Countable System Research Articles

MIXED PROBLEM FOR A NONLINEAR IMPULSIVE DIFFERENTIAL EQUATION OF PARABOLIC TYPE

In this paper, we consider a nonlinear impulsive parabolic type partial differential equation with nonlinear impulsive conditions. Dirichlet type boundary value conditions with respect to spatial variable is used, and eigenvalues and eigenfunctions of the spectral problem are founded. The Fourier method of the separation of variables is applied. A countable system of nonlinear functional equations is obtained with respect to the Fourier coefficients of the unknown function. A theorem on a unique solvability of the countable system of nonlinear functional equations is proved by the method of successive approximations. A criteria of uniqueness and existence of a solution for the nonlinear impulsive mixed problem is obtained. A solution of the mixed problem is derived in the form of the Fourier series. The absolute and uniform convergence of the Fourier series is proved.

THE EXISTENCE OF GENERALIZED SOLUTION TO ONEDIMENSIONAL MIXED PROBLEM FOR ONE CLASS OF THIRD ORDER NONILINEAR PSEUDOPARABOLIC EQUATIONS

Abstract. This work is dedicated to the study of existence in small of generalized solution of onedimensional mixed problem for one class of third order nonlinear pseudoparabolic equations. Conception of generalized solution for mixed problem under consideration is introduced. After applying Fourier method, the solution of original problem is reduced to the solution of some countable system of nonlinear inteqro-differential equations in unknown Fourier coefficients of the sought solution. Besides, existence theorem in small of generalized solution of the mixed problem is proved by contracted mappings principle.

Solvability of a Mixed Problem for an Integro-Differential Equation with the Hilfer Type Fractional Analogue of the Barenblatt–Zheltov–Kochina Operator

In the article a partial integro-differential equation with degenerate kernel and Hilfer fractional operator is considered. The issues of unique classical solvability and construction of a solution to a mixed problem for a homogeneous integro-differential equation containing a Hilfer fractional analogue of the Barenblatt–Zheltov–Kochina operator are studied. The Fourier series method based on the separation of variables was used. By the aid of the Mittag–Leffler function is obtained a countable system. Sufficient coefficient conditions for unique classical solvability of the mixed problem are established. The absolute and uniform convergence of the obtained series is proved.

Comments on “Fractal set of generalized countable partial iterated function system with generalized contractions via partial Hausdorff metric”

The purpose of this work is to fill a gap in the article Priya and Uthayakumar (2022) [13]. To prove fractals (fixed points), the authors of that article tried to connect two independent conceptual domains. This results in their main theorem being wrong and becoming a conjecture. This paper scrutinizes and presents both the concepts above to demonstrate fractals in the settings posed in that article. Using a proper definition, the correct theorem to achieve the purpose of that article is discussed.

Bending Vibrations of an Elastic Rod Controlled by Piezoelectric Forces

Bending vibrations of a thin elastic rod of rectangular cross-section are studied. A number of piezoelectric actuators (elements) is symmetrically attached without gaps to two opposite sides of the rod. Each element is glued to the neighboring ones, forming with the rod a single elastic body in the form of a rectangular parallelepiped. The body is hinged at both ends relative to the cross-sectional axis parallel to the piezoelectric layers. In opposite piezoelements, homogeneous fields of normal stresses are set antisymmetrically as functions of time. These stresses are parallel to the axis of the rod and force the elastic system to perform bending motions. Within the framework of the linear theory of elasticity for the considered system, generalized formulations of the initial-boundary value problem and the corresponding eigenvalue problem are given. These problems are defined through unknown displacements and the time integrals of mechanical stresses. An approximation of the displacement and stress fields, which is polynomial in transverse coordinates, is proposed. This approximation exactly satisfies the homogeneous boundary conditions for stresses on the lateral sides and takes into account the symmetry properties of the bending motions. For the chosen approximation, the boundary value problem for eigenvalues is exactly solved. Two branches of eigenvalues are found and used to reduce the initial-boundary value problem to a countable system of first-order ordinary differential equations with respect to complex variables. The dynamical system is decomposed into independent infinite-dimensional subsystems with a scalar control input. One of these subsystems is not controllable. For the remaining subsystems, each corresponding to a pair of piezoelectric elements, a control law for vibration damping is proposed for a specific number of the lower modes associated with the lower branch.

Approximate Solution of Two Dimensional Disc-like Systems by One Dimensional Reduction: An Approach through the Green Function Formalism Using the Finite Elements Method

We present a comprehensive study for common second order PDE’s in two dimensional disc-like systems and show how their solution can be approximated by finding the Green function of an effective one dimensional system. After elaborating on the formalism, we propose to secure an exact solution via a Fourier expansion of the Green function, which entails solving an infinitely countable system of differential equations for the Green–Fourier modes that in the simplest case yields the source-free Green distribution. We present results on non separable systems—or such whose solution cannot be obtained by the usual variable separation technique—on both annulus and disc geometries, and show how the resulting one dimensional Fourier modes potentially generate a near-exact solution. Numerical solutions will be obtained via finite differentiation using Finite Difference Method (FDM) or Finite Element Method (FEM) with the three-point stencil approximation to derivatives. Comparing to known exact solutions, our results achieve an estimated numerical relative error below 10−6. Solutions show the well-known presence of peaks when r=r′ and a smooth behavior otherwise, for differential equations involving well-behaved functions. We also verified how the Green functions are symmetric under the presence of a “weight function”, which is guaranteed to exist in the presence of a curl-free vector field. Solutions of non-homogeneous differential equations are also shown using the Green formalism and showing consistent results.

Domains of attraction of invariant distributions of the infinite Atlas model

The infinite Atlas model describes a countable system of competing Brownian particles where the lowest particle gets a unit upward drift and the rest evolve as standard Brownian motions. The stochastic process of gaps between the particles in the infinite Atlas model does not have a unique stationary distribution and in fact for every a≥0, πa:=⨂ i=1∞Exp(2+ia) is a stationary measure for the gap process. We say that an initial distribution of gaps is in the weak domain of attraction of the stationary measure πa if the time averaged laws of the stochastic process of the gaps, when initialized using that distribution, converge to πa weakly in the large time limit. We provide general sufficient conditions on the initial gap distribution of the Atlas particles for it to lie in the weak domain of attraction of πa for each a≥0. The cases a=0 and a>0 are qualitatively different as is seen from the analysis and the sufficient conditions that we provide. Proofs are based on the analysis of synchronous couplings, namely, couplings of the ranked particle systems started from different initial configurations, but driven using the same set of Brownian motions.

On a mixed problem for Hilfer type differential equation of higher order

The study considers the solvability of a mixed problem for a Hilfer type partial differential equation of the even order with initial value conditions and small positive parameters in mixed derivatives in threedimensional domain. It studies the solution to this fractional differential equation of higher order in the class of regular functions. The case, when the order of fractional operator is 1 < α < 2, is examined. During this study the authors use the Fourier series method and obtain a countable system of ordinary differential equations. The initial value problem is integrated as an ordinary differential equation and the integrated constants find by the aid of given initial value conditions. Using the Cauchy–Schwarz inequality and the Bessel inequality, it is proved the absolute and uniform convergence of the obtained Fourier series. The stability of the solution to the mixed problem on the given functions is studied.

On a nonlocal boundary value problem for a partial integro-differential equations with degenerate kernel

On a nonlocal boundary value problem for a partial integro-differential equations with degenerate kernel\Abstracteng{In this article the problems of the unique classical solvability and theconstruction of the solution of a nonlinear boundary value problem for a fifth orderpartial integro-differential equations with degenerate kernel are studied. Dirichletboundary conditions are specified with respect to the spatial variable. So, the Fourierseries method, based on the separation of variables is used. A countable system of~thesecond order ordinary integro-differential equations with degenerate kernel is obtained.The method of degenerate kernel is applied to this countable system of ordinaryintegro-differential equations. A system of~countable systems of algebraic equations isderived. Then the countable system of nonlinear Fredholm integral equations is obtained.Iteration process of solving this integral equation is constructed. Sufficient coefficientconditions of the unique solvability of the countable system of nonlinear integralequations are established for the regular values of parameter. In proof of uniquesolvability of the obtained countable system of nonlinear integral equations the method ofsuccessive approximations in combination with the contraction mapping method is used.In the proof of the convergence of Fourier series the Cauchy--Schwarz and Besselinequalities are applied. The smoothness of solution of the boundary value problemis also proved.

An inverse problem for Hilfer type differential equation of higher order

In three-dimensional domain, an identification problem of the source function for Hilfer type partial differential equation of the even order with a condition in an integral form and with a small positive parameter in the mixed derivative is considered. The solution of this fractional differential equation of a higher order is studied in the class of regular functions. The case, when the order of fractional operator is 0 < α < 1, is studied. The Fourier series method is used and a countable system of ordinary differential equations is obtained. The nonlocal boundary value problem is integrated as an ordinary differential equation. By the aid of given additional condition, we obtained the representation for redefinition (source) function. Using the Cauchy–Schwarz inequality and the Bessel inequality, we proved the absolute and uniform convergence of the obtained Fourier series.

Fractal set of generalized countable partial iterated function system with generalized contractions via partial Hausdorff metric

The primary interest of this article is to extend the fixed point result proved for a complete partial metric space (simply, pms) with a generalized contraction to a bivariate case. Taking this as a basic note, a generalized countable partial iterated function system (GCPIFS, for short) is introduced with the generalized contractions, using the partial Hausdorff metric. The attractor of this GCPIFS is proved as the fixed point of the generalized Hutchinson operator concentrated on the complete metric space (CBpm⁎(P⁎)×CBpm⁎(P⁎),Hpm⁎). A certain number of examples are given to show the existence of the theoretical results which we proved.

On Oscillating Solutions of a Countable System of Differential Equations in the Resonance Case

On Oscillating Solutions of a Countable System of Differential Equations in the Resonance Case

A Countable Fractal Interpolation Scheme Involving Rakotch Contractions

The main result of this paper states that for a given countable system of data, there exists a countable iterated function system consisting of Rakotch contractions, such that its attractor is the graph of a fractal interpolation function corresponding to the given countable system of data.

Locally interacting diffusions as Markov random fields on path space

We consider a countable system of interacting (possibly non-Markovian) stochastic differential equations driven by independent Brownian motions and indexed by the vertices of a locally finite graph G=(V,E). The drift of the process at each vertex is influenced by the states of that vertex and its neighbors, and the diffusion coefficient depends on the state of only that vertex. Such processes arise in a variety of applications including statistical physics, neuroscience, engineering and math finance. Under general conditions on the coefficients, we show that if the initial conditions form a second-order Markov random field on d-dimensional Euclidean space, then at any positive time, the collection of histories of the processes at different vertices forms a second-order Markov random field on path space. We also establish a bijection between (second-order) Gibbs measures on (Rd)V (with finite second moments) and a set of (second-order) Gibbs measures on path space, corresponding respectively to the initial law and the law of the solution to the stochastic differential equation. As a corollary, we establish a Gibbs uniqueness property that shows that for infinite graphs the joint distribution of the paths is completely determined by the initial condition and the specifications, namely the family of conditional distributions on finite vertex sets given the configuration on the complement. Along the way, we establish approximation and projection results for Markov random fields on locally finite graphs that may be of independent interest.

Inverse Problem for a Partial Differential Equation with Gerasimov–Caputo-Type Operator and Degeneration

In the three-dimensional open rectangular domain, the problem of the identification of the redefinition function for a partial differential equation with Gerasimov–Caputo-type fractional operator, degeneration, and integral form condition is considered in the case of the 0<α≤1 order. A positive parameter is present in the mixed derivatives. The solution of this fractional differential equation is studied in the class of regular functions. The Fourier series method is used, and a countable system of ordinary fractional differential equations with degeneration is obtained. The presentation for the redefinition function is obtained using a given additional condition. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series is proven.

A Countable System of Fractional Inclusions with Periodic, Almost, and Antiperiodic Boundary Conditions

This article is dedicated to the existence results of solutions for boundary value problems of inclusion type. We suggest the infinite countable system to fractional differential inclusions written byDαABCνit∈Yit,νiti=1∞. The mappingsyit,νiti=1∞are proposed to be Lipschitz multivalued mappings. The results are explored according to boundary conditionσνi0=γνiρ, σ,γ∈ℝ. This type of condition is the generalization of periodic, almost, and antiperiodic types.

Mixed Problem for a Higher-Order Nonlinear Pseudoparabolic Equation

We examine the unique generalized solvability of the mixed problem for a higher-order nonlinear pseudoparabolic equation with two parameters in mixed derivatives. Using the Fourier variable separation method, we reduce the problem to a countable system of nonlinear integral equations whose unique solvability can be proved by the method of successive approximations. We prove the continuous dependence of a generalized solution to the mixed problem on the initial functions and the positive parameters.

A Concretization of an Approximation Method for Non-Affine Fractal Interpolation Functions

The present paper concretizes the models proposed by S. Ri and N. Secelean. S. Ri proposed the construction of the fractal interpolation function (FIF) considering finite systems consisting of Rakotch contractions, but produced no concretization of the model. N. Secelean considered countable systems of Banach contractions to produce the fractal interpolation function. Based on the abovementioned results, in this paper, we propose two different algorithms to produce the fractal interpolation functions both in the affine and non-affine cases. The theoretical context we were working in suppose a countable set of starting points and a countable system of Rakotch contractions. Due to the computational restrictions, the algorithms constructed in the applications have the weakness that they use a finite set of starting points and a finite system of Rakotch contractions. In this respect, the attractor obtained is a two-step approximation. The large number of points used in the computations and the graphical results lead us to the conclusion that the attractor obtained is a good approximation of the fractal interpolation function in both cases, affine and non-affine FIFs. In this way, we also provide a concretization of the scheme presented by C.M. Păcurar.

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.

Exploring the mechanism of social media addiction: an empirical study from WeChat users

PurposeThe problematic use of social media progressively worsens among a large proportion of users. However, the theory-driven investigation into social media addiction behavior remains far from adequate. Among the countable information system studies on the dark side of social media, the focus lies on users' subjective feelings and perceived value. The technical features of the social media platform have been ignored. Accordingly, this study explores the formation of social media addiction considering the perspectives of users and social media per se on the basis of extended motivational framework and attachment theory.Design/methodology/approachThis study investigates the formation of social media addiction with particular focus on WeChat. A field survey with 505 subjects of WeChat users was conducted to investigate the research model.FindingsResults demonstrate that social media addiction is determined by individuals' emotional and functional attachment to the platform. These attachments are in turn influenced by motivational (perceived enjoyment and social interaction) and technical (informational support, system quality and personalization) factors.Originality/valueFirst, this study explains the underlying mechanism of how users develop social media addiction. Second, it highlights the importance of users' motivations and emotional dependence at this point. It also focuses on the technical system of the platform that plays a key role in the formation of addictive usage behavior. Third, it extends attachment theory to the context of social media addictive behavior.