with a boundary varying with time are reduced by the method of thermal potentials. The peculiarity of such problems is that the domain degenerates into a point at the initial moment of time. Accordingly, a distinctive feature of the integral equation under study is that the integral of the kernel, as the upper limit of integration tends to the lower one, is not equal to zero. This circumstance does not allow solving this equation by the method of successive approximations. We constructed the general solution of the corresponding characteristic equation and found the solution of the complete integral equation by the Carleman–Vekua method of equivalent regularization. It is shown that the corresponding homogeneous integral equation has a nonzero solution.

Interaction of acoustic waves with moving boundary surfaces

A quasi-static approximation is considered for the interaction of a probing ultrasonic beam with a vibrating boundary surface. The model is considered in the form of a boundary value problem, presented in the form of d'Alembert. The method of successive approximations was used for the solution. The error arising from this interaction is established. Keywords: quasi-static approximation, boundary value problem, d'Alembert form, Doppler effect, rheological medium. [email protected]; [email protected]

Solving Some Derivative Equations Fractional Order Nonlinear Partials Using the Some Blaise Abbo Method

In this paper, we propose the solution of some nonlinear partial differential equations of  fractional order that modeled diffusion, convection and reaction problems. For the solution of these equations we will use the SBA method which is a method based on the combination of the Adomian Decomposition Method (ADM), the Picard's principle  and the method of successive approximations.   

On solvability of one class of third order differential equations

UDC 517.9 One-dimensional mixed problem for one class of third order partial differential equation with nonlinear right-hand side is considered. The concept of generalized solution for this problem is introduced. By the Fourier method, the problem of existence and uniqueness of generalized solution for this problem is reduced to the problem of solvability of the countable system of nonlinear integro-differential equations. Using Bellman's inequality, the uniqueness of generalized solution is proved. Under some conditions on initial functions and the right-hand side of the equation, the existence theorem for the generalized solution is proved using the method of successive approximations.

Constructing C0-Semigroups via Picard Iterations and Generating Functions: An Application to a Black–Scholes Integro-Differential Operator

An alternative approach is proposed for constructing a strongly continuous semigroup based on the classical method of successive approximations, or Picard iterations, together with generating functions. An application to a Black–Scholes integro-differential operator which arises in the pricing of European options under jump-diffusion dynamics is provided. The semigroup is expressed as the Mellin convolution of time-inhomogeneous jump and Black–Scholes kernel functions. Other applications to the heat and transport equations are also given. The connection of the proposed approach to the Adomian decomposition method is explored.