System Of Linear Equations Research Articles

Profil kemampuan komunikasi peserta didik materi SPLDV

This research aims to find out and describe how high the mathematical communication skills of Grade IX Middle School (SMP) students are on the material Systems of Linear Equations in Two Variables (SPLDV). This study involved students in class IX-D at one of the State Middle Schools in Karawang Regency for the 2023/2024 academic year as research subjects. Purposive sampling technique was used to select subjects and obtained 32 students. Qualitative descriptive research was used. The data collection method was carried out using a mathematical communication ability evaluation tool which consisted of three descriptive questions. The data processing technique used descriptive statistics to determine the high, medium and low categories of all students in classes IX-D. It is proven from the result of the students’ mathematical communication skills test that they are in the low classification because they show an average score of 32,00957.

Approximation of one and two dimensional nonlinear generalized Benjamin-Bona-Mahony Burgers' equation with local fractional derivative

This study presents, a numerical method for the solutions of the generalized nonlinear Benjamin-Bona-Mahony-Burgers' equation, with variable order local time fractional derivative. This derivative is expressed as a product of two functions, the usual integer order time derivative, and a function of time having a fractional exponent. Then, forward difference approximation is used for time derivative. The unknown solution of the differential problem and corresponding derivatives are estimated using Haar wavelet approximations (HWA). The collocation procedure is then implemented in HWA, to transform the given model to the system of linear algebraic equations for the determination of unknown constant coefficient of the Haar wavelet series, which update the derivatives and the numerical solutions. The sufficient condition is established for the stability of the proposed technique, and then verified computationally. To check the performance of the scheme, few illustrative examples in one and two dimensions along with l∞ and l2 error norms are also given. Besides this, the computational convergence rate is calculated for both type equations. Additionally, computed solutions are compared with available results in literature. Simulations and graphical data discloses, that suggested scheme works well for such complex problems.

Optimal Synthesis of a Satellite Attitude Control System under Constraints on Control Torques and Velocities of Reaction Wheels

The problem of optimal synthesis of a nonlinear system’s control law parameters of satellite attitude control under constraints is considered. The maximum stability degree of a system is considered as an optimality criterion. The reaction wheels’ control moments and angular velocities achievable within the drives’ physical characteristics are considered as constraints. A linear model of the system converted from the original nonlinear model was used to solve the problem. The decomposition method, based on the transition from real time to relative time, is used to separate the tasks of optimality criterion maximization and constraints consideration. A method of synthesizing the optimal form of system transient process according to the maximum stability degree criterion is developed. An analytical method for determining the optimal values of control law parameters is proposed, based on the structural features of the linear differential equations system. A method that takes into account constraints on control moments and angular velocities of reaction wheels, based on transition scale from relative time to real time and its calculation algorithm, ensuring that conditions of all constraints are met, is developed. An example of solving the problem of optimal system synthesis is considered. The results demonstrate the effectiveness and simplicity of the proposed methods and algorithm for targeted selection of a system’s technical characteristics, taking into account constraints on reaction wheels’ maximum angular velocities and control moments values.

Analysis of Students' Mathematical Communication Skills On The Material of Two-Variable Linear Equation System With The Application of Bruner Theory

The subjects of this qualitative descriptive research were class VIII students of SMP Negeri 4 Tarutung located in North Tapanuli Regency. The aim of this research is to determine and explain students' mathematical communication abilities when solving problems using Bruner's Theory. The data is in the form of written test results given to students to assess their mathematical communication skills in solving problems using Bruner's Theory. The results of the interview also consist of students' answers to verbal questions given to obtain additional information. Documentation data in the form of pictures and archives of test answers taken by class VIII students of SMP Negeri 4 Tarutung. Data collection methods include written tests, interviews, and documentation. Data analysis is carried out by reducing data, presenting and drawing conclusions. The results of the research show that students' mathematical communication skills when solving problems use Bruner's Theory 43% of students with high mathematical communication skills were able to solve questions correctly and well, while 36% of students with moderate abilities could only meet some indicators, and 21% of students with low abilities could not meet all the indicators.

An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations

In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.

Solution of an Algebraic Linear System of Equations Using Fixed Point Results in C∗-Algebra Valued Extended Branciari Sb-Metric Spaces

Solution of an Algebraic Linear System of Equations Using Fixed Point Results in C∗-Algebra Valued Extended Branciari Sb-Metric Spaces

Micromechanical analysis of magnetoelectric coupling of composites containing nonlinear magnetostrictive and piezoelectric constituents

This paper is a reformulation of the simplified unit-cell and Mori-Tanaka micromechanics theories in their applications to nonlinear magnetoelectric analysis of two-phase composite materials with 0–3, 1–3, and 2–2 connectivity types, respectively. We elucidate the similarity between the two models insofar as concentration-factor matrices are concerned. The representations of the bulk magnetostrictive and piezoelectric phases are based on nonlinear constitutive equations. Due to material nonlinearity, the mathematical frameworks are accomplished via incremental formulation that provides a system of linear algebraic equations at each increment which is obviously a great advantage over nonlinear one. The derived nonlinear composite constitutive relations that govern the hysteresis behavior of composites are implemented to study the composite composed of Terfenol-D and PZT constituents. The responses of this composite to a complete cycle of magnetic field are shown and the micromechanics predictions are compared in light of existing experimental data.

Modeling of charged self-gravitating compact configurations using conformal killing vector

This manuscript investigates sources that are spherically symmetric when subjected to an electromagnetic field (EMF), specifically concentrating the fluid distribution in non-static spacetime. We analyzed the source in f(R) gravity, where R is the Ricci scalar. We investigate the fluid’s collapse by using the strategy that the metric satisfies the conformal killing vector (CKV) equation. To make our system solvable, we imposed some limitations on it, i.e., the homologous collapse and the diminishing of complexity factor. It is analyzed that the electric field intensity has an effect on physical variables like energy density and stress components. It is also concluded that temperature distribution implicitly depends on electric field intensity.

Two-Level method for blind image deblurring problems

Blind image deblurring (BID) is a procedure for reducing blur and noise in a deteriorated image. In this process, the estimation of the original image, as well as the blurring kernel of the degraded image, is done without or with only partial information about the imaging system and degradation. This is an inverse problem (ill-posed) that corresponds to the direct problem of deblurring. To overcome the ill-posedness of BID and attain useful solutions, the regularization models based on mean curvature (MC) are utilized. The discretization of MC-based models often leads to a large ill-conditioned nonlinear system of equations, which is computationally expensive. Moreover, the existence of MC functionals in the governing equations of the BID model complicates the calculation of the nonlinear system. To overcome these problems, in this paper, we propose the Two-Level blind image deblurring method (TLBID). First, on the coarse-grid, we solve a small nonlinear system (with a small number of pixels) for a mesh size of H, followed by solving a large linear system of equations on the finer grid (with a large number of pixels) of size h (h≤H). On the coarse mesh, we solve the BID problem utilizing the computationally expensive MC regularization functional. After this, we interpolate the results to the finer mesh. On the finer mesh, we solve the BID problem with less computationally expensive regularization functionals such as total variation (TV) or Tikhonov. This approach produces an approximate solution of the BID equations with high accuracy, which is cost-effective. The TLBID algorithm is implemented with MATLAB, and verification and validation are carried out using benchmark problems and medical digital images.

Converging Time-Dependent Density Functional Theory Calculations in Five Iterations with Minimal Auxiliary Preconditioning.

Eigenvalue problems and linear systems of equations involving large symmetric matrices are commonly solved in quantum chemistry using Krylov space methods, such as the Davidson algorithm. The preconditioner is a key component of Krylov space methods that accelerates convergence by improving the quality of new guesses at each iteration. We systematically design a new preconditioner for time-dependent density functional theory (TDDFT) calculations based on the recently introduced TDDFT-ris semiempirical model by retuning the empirical scaling factor and the angular momenta of a minimal auxiliary basis. The final preconditioner produced includes up to d-functions in the auxiliary basis and is named "rid". The rid preconditioner converges excitation energies and polarizabilities in 5-6 iterations on average, a factor of 2-3 faster than the conventional diagonal preconditioner, without changing the converged results. Thus, the rid preconditioner is a broadly applicable and efficient preconditioner for TDDFT calculations.

Two-mode weakly linear systems of fuzzy relation equations: Structures of solutions, computation methods, and applications

The problem of aggregating large fuzzy relational structures is of major importance. One of the main issues is seeking a simpler fuzzy relational structure, while keeping the structure properties of the original fuzzy relational structure as much as possible. This paper covers the mathematical foundations of the described problem, and deals with the solvability issues of two-mode weakly linear systems over complete residuated lattices. Within these systems, our objective is to identify a pair of unknown fuzzy relations that are present on both sides of the equation or inequality. We provide a characterization of the set of all pairs of fuzzy relations that are solutions to two-mode weakly linear systems to a desired degree, and develop an algorithm to compute the greatest such pair. We also present an algorithm to determine a pair that solves a two-mode weakly linear system to a given extent, such that each component is a fuzzy preorder (resp. fuzzy equivalence). Ultimately, we demonstrate the application aspects of aggregating fuzzy relational structures.

Numerical solution of optimal control problems for linear systems of ordinary differential equations

Abstract An original numerical matrix algorithm aimed at solving the optimal control problems for linear systems of ordinary differential equations with constant coefficients is proposed. The work of the algorithm is demonstrated with the problem, which consists in generating a given small disturbance of the Poiseuille flow in an infinite duct by blowing and suction through the walls. The costs of creating the leading modes and optimal disturbances are compared, which is of independent interest.

Accurate Computations with Generalized Green Matrices

We consider generalized Green matrices that, in contrast to Green matrices, are not necessarily symmetric. In spite of the loss of symmetry, we show that they can preserve some nice properties of Green matrices. In particular, they admit a bidiagonal decomposition. Moreover, for convenient parameters, the bidiagonal decomposition can be obtained efficiently and with high relative accuracy and it can also be used to compute all eigenvalues, all singular values, the inverse, and the solution of some linear system of equations with high relative accuracy. Numerical examples illustrate the high accuracy of the performed computations using the bidiagonal decompositions. Finally, nonsingular and totally positive generalized Green matrices are characterized.

Numerical Analysis of Dynamic Stability of Flutter Panels in Supersonic Flow

This paper introduces a novel numerical method for investigating the dynamic stability of a flutter panel exposed to a supersonic gas flow and a fluctuating axial excitation force. Initially, the system equation of motion was derived using Lagrange’s equation, where the first two modes are coupled Mathieu–Hill equations with damping, constituting a system of linear second-order differential equations with periodically variable coefficients. Subsequently, a new numerical method was proposed to analyze the dynamic stability of coupled Mathieu–Hill equations with damping. This method involves breaking down an arbitrary parametric load into discrete segments to approximate the variable excitation function using a step function. The system responses of each segment are then accumulated in matrix form. The proposed numerical method proves particularly effective for dynamic systems whose parameters cannot be treated as small. In practical application, the method allows the construction of instability regions corresponding to natural frequencies, subharmonics, and combination frequencies. Dynamic stability diagrams were generated based on dynamic pressure ratio, air/panel density ratio, Mach number, panel thickness—length ratio, and excitation frequency. The results demonstrated general agreement with those obtained through Hsu’s perturbation method, however, our numerical results have proven more accurate. The paper concludes by offering suggestions for suppressing panel flutter through appropriate parameter combinations.

Solution of Fuzzy System of Linear Equation Under Different Fuzzy Difference Ideology

In this article, we solve fuzzy system of linear equations under Hukuhara difference and generalized Hukuhara difference using analytical technique. We also give necessary condition for existence and uniqueness of the fuzzy solution of system in parametric form. Under this proposed concept, nonnegativity of parameters is also taken care. Lastly, a numerical illustration is solved under both differences and graphically displayed.

Hybrid deconvolution method based on mode composition beamforming for separating sound sources with different motion modes

The presence of sound sources that rotate can cause interference with stationary acoustic imaging, especially if the intensity of the rotating sound source is strong. Vice versa, stationary sound sources can also interfere with rotational acoustic imaging. Current algorithms for separating and locating rotating and stationary sound sources have problems such as weak separation capabilities, complex calculations, and limited array layouts. To address these challenges more effectively, a frequency-domain hybrid deconvolution method based on modal composition beamforming is proposed to separate and localize rotating and stationary sound sources. This algorithm operates within the frequency domain and is not restricted by array layout or the tonal single-frequency nature of sound sources. First, the point spread function and cross-point spread function matrices of the array are derived based on the modal transfer function. Then, a system of linear equations is constructed to separate rotating and stationary sound sources. Finally, the sound source separation and localization problems are solved through Gaussian Seidel and sparse-constrained deconvolution, respectively. Compared with the virtual rotating array method combined with matrix completion, simulation and experimental results demonstrate that this method can accurately locate and separate rotating and stationary sound sources, even when there is a substantial difference in intensity between rotating and stationary sound sources.

Random walk algorithms for solving nonlinear chemotaxis problems

Abstract Random walk based stochastic simulation methods for solving a nonlinear system of coupled transient diffusion and drift-diffusion equations governing a two-component chemotaxis process are developed. The nonlinear system is solved by linearization, the system is evolved in time, by small time steps, where on each step a linear system of equations is solved by using the solution from the previous time step. Three different stochastic algorithms are suggested, (1) the global random walk on grid (GRWG), (2) a randomized vector algorithm (RVA) based on a special transformation of the original matrix to a stochastic matrix, and (3) a stochastic projection algorithm (SPA). To get high precision results, these methods are combined with an iterative refinement method.

Growth in Mathematical Understanding of a Junior High School Student on the Topic of System of Linear Equations in Two Variables (SPLDV)

Growth in Mathematical Understanding of a Junior High School Student on the Topic of System of Linear Equations in Two Variables (SPLDV)

Numerical simulation for an initial-boundary value problem of time-fractional Klein-Gordon equations

This paper mainly presents numerical solutions to an initial-boundary value problem of the time-fractional Klein-Gordon equations. We developed a numerical scheme with the help of the finite difference methods and the predictor-corrector methods to find numerical solutions of the considered problems. The proposed scheme is based on discretizing the considered problems with respect to spatial and temporal domains. Numerical results are derived for some illustrative problems, and the outputs are compared with the exact solution in the integer order case. The solution behavior and 3D graphics of the discussed problems are demonstrated using the proposed scheme. Finally, the proposed scheme, which does not require solving large systems of linear equations, can be extended and modified to handle other classes of time-fractional PDEs.

Generalized and novel iterative scheme for best approximate solution of large and sparse augmented linear systems

This article presents a generalized novel iterative technique for obtaining the approximate solution of systems of linear equations(ABBT0)(xy)=(pq). Parameters involved in the scheme play a vital role to obtain the rapid and best approximate solution. The technique is applied to some important problems and the results are presented in detail. The article compares the effectiveness of the new presented iterative scheme with modified SOR-like method, generalized SOR and SOR-like method for solving large systems of linear equations. The convergence criteria for the iterative scheme are also analysed and it can be observed in the article that the theoretical derivation is supported by numerical calculations and graphical exhibition. Overall, the article provides a new and effective method with a detailed analysis of its performance and convergence criteria for solving systems of linear equations of the form presented. The newly derived technique has potential applications in various fields of science and engineering where systems of linear equations arise and need to be solved.