Problems For Linear Equations Research Articles

Substantiation of amplitude-frequency characteristics and design parameters of the vibration exciter of the separator of volume vibrations

Evaluation of the influence of amplitude-frequency and power parameters of external technological action on bulk products in the process of separation of the impact, the way of lowfrequency oscillations of the working bodies of the separator and the angle of inclination of the container during processing determines the main indicators of technical and economic efficiency of the explored process, which is the relevance of the research. The purpose of the study is to substantiate the operating parameters of the explored separator of volume vibrations by determining the patterns of change in the kinematic and power characteristics of the vibration drive. To determine the rational parameters of the vibration screening process, the equations of motion of the working bodies in the form of a conical sieve surface were obtained using the method of Lagrange equations of the 2nd kind. Using the solution of the Cauchy problem for linear inhomogeneous differential equations, the solution of the latter was obtained. Using the Math CAD mathematical environment, the dependences of the amplitude of oscillations, vibration velocity and vibration acceleration, and the intensity of oscillatory motion were obtained, which allowed performing a mathematical analysis of the power and energy characteristics of the vibration drive of the explored separator. The main effects of the developed design of the vibrating separator are an increase in the driving force of the process of separation of bulk solids in this work, which was achieved by providing the working cylindrically-conical container with vibratory motion; improvement of the conditions for the passage of product particles through the perforations, which was achieved by providing the sieve surface with volumetric vibrations; reduction of energy consumption and improvement of the operating conditions of the supporting units during the operation of the designed vibrating screen, which was achieved by installing additional elastic elements between the separator body The inclined arrangement of the conical sieve surface allows for spatial gyratory or circular translational motion, which allows implementing of the advantages of bulk separation of bulk solids. The results of the analytical study allowed substantiation of the optimal angle of inclination of the working sieve surface. Based on the analysis, the design parameters of the vibration exciter were substantiated and specified and the design of this technical system was presented. The practical value of the conducted research can be attributed to using the designed kinematic combined vibration exciter of volumetric oscillations in the separator, which allows reducing the weight of the oscillating parts of the drive and, accordingly, the energy consumption for the separation process

Student Deductive Reasoning in Solving Problems of One-Variable Linear Equations Through Problem-Based Learning

Learning is a two-way communication process, and teaching by the teacher as an educator while learning is done by learners or students. Mathematics learning conducted by teachers places less emphasis on aspects of student thinking. Especially using reason in solving various problems faced by students in the learning process. Teachers in learning only focus on the aspect of grades, namely good learning outcomes must be achieved by students. The purpose of this research is to find out the reasoning ability of students on the subject of the one-variable linear equation system through problem-based learning of class VII of SMP Negeri 11 Seram Barat. The type of research used is qualitative descriptive, which is research that aims to obtain data and describe students' deductive reasoning skills on the subject of the one-variable linear equation system through problem-based learning of class VII of SMP Negeri 11 Seram Barat. The number of subjects in this study amounted to 1 person. The instruments used are tests and interviews. The data analysis technique used is qualitative descriptive. Based on the results of the data analysis obtained that the reasoning ability of the subject of class VII SMP Negeri 11 Seram Barat on the material of the linear equation system of one variable obtained the tendency of the subject to have reasoning ability. This is known by the way the problem is used by the subject which starts from general things that have been known the truth and then obtains a more specific conclusion. The subject in his learning has three reasoning indicators used in this study, submitting allegations based on the way the problem is solved by the subject, the subject can give reasons for the arguments presented at the conclusion of the problem in the interview process, and the subject can check the validity of an argument based on the resolution of the problem

Assessing the Effect of an Ethnomathematics Teaching Material on Students’ Understanding of Mathematics

The integration of cultural context in mathematics teaching materials is a form of implementing ethnomathematics-based teaching. It is an effort of making mathematics learning easier and meaningful. This paper reports the effect of an ethnomathematics teaching material on students’ understanding of linear equations at the Department of Primary Education Universitas Syiah Kuala. 105 students participated in the study. Students’ understanding was assessed using a test. Data analysis was conducted using descriptive qualitative method. Students’ understanding of linear equation was analyzed based on 3 aspects of solving linear equation problems: correctly translate word problems into linear equation, perform calculation correctly, and give correct final answer. Results showed that more than 83% of the students could answer the test satisfactorily. Therefore, it is concluded that the implementation of the ethnomathematics teaching material helps students understand linear equation. This finding implies that the implementation of ethnomathematics teaching material is beneficial for mathematics learning.

Asymptotics and Quotient Spaces of Solutions of Operator-Difference Equations and Differential Equations with Small Delay

Formerly, in order to conduct the in-depth study of differential equations with delay, the author proposed the method of splitting the solution space reducing such equations to the systems of operator-difference equations. Using this method, the author assumed new conditions, i.e. the absolute domains for coefficients sufficient for the existence of special (slowly changing) solutions, and proved the presence of approximating and asymptotically approximating properties in them, as well as the asymptotic one-dimensional space of solutions of the initial problems for linear scalar differential equations with insignificantly retarded argument and the corresponding operator-difference equation systems (special solutions correspond, to the solutions with a slowly changing first component and a relatively small second component). For the purposes of the single-point representation of the obtained results and other data related to the theory of dynamic systems (the distance between the solution values tends to zero alongside the unlimited increase in argument), throughout this research paper the author uses the concept of the asymptotic equivalence of solutions for dynamic systems, as it was introduced by the author in their previous research. In order to shape the new mathematical objects, the concept of asymptotic Hausdorff equivalence of solutions for dynamic systems is introduced (the distance between solution values tends to zero with unlimited increase in argument of one solution and monotonic transformation of argument of another solution).

The Mathematic Model Study of Optimum Align Method in Compass

In this paper, based on the study of the characteristics of horizontal well designed by optimum align method in compass software, we speculated that, the optimum align method is one kind of space circular arc method. Therefore, we built a horizontal well track design model based on spatial arc curve optimization alignment hypothesis. The orbit design problem of horizontal wells is transformed into the problem solving of multiple linear equations with trigonometric functions. If the kick off point, the target point and the dog leg degree of the first and second inclined sections are determined, the solution of the track parameters of the horizontal well is unique, and the track parameters of the horizontal well can be calculated accurately. The model result is the same as the optimum align method in compass program. Therefore, the algorithm essence of Compass Optimum Align design method is to solve the problem of multiple linear equations based on the optimal alignment of arc curves in space.

Time scale observability and constructibility of linear dynamic equations

This paper investigates the observability and constructibility problems of time-varying linear dynamic equations using time scale theory. First, we define observability, reachability and constructibility operators on time scales. Some necessary and sufficient conditions are proposed to ensure the observability on non-uniform time domains based on some linear algebra tools. Then, constructibility is also examined using the same approach. Moreover, the link between observability and constructibility concepts on arbitrary time sets is discussed. Further, the observability and reachability duality relationship for time-varying linear systems on time scales is established. The current work unifies and extends some existing results given for standard cases (i.e. the continuous line and the discrete time domain) to non-uniform time domains. Finally, the obtained results are described with an illustrative example.

Boundary-value problems for singular p– and p(x)– Laplacian equations in a domain with conical point on the boundary

This paper is a survey of our last results about solutions to the Dirichlet and Robin boundary problems, the Robin transmission problem for an elliptic quasilinear second-order equation with the constant p– and variable p(x)-Laplacians, as well as to the degenerate oblique derivative problem for elliptic linear and quasilinear second-order equations in a conical bounded n-dimensional domain.

Assessing the Effect of an Ethnomathematics Teaching Material on Students’ Understanding of Mathematics

The integration of cultural context in mathematics teaching materials is a form of implementing ethnomathematics-based teaching. It is an effort of making mathematics learning easier and meaningful. This paper reports the effect of an ethnomathematics teaching material on students’ understanding of linear equations at the Department of Primary Education Universitas Syiah Kuala. 105 students participated in the study. Students’ understanding was assessed using a test. Data analysis was conducted using descriptive qualitative method. Students’ understanding of linear equation was analyzed based on 3 aspects of solving linear equation problems: correctly translate word problems into linear equation, perform calculation correctly, and give correct final answer. Results showed that more than 83% of the students could answer the test satisfactorily. Therefore, it is concluded that the implementation of the ethnomathematics teaching material helps students understand linear equation. This finding implies that the implementation of ethnomathematics teaching material is beneficial for mathematics learning.

Misconception in Linear Equation System: The Case of Students Using Imitative Reasoning

Imitative reasoning is one of the strategies students used in solving mathematical problems. The imitative reasoning links to the ways in which students using their prior knowledge on solving a similar problem. Those using imitative reasoning, in fact, could only comprehend a problem on its surface level without deeply conceptualise it. The issue leads to errors and misconception in answering linear equation problems. This study aims to analyse students’ misconception when solving linear equation problems, specifically in those using imitative reasoning. This is a qualitative study with 64 students participated in the study. The study reveals that the misconceptions can be found in erroneous calculation, the application of problem solving strategies, the correlation between procedure and concept as well as the solutions of linear equations in intersecting, parallel, and coinciding lines.

On numerical resolution of an inverse Cauchy problem modeling the airflow in the bronchial tree

This paper is devoted to the numerical resolution of an inverse Cauchy problem governed by Stokes equation modeling the airflow in the lungs. It consists in determining the air velocity and pressure on the artificial boundaries of the bronchial tree. This data completion problem is one of the highly ill-posed problems in the Hadamard sense (Hadamard in Lectures on Cauchy’s problem in linear partial differential equations. Dover, New York, 1953). This gives great importance to its numerical resolution and in particular to carry out stable numerical approaches, mostly in the case of noisy data. The main idea of this work is to extend some regularizing, stable and fast iterative algorithms for solving this problem based on the domain decomposition approach (Chakib et al. in Inverse Prob 35(1):015008, 2018). We discuss the efficiency and the feasibility of the proposed approach through some numerical tests performed using different domain decomposition algorithms. Finally, we opt for the Robin–Robin algorithm, which showed its performance, for the numerical simulation of the airflow in the bronchial tree configuration.

THE DARBOUX AND KOSHI PROBLEM FOR LINEAR HYPERBOLIC EQUATIONS WITH CONSTANT COEFFICIENTS

Many phenomena of mechanics, physics, and biology are reduced to the study of hyperbolic equations. In order to describe these phenomena completely, the Darboux problem is posed for hyperbolic equations, and for further studies, an explicit representation of the problem under consideration is necessary.
 In this article discusses, we study the Darboux and Koshi problems for linear hyperbolic equations with constant coefficients.

Представление решения задачи Гурса для линейных стохастических гиперболических дифференциальных уравнений второго порядка

The article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way under the assumption of sufficient smoothness of the coefficients of the terms included in the right-hand side of the equation were obtained. Meanwhile, study of many stochastic applied optimal control problems described by linear or nonlinear second-order stochastic differential equations, in partial derivatives hyperbolic type, the assumptions of sufficient smoothness of these equations are not natural. Proceeding from this, in the considered Goursat problem, in contrast to the known works, the smoothness of the coefficients of the terms in the right-hand side of the equation is not assumed. They are considered only measurable and bounded matrix functions. These assumptions, being natural, allow us to further investigate a wide class of optimal control problems described by systems of second-order stochastic hyperbolic equations. In this work, a stochastic analogue of the Riemann matrix is introduced, an integral representation of the solution of considered boundary value problem in explicit form through the boundary conditions is obtained. An analogue of the Riemann matrix was introduced as a solution of a two-dimensional matrix integral equation of the Volterra type with one-dimensional terms, a number of properties of an analogue of the Riemann matrix were studied.

The regularity properties and blow-up of the solutions for improved Boussinesq equations

In this paper, we study the Cauchy problem for linear and nonlinear Boussinesq type equations that include the general differential operators. First, by virtue of the Fourier multipliers, embedding theorems in Sobolev and Besov spaces, the existence, uniqueness, and regularity properties of the solution of the Cauchy problem for the corresponding linear equation are established. Here,Lp-estimates for a~solution with respect to space variables are obtained uniformly in time depending on the given data functions. Then, the estimates for the solution of linearized equation and perturbation of operators can be used to obtain the existence, uniqueness, regularity properties, and blow-up of solution at the finite time of the Cauchy for nonlinear for same classes of Boussinesq equations. Here, the existence, uniqueness,Lp-regularity, and blow-up properties of the solution of the Cauchy problem for Boussinesq equations with differential operators coefficients are handled associated with the growth nature of symbols of these differential operators and their interrelationships. We can obtain the existence, uniqueness, and qualitative properties of different classes of improved Boussinesq equations by choosing the given differential operators, which occur in a wide variety of physical systems.

К теории переопределенных систем дифференциальных уравнений в частных производных второго порядка с одной сингулярной точкой и одной сингулярной линией

In this paper, a over determined system of second-order partial differential equations with one singular point and one singular line is investigated. A compatibility condition is found for over determined systems of second-order partial differential equations with one singular point and one singular line. If the compatibility condition is met, integral representations of the variety of solutions are found explicitly in terms of three arbitrary constants, when the singular line is in the boundaries of the domain for which initial data problems (Cauchy-type Problems) can be set. In this paper considers a redefined system of second-order partial differential equations, when the coefficients and right parts have one singular point and one singular line. Obtaining a variety of solutions and studying boundary value problems for linear differential equations of the hyperbolic type of the second order, some linear redefined systems of the first and second order with one and two supersingular lines and supersingular points is devoted to the monograph of academician of the National Academy of Sciences of the Republic of Tatarstan Rajabov N. - 1992 "Introduction to the theory of partial differential equations with supersingular coefficients" [6, p.126]. Using the obtained results of The monograph of Rajabov N., a variety of solutions of redefined systems of partial differential equations of the second order with one singular point and one singular line in an explicit form, through three arbitrary constants, was found.

ON THE SOLUTION OF SOME HIGHER-ORDER INTEGRO-DIFFERENTIAL EQUATIONS OF SPECIAL FORM

The article is devoted to the solution of boundary value problems for higher-order linear integro-differential equations of Fredholm type with differential and integral operators encompassing powers of an ideal bijective linear differential operator whose inverse is known explicitly. The conditions for existence and uniqueness of solutions are derived and the solutions are delivered in closed form. The approach is based on the view that an integro-differential operator is a perturbed differential operator. The results obtained are of both theoretical and practical importance. The method is elucidated by solving two illustrative problems.

Global error estimation for linear ordinary differential equations and their numerical optimal solutions

Solving initial value problems (IVPs) for linear ordinary differential equations (ODEs) plays an important role in many applications. There are variousnumerical methods and solvers to obtain approximate solutions typically represented by points. However, few works about estimation of global errors canbe found in literature. In this paper, we first use Hermite cubic spline interpolation at mesh points to represent the solution, and then we define this residual obtained by substituting the interpolation solution back to ODEs. Then the global error between the exact solution and an approximate solution can be bounded by using the residual. Moreover, solving ODEs can be reduced to an optimization problem of the residual in certain solution space which can be solved by the conjugate gradient method by taking advantage of sparsity of the corresponding matrix. The examples in the paper show that our estimation works well for linear ODE models and the refinement can find solutions with smaller global errors than some popular methods in MATLAB without additional mesh points.

On the Boundedness of Solution of the Second Order Ordinary Differential Equation with Involution

In the present paper, the initial value problem for the second order ordinary differential equation with involution is investigated. We obtain equivalent initial value problem for the fourth order ordinary differential equations to the initial value problem for second order linear differential equations with involution. Theorem on stability estimates for the solution of the initial value problem for the second order ordinary linear differential equation with involution is proved. Theorem on existence and uniqueness of bounded solution of initial value problem for second order ordinary nonlinear differential equation with involution is established.

Boundary value problems for singular p- and p(x)- Laplacian equations in a domain with conical point on the boundary

This paper is a survey of our last results about solutions to the Dirichlet and Robin boundary problems, the Robin transmission problem for an elliptic quasilinear second-order equation with the constant p- and variable p(x)-Laplacians, as well as to the degenerate oblique derivative problem for elliptic linear and quasilinear second-order equations in a conical bounded n-dimensional domain.

RETRACTED: Asymptotic behavior of the solution of a singularly perturbed general boundary value problem with boundary jumps

AbstractIt is known that the study of boundary value and mixed problems for integrable linear equations encounters significant difficulties of a fundamental nature. Exceptions are problems with boundary conditions of a special type, which are often called integrable or linearizable. The purpose of this article is to study the asymptotic behaviors of solutions of singularly perturbed general boundary value problems with boundary jumps for higher‐order equations. Using the Schlesinger–Birghof theorem, we constructed a fundamental system of solutions of a homogeneous perturbed equation of conditionally stable type in the critical case. Initial boundary functions are constructed based on the fundamental system of solutions. An analytical representation is found, the existence and uniqueness of a solution to this boundary value problem are proved. Asymptotic estimates of the solution and its derivatives are derived from the analytical representation of the solution of the given boundary value problem. The limit passage of solution of the perturbed problem to the solution of the unperturbed problem is proved. The conditions of the existence of jumps are found. The values of boundary jumps are determined. As a result, a class of boundary value problems is highlighted that has possessing of phenomenon of boundary jumps.

Global stability result for parabolic Cauchy problems

Abstract Uniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard [Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1953], these kind of problems are known to be ill-posed and even severely ill-posed. Until now, there are only few partial results concerning the quantification of the stability of parabolic Cauchy problems. We bring in the present work an answer to this issue for smooth solutions under the minimal condition that the domain is Lipschitz.