Search

This paper proposes an approach for the space-fractional diffusion equation in one dimension. Since fractional differential operators are non-local, two main difficulties arise after discretization and solving using Gaussian elimination: how to handle the memory requirement of O(N2) for storing the dense or even full matrices that arise from application of numerical methods and how to manage the significant computational work count of O(N3) per time step, where N is the number of spatial grid points. In this paper, a fast iterative finite difference method is developed, which has a memory requirement of O(N) and a computational cost of O(N logN) per iteration. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.

The paper considers the Gaussian method and its application in solving the main geodetic problems on the Earth's ellipsoid. There are a lot of methods for solving the main geodetic tasks on the ellipsoid. They can be grouped into three categories, ie methods: small distances (up to 200 km), medium range (from 200 to 1000 km) and long distances (over 1.000 km). The aim of this paper is to analyze the application of the Gaussian method, ie comparison of obtained values for length Gaussian model for small and medium distances, with results that were obtained Bessel's universal method. From the results obtained and the values for length, can perform certain analyzes and conclusions to which values can be practically applied Gaussian method. Specifically, the results show that the Gaussian solutions suitable for medium distances (up to 400 km), except that for short distances, for example to 150 km, to be ignored members of the fourth order.

In this paper, an approach based on matrix polynomials is introduced for solving linear systems of partial dierential equations. The main feature of the proposed method is the computation of the Smith canonical form of the assigned matrix polynomial to the linear system of PDEs, which leads to a reduced system. It will be shown that the reduced one is an independent system of PDEs having only one unknown in each equation. A comparison of the results for several test problems reveals that the method is very eective and convenient. The basic idea described in this paper can be employed to solve other linear functional systems.

The paper deals with the issues of electronic rangefinders correction. In the course of our study an algorithm of research data processing of constant correction of electronic rangefinders has been developed. Its essence is an application of Gaussian method to find the minimum of some function for determining the errors of measured distances. Firstly, the number of required measurements for a device was investigated for consideration of correlation influence between instrumental and random errors. Next, a system of equations is compiled based on relations between the measured target length and its segments in all combinations, taking into account the constant correction of the rangefinder. Thus, conditional equations are obtained with one unknown value – the constant correction of the rangefinder. Due to the large number of measurements, the solution of such system of equations is to find the minimum of some function for determining the errors of the measured distances. In this case, the most appropriate choice is the criterion of minimizing the maximum deviation to reject gross errors of the measured values. Thus, the most probable value of the constant correction of the rangefinder and lengths of the measured target and its segments for all possible combinations has been obtained. The research was carried out on a geodetic landfill under optimal meteorological conditions. The research has also revealed that the proposed technological solutions allow increasing constant correction balancing accuracy of the rangefinder, measured line and its segments lengths in comparison with the known method of correlate by at least 10 %. On the example of the SOUTH NTS-350 rangefinder, the accuracy of the obtained approximate values of line lengths and its segments was defined to be almost three times higher than line lengths declared in the technical specifications of the root mean square error measurement for the same device.

In the paper we study how to integrate numerically large-scale systems of semi-explicit index 1 differential-algebraic equations by implicit Runge-Kutta methods. In this case, if Newton-type iterations are applied to the discrete problems we need to solve high dimension linear systems with sparse coefficient matrices. Therefore we develop an effective way for packing such matrices of coefficients and derive special Gaussian elimination for parallel factorization of nonzero blocks of the matrices. As a result, we produce a new efficient procedure to solve linear systems arising from application of Newton iterations to the discretizations of large-scale index 1 differential-algebraic equations obtained by implicit Runge-Kutta methods. Numerical tests support theoretical results of the paper.

The quality of measuring systems of the microwave range, including vector network analyzers, largely depends on the reliability of calibration procedures and direct measurements, which allow to take into account information about the reproducible errors of measuring systems for subsequent correction. The aim of the paper is mathematical modeling of the errors of the measuring system for the generalized case for 2-n pole device.The problems of increasing the accuracy of measuring microwave systems due to compensation of systematic errors determined during calibration are considered. Calibration of measuring systems and correction of the results of direct measurements based on calibration results require the use of appropriate mathematical models of errors. Mathematical models of errors are represented in the form of multipolar errors, included between the object of measurement and the measuring system, which is assumed to be ideal, free of errors. The article proposes a generalized mathematical model of errors, described by a network of errors containing n ports connected to the n-port measuring system, and n ports connected to the n port of the measurement object. To obtain in general form the calibration equation for the 2n-port model of the error multipolar network, its transmission wave matrix [T], recorded in a cellular form, was used, and then a relationship was found between the measurement result in a matrix form with the cellular wave matrix T. A solution for finding the error matrix of matrix equation that connects the matrices known from the results of the corresponding attestation for the standards with the results of measurements during calibration in the matrix form. When solving this equation, a matrix product of «sandwich» type appears due to the cellular wave matrix [T]. The solution is possible when using the Kronecker product of two matrix, the matrix translation operator, the RS operator of the matrix, and the Gaussian elimination method. An equation is obtained for reconstructing the actual values of the scattering matrix of the measurement object, starting from the results of direct measurements in the matrix form and the error matrix. When solving the reconstruction equation, it is advisable to use a matrix inverse to the transmission matrix [T].The developed generalized mathematical model can be used, for example, when it is necessary to measure the parameters of complex microwave devices made on boards (wafer), with probe transitions to measuring ports, where it is important to consider the presence of additional microwave power leaks between ports.

In this paper, we consider a type of fractional diffusion equation (FDE) with variable coefficients on a finite domain. Firstly, we utilize a second-order scheme to approximate the RiemannLiouville fractional derivative and present the finite difference scheme. Specifically, we discuss the CrankNicolson scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the scheme and conclude that the scheme is unconditionally stable and convergent with the second-order accuracy of O(2+h2). Furthermore, we develop a fast accurate iterative method for the CrankNicolson scheme, which only requires storage of O(m) and computational cost of O(mlogm) while retaining the same accuracy and approximation property as Gauss elimination, where m=1/h is the partition number in space direction. Finally, several numerical examples are given to show the effectiveness of the numerical method, and the results are in excellent agreement with the theoretical analysis.

In this paper, schemes for constructing solutions to boundary value problems for static calculation of flexible circular plates with the nonlinear theory of Lyava and Volmyr are presented. From the equations of the equilibrium system of the plates, given in curvilinear coordinates, the system of equilibrium equations for flexible round plates is obtained. Substituting the expressions for the efforts and shearing forces and introducing dimensionless quantities, we obtain a system of quasilinear quantities in displacements. To develop an automated system for static calculation of flexible round plates, we use central finite-difference schemes that approximate derivatives with second-order accuracy, we obtain a system of quasilinear algebraic equations. To test the constructed automatic system for static calculation, the difference equations are reduced to vector form. An implicit iterative process combined with the Gaussian elimination method is applied to the solution of the system of equations. When calculating iterative processes, it continues until the above conditions are met. After determining the required functions by the finite difference method, we calculate the calculated values. Using the obtained numerical results, we will construct their graphs.

Fractional advection-dispersion equations, as generalizations of classical integer-order advection-dispersion equations, are used to model the transport of passive tracers carried by fluid flow in a porous medium. In this paper, we develop an implicit finite difference method for fractional advection-dispersion equations with fractional derivative boundary conditions. First-order consistency, solvability, unconditional stability, and first-order convergence of the method are proven. Then, we present a fast iterative method for the implicit finite difference scheme, which only requires storage of O(K) and computational cost of O(Klog⁡K). Traditionally, the Gaussian elimination method requires storage of O(K2) and computational cost of O(K3). Finally, the accuracy and efficiency of the method are checked with a numerical example.

In this paper, we report new three level implicit super stable methods of order two in time and four in space for the solution of hyperbolic damped wave equations in one, two and three space dimensions subject to given appropriate initial and Dirichlet boundary conditions. We use uniform grid points both in time and space directions. Our methods behave like fourth order accurate, when grid size in time-direction is directly proportional to the square of grid size in space-direction. The proposed methods are super stable. The resulting system of algebraic equations is solved by the Gauss elimination method. We discuss new alternating direction implicit (ADI) methods for two and three dimensional problems. Numerical results and the graphical representation of numerical solution are presented to illustrate the accuracy of the proposed methods.

Linear matrix equations are encountered in many systems and control applications. In this paper, we consider the general coupled matrix equations (including the generalized coupled Sylvester matrix equations as a special case) #### over the generalized reflexive matrix group (Y1, Y2, ?, Yi). We derive an efficient gradient-iterative (GI) algorithm for finding the generalized reflexive solution group of the general coupled matrix equations. Convergence analysis indicates that the algorithm always converges to the generalized reflexive solution group for any initial generalized reflexive matrix group (Y1(1), Y2(1), ?, Yl(1)). Finally, numerical results are presented to test and illustrate the performance of the algorithm in terms of convergence, accuracy as well as the efficiency.

The boundary-volume integral equation numerical technique can be a powerful tool for piecewise heterogeneous media, but it is limited to small problems or low frequencies because of great computational cost. Therefore, a restarted GMRES method is applied to solve large-scale boundary-volume scattering problems in this paper to overcome the computational barrier. The iterative method is firstly applied to responses of dimensionless frequency to a semicircular alluvial valley filled with sediments, compared with the standard Gaussian elimination method. Then the method is tested by a heterogeneous multilayered model to show its applicability. Numerical experiments indicate that the preconditioned GMRES method can significantly improve computational efficiency especially for large Earth models and high frequencies, but with a faster convergence for the left diagonal preconditioning.

This paper addresses the problem of shape preserving spline interpolation formulated as a differential multipoint boundary value problem (DMBVP for short). Its discretization by mesh method yields a five-diagonal linear system which can be ill-conditioned for unequally spaced data. Using the superposition principle we split this system in a set of tridiagonal linear systems with a diagonal dominance. The latter ones can be stably solved either by direct (Gaussian elimination) or iterative methods (SOR method and finite-difference schemes in fractional steps) and admit effective parallelization. Numerical examples illustrate the main features of this approach.

In the present paper, we propose a hierarchical identification method (SSHI) for solving Lyapunov matrix equations, which is based on the symmetry and skew-symmetry splitting of the coefficient matrix. We prove that the iterative algorithm consistently converges to the true solution for any initial values with some conditions, and illustrate that the rate of convergence of the iterative solution can be enhanced by choosing the convergence factors appropriately. Furthermore, we show that the method adopted can be easily extended to study iterative solutions of other matrix equations, such as Sylvester matrix equations. Finally, we test the algorithms and show their effectiveness using numerical examples.

In the present paper, the nonlocal boundary value problem { i d u d t + A u = f ( t ) , 0 t T , u ( 0 ) = ∑ m = 1 p α m u ( λ m ) + φ , 0 λ 1 λ 2 ⋯ λ p ≤ T for the Schrodinger equation in a Hilbert space H with the self-adjoint operator A is considered. Stability estimates for the solution of this problem are established. Two nonlocal boundary value problems are investigated. The first and second order of accuracy difference schemes for the approximate solutions of this nonlocal boundary value problem are presented. The stability of these difference schemes is established. In practice, stability inequalities for the solutions of difference schemes for the Schrodinger equation are obtained. A numerical method is proposed for solving a one-dimensional Schrodinger equation with nonlocal boundary condition. A procedure involving the modified Gauss elimination method is used for solving these difference schemes. The method is illustrated by giving numerical examples.

Summary. In this paper I present the Kronecker-Capelli theorem which states that a system of linear equations has a solution if and only if the rank of its coecient matrix is equal to the rank of its augmented matrix.

In this paper, we consider the accuracy and stability of implementing the method of fundamental solutions. In contrast to the results shown in [5], we find that Gaussian elimination can be used reliably to solve the MFS equations and the use of the singular value decomposition shows no improvement over Gaussian elimination provided that the boundary condition is non-noisy. However, for noisy boundary conditions, there is evidence that the singular value decomposition with truncation is more accurate than Gaussian elimination.

This paper is concerned with the rank of the matrix radius of the limiting set for a singular Hamiltonian system with one singular end point. The exact relationship between the rank of the matrix radius and the number of square integrable solutions is obtained and then the defect index of the corresponding minimal operator can be represented in terms of the rank of the matrix radius. So two results obtained by Allan M. Krall [SIAM J. Math. Anal. 20 (1989) 664] are improved. In addition, it is discussed that the rank of the matrix radius is independent of the spectral parameter and a certain matrix. Especially, the classification of singular linear Hamiltonian systems is present and several sufficient and necessary conditions for the limit point and limit circle cases are established.

In this paper, based on matrix structure analysis, we derive and analyze efficient algorithms to solve nonlinear matrix equations of the form X +∑ 1⩽ i ⩽ d A i X −1 D i = C . This class of equations is encountered in the solution of Tree-Like stochastic processes which are a generalization of Quasi-Birth-and-Death (QBD) processes.

The analytical approach of unsteady conduction heat transfer under actual conditions represent a very difficult (if not insurmountable) problem due to the issues related to finding analytical solutions for the conduction heat transfer equation. Various techniques have been developed in order to overcome these difficulties, among which the alternate directions method and the decomposition method. Both of them are particularly suited for two-dimension heat propagation. The paper deals with both techniques in order to verify whether the results provided are in good accordance. The studied case consists of a long hollow cylinder, and considers that the time-dependent temperature field varies both in the radial and the axial directions. The implicit technique is used in both methods and involves the simultaneous solving of a set of equations for all of the nodes for each time step successively for each of the two directions. Gauss elimination is used to obtain the solution of the set, representing the nodal temperatures. After using the two techniques the results show a very good agreement, and since the decomposition is easier to use in terms of computer code and running time, this technique seems to be more recommendable.

This paper proposes four geometric invariants to be used to eliminate joint variables in closure equations for a 6R manipulator instead of the Gaussian elimination used in other studies. The geometric invariants are determined by the structure of any three consecutive joints in space. They have specific geometric meanings such as angle, length, area, volume. For a general 6R manipulator, its four basic closure equations containing only three angular variables can be directly constructed in a few minutes from the geometric invariants using Maple on a general personal computer. These resulting equations have no extraneous roots and are algebraically independent. Because the basic closure equations are obtained from the geometric invariants, they have the most simple forms and provide a very good chance to solve the input-output equation for the inverse kinematics problem. As a result, we use the set of basic closure equations to derive the symbolic 16th degree input-output equation and compute the input-output equation for a special case with 16 different real solutions. All position and orientation coordinates of the end-effector may be symbolic parameters. The definition of the geometric invariants is independent of joint types and can be applied to manipulators with any serial geometry.

The large sparse linear systems arising from the finite element or finite difference discretization of elliptic PDEs can be solved directly via, e.g., nested dissection or multifrontal methods. Such techniques reorder the nodes in the grid to reduce the asymptotic complexity of Gaussian elimination from O(N 2) to O(N 1.5) for typical problems in two dimensions. It has recently been demonstrated that the complexity can be further reduced to O(N) by exploiting structure in the dense matrices that arise in such computations (using, e.g., ? $\mathcal {H}$ -matrix arithmetic). This paper demonstrates that such accelerated nested dissection techniques become particularly effective for boundary value problems without body loads when the solution is sought for several different sets of boundary data, and the solution is required only near the boundary (as happens, e.g., in the computational modeling of scattering problems, or in engineering design of linearly elastic solids). In this case, a modified version of the accelerated nested dissection scheme can execute any solve beyond the first in O(N boundary) operations, where N boundary denotes the number of points on the boundary. Typically, N boundary ~ N 0.5. Numerical examples demonstrate the effectiveness of the procedure for a broad range of elliptic PDEs that includes both the Laplace and Helmholtz equations.

Standard approaches to the modeling of large flexible structures, such as the Finite Element Method, are essentially based on small-scale approximation: the structure is thought of as consisting of many pieces that are small enough to have their dynamics approximated by a low-order model. This modeling philosophy is suitable for complicated and inhomogeneous structures, but becomes awkward when one has to do with structures that are built up from highly flexible but homogeneous parts. In the latter case, it would be much more natural to use partial differential equations. In this paper, we discuss some possible analytic approaches to the modeling of flexible structures, with an emphasis on the computation of the natural frequencies.

In the paper we further develop the idea of parallel factorization of nonzero blocks of sparse coefficient matrices of the linear systems arising from discretization of large-scale index 1 differential-algebraic problems by Runge-Kutta methods and their following solving by Newton-type iterations. We formulate a number of theorems that give estimates for the local fill-in of such matrices on some stages of Gaussian elimination. As the result, we derive that only the suggested modification of Gauss method appeared to be effiective and economical one from the standpoint of CPU time and RAM.

This paper presents a package of FOTRAN IV subroutines for the out-of-core assembly and solution of a system of linear equations with a non-symmetric coefficient matrix, as met in many finite element applications. A skyline block storage mode is used for the coefficient matrix and a direct triangular decomposition method is employed to solve the system of linear equations. The block partitioning and storage mode of the coefficient matrix and the data transfer procedures for the assembly and solution of the system Ax = b are discussed in detail. Examples from the field of fluid dynamics are presented.

The quasilinear theory, the lowest order nonlinear theory of plasma turbulence is nowadays one of the most consolidated and useful theoretical tools for the investigation of several important problems in the domain of controlled nuclear fusion research. Perhaps, the best merit of quasilinear theory is that of reducing the complicated phenomenon of the wave-particle interaction within the familiar framework of advection-diffusion processes for which a huge amount of wisdom can be straightforwardly imported from other areas of theoretical physics. A recent application where quasilinear theory has been successfully applied concerns the study of lower hybrid current drive, i.e., the description of the mechanisms by which a steady current can be maintained in thermonuclear plasmas by injecting electromagnetic waves from the exterior with no assist of any applied ohmic voltage or even in opposition to it. In this paper we describe the main features associated with the implementation of the quasilinear model of lower hybrid current drive into a numberical code. Our study is based on ADLER, a finite element code designed to solve the quasilinear equations in two dimensions in velocity and wave number space. Special attention is paid to the algebraic solver, which was found to be the computationally most intensive section of the code. In particular we have compared the direct Gauss elimination technique with a series of vectorized iterative solvers developed at ECSEC. This comparison shows that the iterative solvers do perform successfully and, apart from the obvious savings of storage, they also prove advantageous from the viewpoint of CPU costs especially when the grid resolution is augmented. Finally, some new application of ADLER in the domain of unsteady lower hybrid current drive are presented. A good qualitative agreement with the experimental result is found, showing once more that quasilinear theory is an appropriate tool for the description of the current drive physics.

In this paper we find exact solutions for linear ordinary differential equations of any order when they are given in matrix form, as well as for classes of Riccati matrix equations with two or three arbitrary matrix coefficients. Other nonlinear systems of triangular form are also solved completely.

where G is a rectangle, Au = 82u/8x2 + 82u/8y2, and v, w are known functions. For computational purposes, this partial differential equation is frequently replaced by a finite difference analogue. These discrete models for (1) consist of linear systems of equations of very large dimension, and it is widely recognized that the usual direct methods (e.g., Gaussian elimination) are unsatisfactory for such systems [18, ?? 21.2-21.3]. Theoretical investigation has, therefore, been primarily directed toward the development of effective iterative methods for the solution of these problems [64], [66]. In recent years, however, direct methods that take advantage of the special block structure of these linear equations have appeared. For the rectangular regions under consideration, these methods can be considerably faster than iterative methods. The purpose of this survey paper is to provide brief summaries and a list of references for methods which can be used to directly solve the finite difference equations. Some of these methods can be applied to problems in more general domains. However, the extensions generally include only simple rectilinear regions, such as L-shaped or T-shaped domains. This is basically due to the fact that the direct methods require a great degree of regularity in the block structure of the matrix equation. In our discussion, we will indicate whether the methods are easily adaptable to more general regions, and to more general elliptic partial differential equations.

As a continuation of my paper “New techniques for the analysis of linear interval equations” [ Linear Algebra Appl. 58:273–325 (1984)], the interrelation between interval Gauss elimination and interval iteration is investigated. Main results are a new existence theorem for interval Gauss elimination (in the guise of a perturbation theorem), a convergence and comparison theorem for a general family of interval iteration schemes, and a new method for the calculation of the hull of the solution set of linear interval equations with inverse positive coefficient matrix.

A computational algorithm for solving a system of coupled matrix algebraic equations is presented in this paper. Such equations arise in optimal control problems of linear systems with jump parameters.