Homogeneous Linear Equations Research Articles

Bounded on the semi-axis multiperiodic solution of a linear finite-hereditarity integro-differential equation of parabolic type

The question of the existence of a solution of linear integro-differential systems of parabolic type limited on the semiaxis in a spatial variable and multiperiodic in time variables was considered. Sufficient conditions of multiperiodic oscillations in time variables in a linear homogeneous equation with a boundary condition and in a linear inhomogeneous equation were established. A linear homogeneous and inhomogeneous finitehereditarity integro-differential equation of convective-diffusion type were investigated.

Heat Transfer in Solid by Using Linear Differential Equation

Abstract: Differential equations are fundamental importance in engineering mathematics because any physical law s and relations appear mathematically in the form of such equations. . The rate of heat conduc- tion in a specified direction is proportional to the temperature gradient,which isthe rate of change in temperature with distance in that direction. In this paper we discussed about first order linear homogeneous equations, first order linear non homogeneous and the application of first order differential equationto heat transfer analysis particularly in heat conduction in solids.

Embedding of Unimodular Row Vectors

In this paper, we mainly study the embedding problem of unimodular row vectors, focusing on avoiding the identification of polynomial zeros. We investigate the existence of the minimal syzygy module of the ZLP polynomial matrix and demonstrate that the minimal syzygy module has structural properties that are similar to the fundamental solution system of homogeneous linear equations found in linear algebra. Finally, we provide several embedding methods for unimodular vectors in certain cases.

Strong solutions of a semilinear impulsive pseudoparabolic equation with an infinitesimal initial layer

We study the multi-dimensional initial-boundary value problem for the semilinear pseudoparabolic equation with a regular nonlinear minor term, which, in general, may be superlinear. This term models a non-instantaneous but a very rapid absorption with q(x)-growth. The minor term depends on a positive integer parameter n and, as n→+∞, converges weakly⋆ to the expression incorporating the Dirac delta function, which, in turn, models an instant absorption at the initial moment. We prove that an infinitesimal initial layer, associated with the Dirac delta function, is formed as n→+∞, and that the family of regular weak solutions to the original problem converges to the strong solution of a two-scale microscopic-macroscopic model. This model consists of two equations and the set of initial, boundary, and matching conditions, so that the ‘outer’ macroscopic solution beyond the initial layer is governed by the linear homogeneous pseudoparabolic equation at the macroscopic (‘slow’) timescale, while the initial layer solution is defined at the microscopic level and obeys the semilinear pseudoparabolic equation at the microscopic (‘fast’) timescale. The latter equation inherits the full information about the profile of the original non-instantaneous absorption. In general, the research is devoted to pseudoparabolic equations with measure data depending on an unknown solution.

On a point kinetic with explicit presence of neutron poisons: reactivity decomposition according to long time scales

The present work consists of deducing the point kinetics model considering neutron absorber poisons from the neutron diffusion equation. When considering poisons products, the non-linear system has different orders of magnitude in short and long time scales. The first one represents operational reactor control, whereas the second is due to the change of the chemical composition of the nuclear fuel as a result of burn-up. This represents a first step in a new direction, as it shows how we arrived at the extension of the point kinetics model considering Xenon-135 and Samarium-149 poisons. The non-linear system is solved using the Adomian decomposition method that expands the non-linear terms into an infinite series, obtaining a recursive system, where the recursion initialization is a homogeneous linear equation and the subsequent recursion steps consider the non-linear contributions as source terms constructed from previous recursion steps. The formulation of the source terms of the decomposition method is shown. In order to demonstrate the robustness of the current approach to this type of problem, case studies are presented where nuclear fuel uses reused elements up to third generation, taking a step in a new direction where poisons are already present in the reactor initially.

APPLICATION OF THE CASE METHOD IN THE PROCESS OF STUDYING THE COURSE “METHODS OF MATHEMATICAL PHYSICS”

The article considers the use of case-study method while studying of the topic “Equations with partial derivatives of the hyperbolic type” from the course “Methods of mathematical physics”. The requirements for the educational institutions are outlined in accordance with the concept of mathematical education in Ukraine. The emphasis is placed on the system-activity approach to the learning process. Attention is focused on the necessity to use the method of problem-based learning and one of its forms – the case method. There has been analyzed the concept of a case. There have been described the advantages of the case-study method and the system of requirements for it. There has been revealed the importance of choosing the topic “Equations with partial derivatives of the hyperbolic type” related to its applied nature. The article demonstrates the main stages of the research work of students of the Faculty of Physics and Mathematics in establishing the minimum conditions for the solvability of a mixed problem for a linear homogeneous hyperbolic equation. The paper also identifies and characterizes the main groups of results to be processed. The investigation indicates the selection criteria that help to establish the minimum conditions for the existence of a solution of a mixed problem for a linear homogeneous hyperbolic equation. The results of the research are presented in the form of a table. The research presents the main statements recognized as the best conditions corresponding to the set goal of the study. The originality of the method of proving statements and the skillful combination of two important topics of the course “Methods of mathematical physics” – “Equations with partial derivatives of the hyperbolic type” and “Integral equations” are noted. Application of the case-study method makes it possible to activate the learning process, to form positive motivation and to renew the creative potential of teachers. Proven research work within the course “Methods of mathematical physics” provides a solid basis for using the case method in the further professional activities of teachers. Along with this, the theoretical and methodical preparation of case technologies in institutions of higher education requires further improvement.

Integrated Resolving Functions for Equations with Gerasimov–Caputo Derivatives

The concept of a β-integrated resolving function for a linear equation with a Gerasimov–Caputo fractional derivative is introduced into consideration. A number of properties of such functions are proved, and conditions for the solvability of the Cauchy problem to linear homogeneous and inhomogeneous equations are found in the case of the existence of a β-integrated resolving function. The necessary and sufficient conditions for the existence of such a function in terms of estimates on the resolvent of its generator are obtained. The example of a β-integrated resolving function for the Schrödinger equation is given. Thus, the paper discusses some aspects of the symmetry of the concepts of integrability and differentiability. Namely, it is shown that, in the absence of a sufficiently differentiable resolving function for a fractional differential equation, the problem of the existence of a solution can be solved by an integrated resolving function of the equation.

Growth properties of solutions of linear difference equations with coefficients having $\varphi$-order

"In this paper, we investigate the relations between the growth of entire coefficients and that of solutions of complex homogeneous and non-homogeneous linear difference equations with entire coefficients of $% \varphi $-order by using a slow growth scale, the $\varphi $-order, where $% \varphi $ is a non-decreasing unbounded function. We extend some precedent results due to Zheng and Tu (2011) [15] and others."

Hyers–Ulam Stability of Linear Homogeneous Quaternion-Valued Difference Equations

Hyers–Ulam Stability of Linear Homogeneous Quaternion-Valued Difference Equations

Stability Properties for the Delay Integro-Differential Equation

In this paper stability inequalities for the linear nonhomogeneous Volterra delay integro-differential equation (VDIDE) is being established. The particular problems are encountered to show the applicability of the method and to confirm the predicted theoretical analysis.

Discrete Approximation of Solutions of the Cauchy Problem for a Linear Homogeneous Differential-Operator Equation with a Caputo Fractional Derivative in a Banach Space

In this paper, we construct and examine the time-discretization scheme for the Cauchy problem for a linear homogeneous differential equation with the Caputo fractional derivative of order α ∈ (0, 1) in time and containing the sectorial operator in a Banach space in the spatial part. The convergence of the scheme is established and error estimates are obtained in terms of the step of discretization. Properties of the Mittag-Leffler function, hypergeometric functions, and the calculus of sectorial operators in Banach spaces are used. Results of numerical experiments that confirm theoretical conclusions are presented.

Linear equations on Drinfeld modules

Let L be a finite extension of the rational function field in one variable over a finite field Fq and E be a Drinfeld module defined over L. Given finitely many elements in E(L), this paper aims to prove that linear relations among these points can be characterized by solutions of an explicitly constructed system of homogeneous linear equations over Fq[t]. As a consequence, we show that there is an explicit upper bound for the size of the generators of linear relations among these points. This result can be regarded as an analogue of a theorem of Masser for finitely many K-rational points on an elliptic curve defined over a number field K.

Comparative Study of Guided Electromagnetic Wave Propagation for Two Models of an Open Tape Helix

Rigorous methods to solve the homogeneous boundary value problems and the techniques to resolve illposed mathematical issues arising in the problem formulation of electromagnetic wave propagation of an Open tape helix slow wave structure with anisotropically conducting and completely conducting model is presented. The dispersion equation of both the models require convergent coeefficients of the infinte linear homogeneous equations whose determinant of the coefficent matrix is zero. The coefficient matrix of the anisotropically conducting model is rapidly convergent, while in the case of completely conducting model, the matrix entries which are infinite summation series representations turns out to be divergent, the not so well-posed boundary value problem is regularised with the method of mollification functions. The dispersion characteristics are plotted after truncating the infinite series expansions to adequate number of terms and summing the terms of converging series expansions(after regularization in the case of the completely conducting tape-helix model) in dispersion equations. The tape-current distribution for both models are estimated from the null-space vector of the truncated coefficient matrix corresponding to a particular (β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> – <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ) root of the dispersion equation. A comparison of the numerical results for both models show that the neglect of the perpendicular tape-current density component entails a substantial modification of the dispersion characteristics of guided waves supported by an open tape helix.

Secure and Efficient Smart Healthcare System Based on Federated Learning

The rapid development of smart healthcare system in the Internet of Things (IoT) has made the early detection of many chronic diseases more convenient, quick, and economical. However, when healthcare organizations collect users’ health data through deployed IoT devices, there are issues of compromising users’ privacy. In view of this situation, this paper introduces federated learning technology to solve the problem of data security. In this paper, we consider the two main problems of federated learning applications in IoT smart healthcare system: (1) how to reduce the time overhead of system running and (2) how to authenticate that the user device uploading data is deployed by the system itself. To solve the above problems, we propose the first federated learning scheme based on full dynamic secret sharing. First, we use a two-mask protocol to keep the user’s local model parameters confidential during federated learning. Then, based on homogeneous linear recursive equation, homomorphic hash function, and elliptic curve cryptosystem, the full dynamic secret sharing and user identity authentication are realized. In addition, our scheme allows users to join or quit during training. Finally, we have carried out simulation test on this scheme. The experimental results show that the efficiency of our scheme is improved by about 60% on average in the case of no user dropping and by about 30% in the case of some users dropping.

SODES: Solving ordinary differential equations step by step

In this paper, we introduce SODES (Stepwise Ordinary Differential Equations Solver) a new solver for Ordinary Differential Equations (ODE). SODES can optionally provide the solution displaying all the steps needed to obtain it. This way, SODES is an important tool not only for researchers who need solving ODE but also constitutes an important tool for the teaching and learning process of ODE. SODES has been developed using programming with a Computer Algebra System (CAS). Specifically, we use the CAS Derive but it can be easily adapted to any other CAS supporting programming.SODES provides, step by step, the solution of the following types of ODE: separable, homogeneous, exact, integrating factors, linear, Bernoulli, Riccati, first order ODE of nth degree, Cauchy’s problems of first order ODE, higher order linear homogeneous equations with constant coefficients, Lagrange’s method for particular solutions of higher order linear equations with constant coefficients, higher order linear equations with constant coefficients and Cauchy’s problems of higher order linear equations with constant coefficients. SODES also deals with two generic programs which determine the type or types of a given ODE and provides the solution.In this paper we will also introduce a draft of a Graphical User Interface (GUI) for SODES in a local web application using programming in Python (using its CAS module SymPy) which is a more portable and free CAS. This draft can be used in English, French and Spanish, and can be easily extended to other languages.The code of SODES and the GUI are freely available so that it can be used by users who also will be able to adapt it to their needs.

New exact analytical thermal buckling solutions of composite thin plates with all edges rotationally-restrained

This research explores the finite integral transform approach for analytical thermal buckling solutions of composite thin plates with all edges rotationally-restrained, which is difficult to handle by other analytical approaches. Performing the transformations defined on the governing thermal buckling equation and the boundaries investigated, which yields the transform coefficient for the deflection. Incorporating the inversion formula to satisfy the remaining boundary restraints yields sets of homogenous linear algebraic equations, which determines the critical temperature and corresponding mode shapes for plates. In addition, we provide new exact solutions for non-Levy-type plates via setting the rotational restrained coefficient. The precision and efficiency of the present solutions were well validated by precise FEM analysis. It is promising to develop the present approach for seeking more analytical solutions for intractable moderately-thick/thick plate problems. RESEARCH HIGHLIGHTS Finite integral transform approach for new exact thermal buckling analysis of fully rotationally-restrained composite rectangular thin plates is developed. New analytic exact thermal buckling solutions of plates under non-classical/non-Levy-type boundaries are obtained. Comprehensive benchmark numerical and graphical results are presented. The method is promising to explore new analytic solutions for intractable problems of plates based on different shear deformation theories.

Buckling analysis of super-light composite cylinders with auxetic core and isotropic facing sheets with variable thickness: An analytical approach

An analytical method is presented to determine the axial buckling load of composite cylindrical shells with honeycomb core layer and nonuniform thickness using the first-order shear deformation theory, the nonlinear von Karman theory, and the Hooke law relations. The composite shell consists of two isotropic inner and outer layers with non-uniform thicknesses and one honeycomb core with constant thicknesses and adjustable Poisson's ratio. The equilibrium equations are a system of nonlinear differential equations with variable coefficients, and they are derived by employing the virtual work principle. The equilibrium equations have been solved analytically using the matched asymptotic expansion method of the perturbation technique. The stability equations extracted using the adjacent criterion include a system of linear homogenous equations with variable coefficients. By solving these equations, the buckling load is obtained analytically. A parametric study investigates the effects of different geometrical and mechanical parameters such as the honeycomb structure on the buckling load. The metamaterial honeycomb core layer has an adjustable Poisson's ratio, and the effect of negative/positive Poisson's ratio has been studied on the buckling behavior. The accuracy of the presented method is studied by comparing the results with the finite element method and some other references in special cases.

A variational modification of the Harmonic Balance method to obtain approximate Floquet exponents

We propose a modification of a method based on Fourier analysis to obtain the Floquet characteristic exponents for periodic homogeneous linear systems, which shows a high precision. This modification uses a variational principle to find the correct Floquet exponents among the solutions of an algebraic equation. Once we have these Floquet exponents, we determine explicit approximated solutions. We test our results on systems for which exact solutions are known to verify the accuracy of our method including one‐dimensional periodic potentials of interest in quantum physics. Using the equivalent linear system, we also study approximate solutions for homogeneous linear equations with periodic coefficients.

Mittag-Leffler stability of numerical solutions to linear homogeneous time fractional parabolic equations

&lt;abstract&gt;&lt;p&gt;In a bounded domain, the solution of linear homogeneous time fractional parabolic equation is known to exhibit polynomial type decay rate (the so-called Mittag-Leffler stability) over time, which is quite different from the exponential decay of classical parabolic equation. We firstly use the finite element method or finite difference method to discretize the parabolic equation in space to obtain fractional ordinary differential equation, and then use fractional linear multistep method (F-LMM) to discretize in time to obtain a fully discretized schemes. We prove that the strongly $ A $-stable F-LMM method combined with appropriate spatial discretization can accurately maintain the long-term optimal algebraic decay rate of the original continuous equation. Numerical examples are included to confirm the correctness of our theoretical analysis.&lt;/p&gt;&lt;/abstract&gt;

KINETIC INSTABILITY IN AN ENSEMBLE OF DISLOCATIONS UNDER PLASTIC DEFORMATION OF METALS

Based on a system of evolutionary equations for an ensemble of edge dislocations, the problem associated with the development of instability of a homogeneous distribution of dislocations is considered. Despite the large number of theoretical studies on this topic, most of them are considered within the framework of reaction-diffusion phenomenological models, which contradicts the mathematical structure of the initial evolutionary equations, which are hyperbolic equations. For the initial system of hyperbolic equations, the case is considered when the process of plastic deformation develops along a given axis. The elastic interaction of dislocations is neglected in order to reveal the instability features of the homogeneous state of the system due to the local kinetics of dislocations. The stability analysis of the homogeneous state was carried out for a system of dislocations of two types differing in mobility. As a result of linearization of the initial system, a system of homogeneous linear algebraic equations is obtained. From the condition of the existence of its nontrivial solutions, a dispersion equation is found, using which, on the basis of the Raut – Hurwitz criterion, the range of parameters of the initial system at which kinetic instability is realized is established. In the instability region, an expression is obtained for the increment of the unstable mode, which determines the behavior of the system beyond the bifurcation point, and an expression for the characteristic scale of the inhomogeneous structure. The region of system parameters where the diffusion dynamics of the system is realized is found. It is shown that the Turing instability for the total dislocation density is realized in this parameter range. Within the framework of the initial system of hyperbolic equations, the dynamics of both total and excess dislocation densities is significant. It is the excess density that leads to disorientation in the formation of heterogeneous dislocation structures. This is a fundamental point, since at the advanced stage of plastic deformation, the dislocation structure is basically a system of disoriented cells and ragged sub-boundaries.