О методе коллокации при построении решения уравнения изгиба длинной прямоугольной нанопластины

The paper proposes a matrix implementation of the collocation method for constructing a solution to Volterra integral equations of the second kind using systems of orthogonal Chebyshev polynomials of the first kind and Legendre polynomials. The integrand in the equations considered in this work is represented as a partial sum of a series for these polynomials. The roots of the Chebyshev and Legendre polynomials are chosen as collocation points. Using matrix and integral transformations, properties of finite sums of products of these polynomials and weight functions at the zeros of the corresponding polynomials with degree equal to the number of nodes, integral equations are reduced to systems of linear algebraic equations for unknown values of the sought functions at these points. As a result, solutions to Volterra integral equations of the second kind are found by polynomial interpolations of the obtained function values at collocation points using inverse matrices, the elements of which are written on the basis of orthogonal relations for these polynomials. In the presented work, the elements of integral matrices are also given in explicit form. Error estimates for the constructed solutions with respect to the infinite norm are obtained. The results of computational experiments are presented, which demonstrate the effectiveness of the collocation method used.

Abstract. In this paper a method for solving an inhomogeneous biharmonic equation while modeling elastically deformed states of thin isotropic rectangular plates using a system of orthogonal Chebyshev polynomials of the first kind is proposed. The method is based on representation of a solution to the initial biharmonic equation as a finite sum of Chebyshev series by each independent variable in combination with matrix transformations and properties of Chebyshev polynomials. The problem is examined for the case when a transverse load acts on the plate, and the hinge fastening along the edges of the plate is taken as boundary conditions. Using the extremes and zeros of Chebyshev polynomials of the first kind as collocation points, the boundary value problem is reduced to a system of linear algebraic equations. Decomposition coefficients of desired function with respect to Chebyshev polynomials act as unknowns in this system. As the comparison showed, the results obtained by this method with a high degree of accuracy coincide with similar results derived using analytical approach that are given in the article. The paper also presents the results of calculations using the proposed method in the case when two opposite edges of the plate are pinched and two others are pivotally fixed. The comparison with similar results of modeling the stress-strain states of rectangular plates which are presented in the open sources is carried out.

The numerical solution of integral equations or a system of integral equations (for a system of conductors) consists in their discretization and reduction to a system of linear algebraic equations for the desired function. For discretization, it is possible to use projection methods, for example, the Galerkin method or the collocation method. A system of linear algebraic equations for the class of problems under consideration is characterized by a complete (filled) complex matrix. For conductor antenna structures that are quite arbitrary in geometry and size, systems of linear algebraic equations turn out to be of a high order, and special computational algorithms are required to solve them. The purpose of the work is to study adequate mathematical models for wire antennas with a subsequent description of uniform numerical algorithms based on methods of sampling and approximation of antenna current. With this method, the error in solving integral equations is determined by the error in calculating the elements of the matrix of a system of linear algebraic equations for a given piecewise polynomial approximation of the solution at step h. If the quadrature formula for numerical integration is chosen, then there are two ways to reduce the solution error. An increase in the number of collocation points leads to a rapid increase in the volume of calculations and, consequently, to a rapid increase in the amount of occupied computer memory. Therefore, the question arises of choosing the most profitable method of piecewise polynomial interpolation and discretization step h from the point of view of using computer resources, ensuring an acceptable number of solutions.

A collocation method based on roots of Chebyshev polynomial for solving Volterra integral equations of the second kind

Subject. This article focuses on the problems of solving systems of homogeneous linear algebraic equations in the pricing problems of an alternative financial model. Objectives. The article aims to develop a non-trivial approach to solving the problem of pricing from the scratch in the construction of a national alternative financial model, which boils down to solving large homogeneous systems of linear algebraic equations, complicated by the fuzzy setting of initial parameters (initial data) within the framework of creating a State model of a socially oriented economy, guaranteed to ensure economic, military and other types of integrated security of the Russian Federation. Methods. For the study, we used tensorial notations involving the use of elements of linear algebra, including operations with matrices and vectors, the apparatus of the fuzzy set theory (interval arithmetic), as well as methods and principles of scientific research and complex logical analysis and modeling of pricing and technological processes. Results. The article shows that an effective non-zero (different from zero, i.e. classical, trivial) solution of systems of homogeneous algebraic equations in pricing problems of an alternative financial model from the scratch is possible if the initial parameters are specified by fuzzy sets, which significantly simplifies the procedure of the computational process and the volume of arithmetic operations. Conclusions. The results obtained can be used in structures involved in the creation and testing of financial and economic models both at the federal and the Russian Federation constituent entity levels, as well as in governmental and non-governmental organizations dealing with the issues of pricing, planning and logistics support for business entities not only in the Russian Federation, but also in other countries experiencing problems in the field of financing.

A review of existing methods for stability research of solutions of systems of linear algebraic equations (SLAE) depending on the input data, that is, parameter variations, have been carried out. Methods for stability research of solving systems of linear algebraic equations, such as condition numbers, modular determinants and the construction of a table of signs using the original and improved construction methods, were considered. Software for stability research of systems of linear algebraic equations have been developed. The software is using condition numbers, modular determinants, and construction of table of signs to find accurate estimates of the variations of solutions of SLAEs that depend on parameters variations.It is shown that stability research using condition numbers gives a very rough estimate of possible errors in solutions, but that research is simple to implement, and for SLAEs it can immediately show that some systems are ill-conditioned, which saves research time, especially if SLAEs have a very large dimensionality. The stability research using modular determinants requires large calculations, but they give a fairly reliable upper estimate with respect to possible variations of individual components of solutions of systems of linear algebraic equations. This is a very important feature of the method because the individual components of solutions may experience significant variations that are not taken into account in the research using condition numbers. The study of stability by constructing a table of signs makes it possible to find the maximum variations of individual components of solutions of a system of linear algebraic equations, which in fact can be significantly less than the upper estimate of possible variations found by method of modular determinants. The paper proposes an improved method for constructing a table of signs, which finds a more accurate range of possible variations of solutions of a system of linear algebraic equations.A comparative analysis was conducted between the traditional method of constructing a table of signs using individual determinants and an improved method of constructing a table of signs based on the derivatives of the division of determinants using the Cramer formula. According to the analysis, the improved method in 30% of cases finds variations that are 1.3 times greater than the variations that the previous method finds, and in 5% of cases these variations are 2 or more times greater than the previous ones. This suggests that the traditional method in some cases underestimated the possible deviations of solutions that depend on variations of the input data.

As it is well known the problem of solving the Fredholm integral equation of the first kind belongs to the class of ill-posed problems. The Tikhonov regularization method is well known. This method is usually applied to an integral equation and a system of linear algebraic equations. The authors firstly propose to reduce the integral equation of the first kind to a system of linear algebraic equations. This system is usually extremely ill-posed. Therefore, it is necessary to carry out the Tikhonov regularization for the system of equations. In this paper, to form a system of linear algebraic equations, local polynomial and non-polynomial spline approximations of the second order of approximation are used. The results of numerical experiments are presented.

This paper presents a description of specific properties of determining the values of partial capacitances of insulation gaps in power cables with paper insulation for various ways of forming and solving the system of linear algebraic equations. Possible ways of inspection the insulation of three core power cables for the estimation of values of partial capacitances by applying aggregate measurements which are based on various ways of connection of emittance meter to tested sample of power cable are given. Estimation of partial capacitances by the direct solution of a system of linear algebraic equations, by minimizing the root mean square error of solving an overdetermined system of equations by the least squares method, as well as by finding a normal solution of an indefinite system of equations by the pseudo-inverse matrix, is also considered. It is shown that minimization of the root mean square error by the least squares method and the direct solution of system of equations show quite similar results for the case of estimation of partial capacitances by means of aggregate measurements, at the same time the solution of an indefinite system of equations by the method of a pseudo-inverted matrix allows to reproduce rather accurately only 3 out of 6 values of partial capacitances. The uneven effect of frequency on the electrical capacitance of the insulation gaps between the cores of the power cable and between its cores and the sheath is shown. It was proposed to use the frequency dependence of the electrical capacitance of insulation gaps as an informative parameter about the technical state of insulating gaps between the cores of the power cable and between its cores and its sheath.
 
 Keywords: root mean square error; least squares method; system of linear algebraic equations; dielectric losses; dielectric permittivity.

In this study, a collocation method based on Laguerre polynomials is presented to numerically solve systems of linear differential equations with variable coefficients of high order. The method contains the following steps. Firstly, we write the Laguerre polynomials, their derivatives, and the solutions in matrix form. Secondly, the system of linear differential equations is reduced to a system of linear algebraic equations by means of matrix relations and collocation points. Then, the conditions in the problem are also written in the form of matrix of Laguerre polynomials. Hence, by using the obtained algebraic system and the matrix form of the conditions, a new system of linear algebraic equations is obtained. By solving the system of the obtained new algebraic equation, the coefficients of the approximate solution of the problem are determined. For the problem, the residual error estimation technique is offered and approximate solutions are improved. Finally, the presented method and error estimation technique are demonstrated with the help of numerical examples. The results of the proposed method are compared with the results of other methods

The article presents the results of the approximation of the set of solutions of interval systems of linear algebraic equations. These systems are used in the problems of modeling linear deterministic processes. It is assumed that the modeled process is described by an output variable and a set of input variables, the measurement errors of which are assumed to be set by known intervals symmetric with respect to the zero value. Traditionally, the sets of solutions of interval systems of linear algebraic equations in applied problems are approximated by a hyper-rectangular whose sides are parallel to the axes of the selected coordinate system. In this paper, we propose to use an ellipsoidal approximation of these sets, which is more efficient. The main results of the work include the substantiation of assumptions about the properties of the modeled process, the choice of a mathematical method for constructing an approximating ellipsoid, the proposed method for forming boundary points, and a numerical method for solving the problem. A computer simulation of the problem of estimating the parameters of a linear process is performed in Excel, which is used for a comparative study of approximations of solutions of interval systems of linear algebraic equations by a hyper-rectangular and an ellipse.

The treatment of systems of linear algebraic equations is very often the most time-consuming part when large-scale applications arising in different fields of science and engineering are to be handled on computers. These systems can be very large, but in the most of the cases they are sparse (i.e. many of the elements in their coefficient matrices are zeros). Therefore, it is very important to select fast, robust and sufficiently accurate methods for the solution of large and sparse systems of linear algebraic equations. Tests with ten well-known methods have been carried out. Most of the methods are preconditioned conjugate gradient-type methods. Two important issues are mainly discussed: (i) the problem of finding automatically a good preconditioner and (ii) the development of robust and reliable stopping criteria. Numerical examples, which illustrate the efficiency of the developed algorithms for finding the preconditioner and for stopping the iterations when the required accuracy is achieved, are presented. The performance of the different methods for solving systems of linear algebraic equations is compared. Several conclusions are drawn, the main of them being the fact that it is necessary to include several different methods for the solution of large and sparse systems of linear algebraic equations in software designed to be used in the treatment of large-scale scientific and engineering problems.

This work is devoted to the development of a method for solving the Boltzmann kinetic equation for neutron transport. The method is focused on solving an integral equation for neutron transport in a reactor cell with a complex geometry and different boundary conditions. An algorithm is proposed for solving the problem taking account of anisotropic scattering in the linear-anisotropic approximation. The approach is based on expanding the neutron flux in a system of orthogonal two-dimensional polynomials in each uniform zone of a heterogeneous cell. This expansion reduces the system of linear integral equations to a system of linear algebraic equations. The relations required to calculate the coefficients in the equations (six-fold integrals) as well as the computational algorithm are presented. Calculations of cylindrical and cluster cells are presented. The calculations are compared with the surface pseudosource method. It is shown that the results have advantages over the method of first collisions probabilities.

A new technique for solving the large system of linear algebraic equations associated with implicit differencing of multidimensio nal partial differential equations is presented. The coefficient matrix of the equations is factored, and then approximations to certain terms in the matrix are obtained from series expansions. The resulting system of equations is solved easily. The method is developed and demonstrated using a simple representative two-dimensional equation. Very good results are obtained when one direction is dominant. MPLICIT finite-difference schemes for the solution of multidimensional partial differential equations are usually stable and therefore applicable to a large class of problems. However, they are difficult to implement and may require an excessive amount of computer storage and time. The long computing time arises from the need to solve the large system of linear algebraic equations that result from the differencing. The computing time can be reduced significantly by approximating the coefficient matrix of the linear equations with a matrix that produces a system of equations that are relatively easy to solve. Among such methods are the alternating direction method (ADI)l used by Beam and Warming2 and Stone's strongly implicit method,3 which has been tested by Linetal.4 In this paper, a new technique for solving the large system of linear algebraic equations associated with implict differencing of multidimensional partial differential equations is presented. This method, called the pseudo-elimination method (PE), is shown to be faster than Stone's method for certain problems. The method is directly applicable to linear and linearized nonlinear systems of parabolic or elliptic partial differential equations. In order to discuss the method, a simple linear partial differential equation will be used; however, it should be kept in mind that the PE method is applicable to much more complicated problems. The question of whether the method will work when applied to difficult problems is not addressed. The scope of this paper is limited to presenting the method and illustrating, via a simple problem, that the method has some merit and deserves further study.

A new technique for solving the large system of linear algebraic equations associated with implicit differencing of multidimensio nal partial differential equations is presented. The coefficient matrix of the equations is factored, and then approximations to certain terms in the matrix are obtained from series expansions. The resulting system of equations is solved easily. The method is developed and demonstrated using a simple representative two-dimensional equation. Very good results are obtained when one direction is dominant. MPLICIT finite-difference schemes for the solution of multidimensional partial differential equations are usually stable and therefore applicable to a large class of problems. However, they are difficult to implement and may require an excessive amount of computer storage and time. The long computing time arises from the need to solve the large system of linear algebraic equations that result from the differencing. The computing time can be reduced significantly by approximating the coefficient matrix of the linear equations with a matrix that produces a system of equations that are relatively easy to solve. Among such methods are the alternating direction method (ADI)l used by Beam and Warming2 and Stone's strongly implicit method,3 which has been tested by Linetal.4 In this paper, a new technique for solving the large system of linear algebraic equations associated with implict differencing of multidimensional partial differential equations is presented. This method, called the pseudo-elimination method (PE), is shown to be faster than Stone's method for certain problems. The method is directly applicable to linear and linearized nonlinear systems of parabolic or elliptic partial differential equations. In order to discuss the method, a simple linear partial differential equation will be used; however, it should be kept in mind that the PE method is applicable to much more complicated problems. The question of whether the method will work when applied to difficult problems is not addressed. The scope of this paper is limited to presenting the method and illustrating, via a simple problem, that the method has some merit and deserves further study.

The article presents a comparative analysis of the software implementation performance of numerical methods in solving the problem of finding the equilibrium composition of a complex multicomponent heterogeneous system. The task of finding the equilibrium composition of the system is divided into the following subtasks: 1) taking into account constraints (Lagrange method); 2) finding the maximum function of a nonlinear function: 2.1) converting a function into a system of linear equations (Newton Raphson method); 2.2) using numerical methods to solve a system of linear algebraic equations. An analytical review of the literature data has shown that gradient methods have better performance when solving systems of linear algebraic equations. Therefore, the article compared the performance of the software implementation of the entire algorithm using direct (Gauss, LUP decomposition) and iterative methods (conjugate gradient method, biconjugate gradient stabilized method) for solving a system of linear algebraic equations. The calculation speed of the developed program was also compared using dynamically connected libraries Alglib, ILNumerics, MathNet, Accord to solve the SLAE in the problem of finding the equilibrium composition of a thermodynamic system.