System Of Linear Equations Research Articles

An Exact and Approximate Schur Complement Method for Time‐Harmonic Optimal Control Problems

ABSTRACTTime‐harmonic control problems, constrained by a linear differential equation, can be solved efficiently by utilizing a Fourier time series expansion in the angular frequency variable. Then the optimal solution consists of a series of complex variable space discretization equations, which are uncoupled with respect to the different frequencies. Hence, it suffices to consider a single equation with the angular frequency as a parameter. We consider here methods to solve the so‐arising linear system of equations and describe, analyze and test the performance of two novel approaches based on its exact and approximate Schur complement. The performance of the methods is tested and compared with another existing method.

CRAMER’S RULE IN INTERVAL MIN-PLUS ALGEBRA

A min-plus algebra is a set , where is the set of all real numbers, equipped with the minimum and addition operations. The system of linear equations in min-plus algebra can be solved using Cramer's rule. Interval min-plus algebra is an extension of min-plus algebra, with the elements in it being closed intervals. The set is denoted by equipped with two binary operations, namely minimum and addition . The matrix with notation is a matrix over interval min-plus algebra with size . Since the structure of min-plus algebra and interval min-plus algebra are analogous, the system of linear equations in interval min-plus algebra can be solved using Cramer's rule. Based on the research results, the sufficient conditions of Cramer's rule in interval min-plus algebra are for , and . The Cramer rule is .

Approximate recovery of the Sturm–Liouville problem on a half-line from the Weyl function

Abstract The inverse problem of the reconstruction of the Sturm–Liouville problem on a half-line from its Weyl function is considered. Given the Weyl function, we obtain the potential in the Schrödinger equation and the boundary condition at the origin. If the boundary condition is known, this problem is equivalent to the inverse scattering problem of the recovery of the potential, which is zero on a half-line, from a given reflection coefficient. We develop a simple and direct method for solving the inverse problem. The method consists of two steps. First, the Jost solution is computed from the Weyl function, by solving a homogeneous Riemann boundary value problem. Second, a system of linear algebraic equations is constructed for the coefficients of series representations of three solutions of the Schrödinger equation. The potential and the boundary condition are then recovered from the first component of the solution vector of the system. A numerical illustration of the functionality of the method is presented.

Quantum computing inspired iterative refinement for semidefinite optimization

AbstractIterative Refinement (IR) is a classical computing technique for obtaining highly precise solutions to linear systems of equations, as well as linear optimization problems. In this paper, motivated by the limited precision of quantum solvers, we develop the first IR scheme for solving semidefinite optimization (SDO) problems and explore two major impacts of the proposed IR scheme. First, we prove that the proposed IR scheme exhibits quadratic convergence of the optimality gap without any assumption on problem characteristics. As an application of our results, we show that using IR with Quantum Interior Point Methods (QIPMs) leads to exponential improvements in the worst-case overall running time of QIPMs, compared to previous best-performing QIPMs. We also discuss how the proposed IR scheme can be used with classical inexact SDO solvers, such as classical inexact IPMs with conjugate gradient methods.

Flexible iterative methods for linear systems of equations with multiple right-hand sides

Flexible iterative methods for linear systems of equations with multiple right-hand sides

A variational quantum algorithm for the Poisson equation based on the banded Toeplitz systems

Abstract For solving the Poisson equation it is usually possible to discretize it into solving the corresponding linear system $Ax=b$. Variational quantum algorithms (VQAs) for \textcolor{red}{the discreted} Poisson equation have been studied before. We give a VQA based on the banded Toeplitz systems for solving the Poisson equation with respect to the structural features of matrix $A$. In detail, we decompose the matrix $A$ and $A^2$ into a linear combination of the corresponding banded Toeplitz matrix and sparse matrices with only a few non-zero elements. For the one-dimensional Poisson equation with different boundary conditions and the $d$-dimensional Poisson equation with Dirichlet boundary conditions, \textcolor{red}{the number of decomposition terms is less than the work in [Phys. Rev. A108, 032418 (2023)]}. Based on the decomposition of the matrix, we design quantum circuits that \textcolor{red}{evaluate efficiently} the cost function. \textcolor{red}{Additionally, numerical simulation verifies the feasibility of the proposed algorithm. In the end,} the VQAs for linear systems of equations and matrix–vector multiplications with $K$-banded Teoplitz matrix $T_n^K$ \textcolor{red}{are} given, where $T_n^K\in R^{n\times n}$ and $K\in O(ploy\log n)$.

Solvability of a system of nonlinear integral equations on the entire line

A system of singular integral equations with monotone and concave nonlinearity in the subcritical case is investigated. The specified system and its scalar analogy have direct applications in various areas of physics and biology. In particular, scalar and vector equations of this nature are encountered in the dynamic theory of p-adic strings, in the theory of radiative transfer, in the kinetic theory of gases, and in the mathematical theory of the spread of epidemic diseases. A constructive theorem of existence in the space of continuous and bounded vector functions is proved. The integral asymptotic of the constructed solution is studied. An effective iterative process for constructing an approximate solution to this system is proposed. In a certain class of bounded vector functions, the uniqueness of the solution is proved. In the class of bounded vector functions (with non-negative coordinates) with a zero limit at infinity, the absence of an identically zero solution is proved. Examples of the matrix kernel and nonlinearities are given to illustrate the importance of the obtained results. Some of the examples have applications in the above-mentioned branches of natural science.

Fixed Point Theorems with its Applications in Fuzzy Complete Convex Fuzzy Metric Spaces

In this paper, the basic properties of the convex fuzzy metric space will be presented. In particular, the proof of the fixed-point theorem for the fuzzy contraction single valued functions will be discussed. Furthermore, the solution system of linear equations, Volterra equations and Fredholm integral equations will be obtained as a direct application of the fixed-point theorem.

Analisis Kemampuan Pemecahan Masalah Matematis Siswa SMP Negeri Sekon Pada Materi Sistem Persamaan Linear Dua Variabel

This research aims to describe students' mathematical problem solving abilities in the material on systems of linear equations in two variables (SPLDV) class VIII SMPN Sekon. This research was carried out in the odd semester of the 2023/2024 academic year in Sekon. The population in this study were all students in class VIII of Sekon State Middle School with a sample size of 22 students. The research instrument used was a test of conceptual understanding ability and mathematical problem solving ability. The research method used is a qualitative research design. Data were analyzed using data reduction, data presentation, and drawing conclusions. Based on the research results, it can be concluded that students with high level problem solving abilities are students who can work on and explain again what has been done according to the existing problem solving stages, while students with medium level problem solving abilities are students who can work on half of the questions. given and can explain again according to the stages of the questions that have been worked on and for students with low level problem solving abilities, these are students who can only understand the problem in the question but cannot carry out the calculation process to the final stage or conclusion and are also not able to explain again what was said. already written

Mathematical Model of the Propagation of Radioactive Impurities in the Atmosphere and a Numerical Algorithm for Solving the Problem using the Splitting Method

In this research, the splitting into physical processes approach is used to numerically predict the propagation of radioactive contaminants in the environment. The movement, diffusion, and absorption of radioactive particles inside the air mass are all described by the mathematical model. An implicit finite-difference technique with a second order of accuracy in both time and space is employed to tackle the problem. A number of variables, including wind speed, particle deposition, absorption coefficient, wet deposition and radioactive decay, were taken into consideration for modeling. A thorough algorithm for the numerical solution is provided, which includes employing the sweep method to solve a system of linear algebraic equations and dividing the problem into smaller problems based on physical processes.

A New Spectral Mittag-Leffler Weighted Method to Solve Variable Order Fractional Differential Equations

Background: In many studies, spectral methods based on one of the orthogonal polynomials and weighted residual methods (WRMs) have been used to convert the variable-order fractional differential equation (VO-FDEs) into a system of linear or nonlinear algebraic equations, and then solve this system to obtain the approximate solution. Objective: In this paper, a numerical method is presented for solving VO-FDEs. The proposed method is based on Chelyshkov polynomials (CPs). Methods: WRMs are used to obtain approximate solutions of the governing differential equations. In addition, a new weight function based on Mittag-Leffler functions is proposed. The proposed method is applied to a group of linear and non-linear examples with initial and boundary conditions. Results: Acceptable results are obtained in most of the examples. In addition, the effect of polynomials (such as Chebyshev, Jacobi, Legendre, Gegenbauer, Hermite, Taylor, Mittag-Leffler, and Bernstein polynomials) on the accuracy of the approximate solution is studied, and their effect was found to be minimal in most tests. The effect of the proposed weight function is also studied in comparison with the weight functions presented in WRMs, and it was found that it has a strong effect in most examples. Conclusions: The results obtained indicate that the proposed method is effective for solving equations of variable order.

Solving Volterra-Fredholm integral equations by non-polynomial spline functions

It depends on our information, non-polynomial spline functions have not been applied for solving Volterra- Fredholm integral equations of the second kind yet. In this paper, we want to use such functions for finding approximation solutions of Volterra-Fredholm integral equations. In our approach, the coefficients of the non-polynomial spline were found by solving a system of linear equations. Then, these functions were utilized to reduce the fredholm integral equations to the solution of algebraic equations. Analysis of convergences investigated. Finally, three examples were presented to show the effectiveness of the method. This was done with the help of a computer program that used the Python code program version 3.9.

Two Quasi-combining Real and Imaginary Parts Iteration Methods for Solving Complex Symmetric System of Linear Equations

Two Quasi-combining Real and Imaginary Parts Iteration Methods for Solving Complex Symmetric System of Linear Equations

ЛИНЕЙНЫЕ И УГЛОВЫЕ ПЕРЕМЕЩЕНИЯ ЖЕСТКОЙ ПРЯМОУГОЛЬНОЙ ФУНДАМЕНТНОЙ ПЛИТЫ НА УПРУГОМ СЛОЕ

The present work proposes an approach to determining the linear and angular displacements of a rigid rectangular slab lying without friction on an elastic layer. The approach is based on the orthogonal polynomial method, in which the distribution of contact stresses is sought in the form of a double series of Chebyshev polynomials of the first kind with a weight, which allows one to identify a feature in the contact stresses at the slab faces. The Green’s function for the elastic layer is also expanded into a four-fold series in these polynomials using four uses of the quadrature formula. Using the orthogonality of the adopted Chebyshev polynomials with weight reduces the problem under consideration to an infinite system of linear algebraic equations, which is solved by the truncation method. Two examples are given for a rigid square plate on an elastic layer under loading by concentrated symmetrically applied force and moment. For comparison, the results of M.I. Gorbunov-Posadov are given in the case of the plate being located on an elastic half-space.

Parameter Identification and Prediction of the Rőssler System with Complete and Incomplete Information: Two Known and One Unknown State Variables

Parameters identification algorithms are formulated for the system of equations of Rőssler's type. The problem is solved for the cases of complete and incomplete information about functions of the system. If there is complete information about the state variables, it is possible to apply the algorithms discussed to any system of linear or nonlinear ordinary differential equations of arbitrary order in the Cauchy form that linearly depends on the unknown parameters (or groups of unknown parameters). The problems of parameter identification in the case of incomplete information about the state variables must be solved individually, depending on the possibility (or impossibility) of eliminating unknown steady states from the system of equations. Most real-world problems in fields such as chemical kinetics, mathematical ecology, predator-prey dynamics in game reserves, and the spread of infectious diseases belong to this class of problems, in which the algorithms discussed demonstrate their applicability. In the present paper the case of one unknown function is considered. Lemmas about possibilities on complete and incomplete parameter identification are formulated and proven. The algorithms of the parameter identification are formulated in the process of the constructive proofs of the lemmas. Numerical examples and graphs of solutions are considered which demonstrate efficiency and accuracy of the developed algorithms. In the proposed paper, the integration approach is used instead of the differential approach because it allows for the smoothing of discrete data, thereby reducing the estimation errors of the unknown parameters.

Приближенное решение гиперсингулярного интегрального уравнения с применением рядов Чебышева на классе функций, ограниченных на одном конце и неограниченных на другом конце интервала интегрирования

Approximate methods for solving hypersingular integral equations are an actively developing field of computational mathematics. This is due to the numerous applications of hypersingular integral equations in various fields of mathematics and the fact that analytical solutions of such equations are possible only in exceptional cases. A method for finding a solution, limited at one end and unlimited at the other end of the integration interval [-1,1], of hypersingular integral equations of the first kind using Chebyshev series is proposed and justified. The core and the right side of the equation are also decomposed into Chebyshev series, the decomposition coefficients of which are calculated approximately using quadrature Gauss formulas with the highest degree of accuracy. The coefficients of decomposition of an unknown function into a Chebyshev series are found by solving systems of linear algebraic equations. Methods of functional analysis and the theory of orthogonal polynomials are used to substantiate the computational scheme. The Helder space of functions with corresponding norms is introduced. Given hypersingular and corresponding approximate operators are considered in this space. When the condition for the existence of derivatives up to a certain order belonging to the Helder class for given functions is fulfilled, the calculation error is estimated and the order of its tendency to zero is given.

Modeling of shear wave propagation in piezo-viscoelastic microbeam over quadratic heterogeneous viscoelastic plate with sliding contact

Abstract This study analyzes the phase and attenuation dynamic behavior of piezo-viscoelastic microbeam overlying quadratic heterogeneous viscoelastic plate under sliding contact. Using the Kelvin-Voigt model, the material properties of the system are assumed to be viscoelastic. Maxwell’s relations are used to incorporate the electric potential function. The solutions for both media are derived separately by solving the second-order hyperbolic differential equation using the method of separation of variables and asymptotic expansions of Bessel functions. The system of linear homogeneous equations is obtained by applying admissible boundary conditions to determine fundamental physical quantities. The key contribution of the current work is demonstrating the influence of dissipation factors, sliding contact, micro-length, heterogeneity, and thickness ratio parameters on shear wave propagation. The micro-length effect is found to suppress the attenuation of shear waves through the analysis and discussion of the dispersion and attenuation curves.

The Modelling of Light Absorption and Reflection in a SiOx/Si Structure with Al Nanoparticles for Solar Cells

In this study, the light propagation in a structure consisting of SiOx on Si substrate with Al nanoparticles regularly placed in the SiOx layer is considered. Numerical modelling is performed by solving the Maxwell equations for the electromagnetic waves. In distinction from the well-known finite-difference time-domain (FDTD) simulation technique, we do not solve time-dependent wave equations here; rather, we propose a new numerical technique. This technique allows us to determine the stationary amplitudes of the electromagnetic oscillations directly from the linear algebraic equation system. The obtained results apply to silicon solar cells with an SiOx + Al top layer to maximise their efficiency. We found that 26 nm and 39 nm diameters of spherical Al nanoparticles are nearly optimal for a λ = 435.8 nm wavelength of the incident light. In addition, we evaluated the (nearly) optimal parameters of their placement in the SiOx layer. The results show the possibility of increasing the efficiency of solar cells by increasing the light absorption inside the active Si layer from ≈60% to ≈80%. Future perspectives on the proposed method and its possible applications are discussed.

3D airborne electromagnetic forward modeling based on the multiscale hexahedral finite-element method

Slow forward modeling is the main factor that restricts the practical use of 3D inversion and interpretation of airborne electromagnetic (AEM) data. To improve modeling efficiency with 3D AEM data, we develop a new multiscale finite element (MsFE) method based on unstructured hexahedral meshes. Compared to traditional 3D AEM forward modeling, the main advantage of our newly developed method is that it can simulate complex underground structures in the earth quickly. Because we can fit the earth’s topography or the anomalous bodies underground using a small number of hexahedral grids, we can quickly model them using MsFE. The main idea of the MsFE forward-modeling method is to construct an interpolation operator between a coarse and a dense mesh and use the interpolation operator to map the conventional finite-element coefficient matrix to the MsFE coefficient matrix and thus reduce the number of unknowns in the modeling process. This will vastly reduce the scale of the linear equations system. We validate our method by simulating a typical mountain peak model and determine its effectiveness by simulating numerous synthetic models and a model from Voisey Bay’s Ovoid sulfide deposit, Canada.

Numerical Model of Temperature-Filtration Regime of Earth Dam in Harsh Climatic Conditions

The article addresses the issue of numerical modeling of the process of forming the temperature regime of earth dams, along with their foundations, built and operated in permafrost conditions. A large number of such structures have been constructed in the permafrost regions of the Earth to meet the needs of industry and population. The paper outlines the key principles of designing and constructing such structures. These principles were developed based on years of experience in hydrotechnical construction. Failure to follow these principles leads to structural failures, as confirmed by the presented statistics on accidents. It is essential to ensure the appropriate thermal condition of the structure and its foundation, either frozen or thawed. An unplanned transition of soils from one state to another may lead to an emergency situation. Temperature changes can cause phase transitions of water from liquid to solid (ice), which also affects the formation of the structure’s regime. Numerical methods of calculation allow for the most comprehensive consideration of the influencing factors and processes. The article presents the results of numerical modeling of the filtration-temperature regime of an earth dam with a foundation in permafrost conditions, using two computational programs. The first is based on a locally variational approach (Termic, authored by the researchers), while the second uses a classical linear equation system solution (PLAXIS 2D 2022 software). A comparison of the results obtained from both programs showed good qualitative and quantitative consistency. Under the influence of seepage flow, the zone of frozen ground degradation is spreading in the lower part of the earth dam and its foundation. By September of the 27th year of operation, the thawed ground zone reaches approximately the middle of the structure at the base. The temperature values along the screen axis at the base of the structure are +1.2 °C (according to the Termic program—ver. 1.1) and +1.06 °C (according to PLAXIS 2D PC). Recommendations and future research directions on this topic are also formulated.