Extension of Darbo’s fixed point theorem via shifting distance functions and its application

In this paper, we discuss solvability of infinite system of fractional integral equations (FIE) of mixed type. To achieve this goal, we first use shifting distance function to establish a new generalization of Darbo’s fixed point theorem, and then apply it to the FIEs to establish the existence of solution on tempered sequence space. Finally, we verify our results by considering a suitable example.

Well-posedness of the Cauchy problem for system of oscillators on 2D–lattice in weighted $l^2$-spaces

We consider an infinite system of ordinary differential equations that describes the dynamics of an infinite system of linearly coupled nonlinear oscillators on a two dimensional integer-valued lattice. It is assumed that each oscillator interacts linearly with its four nearest neighbors and the oscillators are at the rest at infinity. We study the initial value problem (the Cauchy problem) for such system. This system naturally can be considered as an operator-differential equation in the Hilbert, or even Banach, spaces of sequences. We note that $l^2$ is the simplest choice of such spaces. With this choice of the configuration space, the phase space is $l^2\times l^2$, and the equation can be written in the Hamiltonian form with the Hamiltonian $H$. Recall that from a physical point of view the Hamiltonian represents the full energy of the system, i.e., the sum of kinetic and potential energy. Note that the Hamiltonian $H$ is a conserved quantity, i.e., for any solution of equation the Hamiltonian is constant. For this space, there are some results on the global solvability of the corresponding Cauchy problem. In the present paper, results on the $l^2$-well-posedness are extended to weighted $l^2$-spaces $l^2_\Theta$. We suppose that the weight $\Theta$ satisfies some regularity assumption. Under some assumptions for nonlinearity and coefficients of the equation, we prove that every solution of the Cauchy problem from $C^2\left((-T, T); l^2)$ belongs to $C^2\left((-T, T); l^2_\Theta\right)$. And we obtain the results on existence of a unique global solutions of the Cauchy problem for system of oscillators on a two-dimensional lattice in a wide class of weighted $l^2$-spaces. These results can be applied to discrete sine-Gordon type equations and discrete Klein-Gordon type equations on a two-dimensional lattice. In particular, the Cauchy problems for these equations are globally well-posed in every weighted $l^2$-space with a regular weight.

On the Stability and Null-Controllability of an Infinite System of Linear Differential Equations

In this work, the null controllability problem for a linear system in ℓ2 is considered, where the matrix of a linear operator describing the system is an infinite matrix with $\lambda \in \mathbb {R}$ λ ∈ ℝ on the main diagonal and 1s above it. We show that the system is asymptotically stable if and only if λ ≤− 1, which shows the fine difference between the finite and the infinite-dimensional systems. When λ ≤− 1 we also show that the system is null controllable in large. Further we show a dependence of the stability on the norm, i.e. the same system considered $\ell ^{\infty }$ ℓ ∞ is not asymptotically stable if λ = − 1.

About Conditions of Spatial Collapse in an Infinite System of Bose Particles

Using the variational principle, we show that the condition of spatial collapse in a Bose gas is not determined by the value of the scattering length of the interaction potential between particles contrary to the result following from the Gross–Pitaevskii equation, where the collapse should take place at a negative scattering length.

An Analytical Approach to Obtaining the Stress Field of a Cylindrical Shell with a Circular Hole under Tension

The problem of a cylindrical shell with a circular hole under uniaxial tension is considered. The main obstacle of solving this problem is the necessity to find such coefficients in the expansion of the solution into a sum of basis functions, for which this solution satisfies the boundary conditions. The study of the classical works led to understanding that none of the so far proposed approaches can be considered successfully, and the results of these approaches differ, so it is not clear, which results can be used as a basis. In the present paper, a new analytical approach to studying this issue is proposed. It allows expanding the range of applicability of the solution and gives the opportunity for the analytical study of the stress state. The idea consists in expanding each of the basis functions in a Fourier series by dividing the variables, which allows obtaining explicitly an infinite system of algebraic equations for finding coefficients. One of the important steps of this research is that the authors were able to prove which exact equation is a linear combination of the others and exclude, which made it possible to compose a reduced system for finding unknown coefficients. The proposed approach, in contrast to most classical works, does not impose mathematical restrictions on the values of the main parameter characterizing the cylindrical shell. The existing restrictions are of mechanical nature, as larger cutouts require another model. Moreover, the numerical results obtained by the new method are presented in a fairly complete manner and they are compared with the results of the classical works.

Vibrations of a Cantilevered Thick Plate

It has been for the first time that an analytical solution to the problem of free vibrations of a cantilevered thick orthotropic plate is presented. This problem is quite cumbersome for using the exact methods of the theory of elasticity; therefore, methods based on the variational approach were developed to solve it. The paper suggests using the superposition method to construct a general solution of the vibration equations of a plate in the series form of particular solutions obtained with the help of a variables separation. The particular solutions of one of the coordinates are built in the form of trigonometric functions of a special type (modified trigonometric system). The constructed solution, in contrast to the solutions known in the literature on the basis of the variational approach, accurately satisfies the equations of vibrations. The use of a modified trigonometric system of functions makes it possible to obtain uniform formulas for even and odd vibration shapes and to reduce the quantity of boundary conditions on the plate sides from twelve to nine ones, while five of the nine boundary conditions are also accurately satisfied. The structure of the presented solution on the plate boundary is such that, each of the kinematic or force characteristics of the plate is represented as a sum of two series, i.e. a trigonometric series and a series in hyperbolic functions. Remaining boundary conditions make it possible to obtain an infinite system of linear algebraic equations with respect to the unknown coefficients of the series representing the solution. The convergence of the solution by the reduction method of the infinite system is investigated numerically. Examples of the numerical implementation are given; numerical studies of the spectrum of natural frequencies of the cantilevered thick plate were carried out based on the obtained solution, both with varying elastic characteristics of the material and with varying geometric parameters.

One approach to the axisymmetric problem of impact of fine shells of the S.P. Timoshenko type on elastic half-space

Refined model of S.P. Timoshenko makes it possible to consider the shear and the inertia rotation of the transverse section of the shell. Disturbances spread in the shells of S.P. Timoshenko type with finite speed. Therefore, to study the dynamics of propagation of wave processes in the fine shells of S.P. Timoshenko type is an important aspect as well as it is important to investigate a wave processes of the impact, shock in elastic foundation in which a striker is penetrating. The method of the outcoming dynamics problems to solve an infinite system of integral equations Volterra of the second kind and the convergence of this solution are well studied. Such approach has been successfully used for cases of the investigation of problems of the impact a hard bodies and an elastic fine shells of the Kirchhoff-Love type on elastic a half-space and a layer. In this paper an attempt is made to solve the axisymmetric problem of the impact of an elastic fine spheric shell of the S.P. Timoshenko type on an elastic half-space using the method of the outcoming dynamics problems to solve an infinite system of integral equations Volterra of the second kind. It is shown that this approach is not acceptable for investigated in this paper axisymmetric problem. The discretization using the Gregory methods for numerical integration and Adams for solving the Cauchy problem of the reduced infinite system of Volterra equations of the second kind results in a poorly defined system of linear algebraic equations: as the size of reduction increases the determinant of such a system to aim at infinity. This technique does not allow to solve plane and axisymmetric problems of dynamics for fine shells of the S.P. Timoshenko type and elastic bodies. This shows the limitations of this approach and leads to the feasibility of developing other mathematical approaches and models. It should be noted that to calibrate the computational process in the elastoplastic formulation at the elastic stage, it is convenient and expedient to use the technique of the outcoming dynamics problems to solve an infinite system of integral equations Volterra of the second kind.

An Infinite System of Fractional Order with p-Laplacian Operator in a Tempered Sequence Space via Measure of Noncompactness Technique

In the current study, a new class of an infinite system of two distinct fractional orders with p-Laplacian operator is presented. Our mathematical model is introduced with the Caputo–Katugampola fractional derivative which is considered a generalization to the Caputo and Hadamard fractional derivatives. In a new sequence space associated with a tempered sequence and the sequence space c0 (the space of convergent sequences to zero), a suitable new Hausdorff measure of noncompactness form is provided. This formula is applied to discuss the existence of a solution to our infinite system through applying Darbo’s theorem which extends both the classical Banach contraction principle and the Schauder fixed point theorem.

Analysis of Directional Properties of Angular Horn Antenna

In this paper, horn antennas are considered to belong to the class of aperture antennas which usually include a sound wave reflector and an electroacoustic transducer. For the variant of technical implementation of the electroacoustic transducer in the form of a corner antenna, the problem of sound emission by such an antenna is solved . taking into account the repeated reflection of emitted sound waves from the antenna design elements. The study of the acoustic properties of such an antenna was carried out taking into account a number of assumptions. 'what material, the thickness of the walls of the mouthpiece is infinitesimal. These conditions are supplemented by the known conditions of radiation at infinity. All the above assumptions make it possible to greatly simplify the solution of the problem of sound radiation by an angular horn antenna. To do this, the Helmholtz equation under boundary conditions was solved by the method of connected fields in multiconnected domains, corresponding to the physical model of the antenna. The radiation field of such an antenna is presented in the form of three partial regions, which in turn, according to the method of partial regions - in the form of Fourier series expansions, the coefficients of which are determined by solving differential equations describing piezoceramic transducer oscillations and wave processes. in acoustic environments in contact with it. The solution of the Helmholtz equation is reduced to the solution of an infinite system of linear algebraic equations taking into account the above assumptions, as well as the conditions of field continuity at the boundary of partial domains, we obtain an infinite system of linear algebraic equations. Based on the system, an approximate expression for the normalized radiation pattern is obtained. Studying the features of the directional properties of the angular antenna by direct analysis of the expression is not possible. Therefore, such a study was carried out on the basis of calculated directivity diagrams obtained using a computer for a wide range of wave sizes and geometrical characteristics of the angular antenna. cylindrical electroacoustic transducers. In all calculations, a uniform distribution of the oscillating velocity on the surface of the emitter was chosen Analyzing the obtained data, there is a pronounced dependence of the shape of the pattern on the magnitude of the wave size of the speaker. This is manifested in the fact that the main petal splits into two or even three petals, as well as in increasing the overall sharpness of the pattern.