Numerical solution of two-dimensional integral-algebraic systems using Legendre functions

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.

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 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.

In this research paper a new modification of Gauss-Seidel method has been presented for solving the system of linear algebraic equations. The systems of linear algebraic equations have an important role in the field of science and engineering. This modification has been developed by using the procedure of Gauss-Seidel method and the concept of substitution techniques. Developed modification of Gauss-Seidel method is a fast convergent as compared to Gauss Jacobi’s method, Gauss-Seidel method and successive over-relaxation (SOR) method. It works on the diagonally dominant as well as positive definite symmetric systems of linear algebraic equations. Its solution has been compared with the Gauss Jacobi’s method, Gauss-Seidel method and Successive over-Relaxation method by taking different systems of linear algebraic equations and found that, it was reducing to the number of iterations and errors in each problem.

There are no strict rules for the numerical treatment of partial differential equations. We concentrate here on linear partial differential equations. As an example of elliptic partial differential equations the two-dimensional Poisson equation with Dirichlet boundary conditions is investigated. The application of the finite difference approximation transforms this equation into a set of algebraic equations which can be solved iteratively. The time dependent heat equation is an example of parabolic partial differential equations. The one-dimensional Dirichlet boundary value problem is transformed using the implicit and explicit Euler method into an inhomogeneous system of linear algebraic equations. Stability problems of the explicit Euler method are discussed and demonstrated numerically. The wave equation is a model for hyperbolic partial differential equations. It can be transformed into a system of algebraic equations using, for instance, the explicit Euler method. Again, stability problems connected with this method are discussed and demonstrated numerically. Finally, the quantum mechanical tunneling effect is studied numerically by solving the time dependent Schrodinger equation by means of a Crank-Nicholson scheme which transforms the Schrodinger equation into a system of linear algebraic equations with tridiagonal matrix.

Stable numerical methods for solving simultaneous systems of linear algebraic equations

Iterative refinement of solution with biparameter for solving ill-conditioned systems of linear algebraic equations

The article deals with high-performance information technology (HPC) for the problems of stress-strain analysis at all stages of the life cycle of buildings and structures: construction, operation and reconstruction. The results of numerical simulation of high buildings using software as a processor component based on a new hybrid algorithm for solving systems of linear algebraic equations [1] with a symmetric positive-definite matrix that combines computation on multi-core processors and graphs. It has been found that to accelerate the calculations, hybrid systems that combine multi-core CPUs with accelerator coprocessors, including GPUs, are promising [5]. To test the effectiveness of the proposed parallel algorithm for solving systems of linear algebraic equations [1], numerical experiments were carried out at the most dangerous loads of a 27-story building. Results of numerical researches with use for preprocessor (input of initial data) and postprocessor (output of results of calculations) of processing of the LIRA-SAPR software complex are presented [2, 4, 6]. The results of numerical studies of the behavior of structures of high buildings have shown a multiple reduction in the time of solving systems of linear algebraic equations with symmetric matrices on multiprocessor (multi-core) computers with graphical accelerators using the proposed hybrid algorithms [1]. High-performance technologies based on parallel calculations give more effect than more complex processes: modeling of life cycle of high buildings, bridges, especially complex structures of NPPs, etc. for static and dynamic loads, including emergencies in normal and difficult geological conditions, which make up 70% of Ukraine's territories.

In this paper we will develop procedures for solving the system of non-linear Volterra integral equations, which describe the relations between the complex wave functions in the exit pupil and the Fourier transforms of two intensity distributions in the image plane, corresponding to two images of the same object taken at different values of the defocusing of the optical system. The first method reduces the Volterra integral equations to a finite system of linear algebraic equations which can be solved successively. The second algorithm is based on the Newton-Kantorovich approximation for functional equations, which gives a system of recurrence relations by means of which the Volterra equations can be solved.

In this article, we apply the Single Term Walsh Series (STWS) method to solve the systems of Volterra integral equations of first kind (SVIEF). Because of the ill-posedness of the first kind problem, the SVIEF is converted into system of Volterra integral equations of second kind (SVIES). The STWS method for the SVIES with variable coefficient is presented. Numerical examples for linear and non-linear problems are experimented to show the efficiency and simplicity of the STWS method. The obtained numerical solutions (results) have been compared with the existing method.

In this paper, we provide a generalized block-by-block method for constructing block-by-block systems to solve the system of linear Volterra integral equations of the second kind, and then deduce some of the special cases. Compared with the expansion method and He’s homotopy perturbation method, respectively, numerical examples are given to certify the effectiveness of the method. The results show that the block-by-block method is very effective, simple, and of high accuracy in solving the system of linear Volterra integral equations of the second kind.

The possibilities of the algorithm for applying the boundary element method to solving boundary value problems are discussed on the example of the two-dimensional Poisson differential equation. The algorithm does not change significantly when the type of boundary conditions changes: the Dirichlet problem, the Neumann problem, or a mixed boundary value problem. The idea of the algorithm is taken from the work of John T. Katsikadelis [1]. The algorithm is described in detail in the next sequence of actions. 1) The boundary- value problem for a two-dimensional finite domain is formulated. The desired function in the domain, its values, and its normal derivative on the boundary contour are connected by means of the second Green formula. 2) We pass from the boundary value problem for the Poisson equation to the boundary value problem for the Laplace equation. This simplifies the process of constructing an integral equation. We obtain the integral equation on the boundary contour using the boundary conditions. 3) In the integral equation, we divide the boundary contour into a finite number of boundary elements. The desired function and its normal derivative are considered constant values on each boundary element. We compose a system of linear algebraic equations considering these values. 4) We modify the system of linear algebraic equations taking into account the boundary conditions. After that, we solve it using the Gauss method. The computer program has been developed according to the developed algorithm. We used it in the learning process. The software implementation of the algorithm takes into account the capabilities of modern computer technology and modern needs of the educational process. The work of the program is shown in the test case. Further modification of the described algorithm is possible

In this paper, the existence and uniqueness of solution of the twodimensional Fredholm integral equation with time, with singular kernel, arediscussed and proved. The Fredholm integral term (FIT) is considered inposition while the Volterra integral term (VIT) is considered in time. Usinga numerical technique we have a system of Fredholm integral equations(SFIEs) in two dimensions . This system of integral equations can be reducedto a linear algebraic system (LAS) of equations by using two differentmethods. These methods are: Toeplitz matrix method and Product Nystrommethod. A numerical examples are considered when the kernel takes thefollowing forms: Carleman function, logarithmic form, Cauchy kernel.

The main aim of this paper is to use the operational matrices of fractional integration of Haar wavelets to find the numerical solution for a nonlinear system of two-dimensional fractional partial Volterra integral equations. To do this, first we present the operational matrices of fractional integration of Haar wavelets. Then we apply these matrices to solve systems of two-dimensional fractional partial Volterra integral equations (2DFPVIE). Also, we present the error analysis and convergence as well. At the end, some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.

In the present article a diffraction problem on an ideally conducting parallelepiped the upper face of which is coated with a ferrite layer is solved by means of the method of integral equations. A general approach to the compilation of a system of linear integral vector equations is formulated, and the process of reducing the latter system to a system of linear algebraic equations for a family of problems of the given type is presented.