Alternative correction equations in the Jacobi‐Davidson method

The correction equation in the Jacobi‐Davidson method is effective in a subspace orthogonal to the current eigenvector approximation, whereas for the continuation of the process only vectors orthogonal to the search subspace are of importance. Such a vector is obtained by orthogonalizing the (approximate) solution of the correction equation against the search subspace. As an alternative, a variant of the correction equation can be formulated that is restricted to the subspace orthogonal to the current search subspace. In this paper, we discuss the effectiveness of this variant. Our investigation is also motivated by the fact that the restricted correction equation can be used for avoiding stagnation in the case of defective eigenvalues. Moreover, this equation plays a key role in the inexact TRQ method [18]. Copyright © 1999 John Wiley & Sons, Ltd.     

A vector potential KdV equation and vector Ito equation: soliton solutions, bilinear Bäcklund transformations and Lax pairs

A vector potential KdV equation and vector Ito equation are proposed based on their bilinear forms. Soliton solutions expressed by Pfaffians are obtained. Bilinear Bäcklund transformations and the corresponding Lax pairs for the vector potential KdV equation and the vector Ito equation are derived.     

Grassmann algorithms for low rank approximation of matrices with missing values

The problem of approximating a matrix by another matrix of lower rank, when a modest portion of its elements are missing, is considered. The solution is obtained using Newton’s algorithm to find a zero of a vector field on a product manifold. As a preliminary the algorithm is formulated for the well-known case with no missing elements where also a rederivation of the correction equation in a block Jacobi-Davidson method is included. Numerical examples show that the Newton algorithm grows more efficient than an alternating least squares procedure as the amount of missing values increases.     

Fokker-planck equation for brownian motion of rotating spherical particles : PMM vol. 40, n 6, 1976, pp. 1127–1131

The Langevin equation to derive the Fokker-Planck equation is used for the Brownian motion of particles in translational motion. The Fokker-Planck equation for the Brownian motion of particles which have, in addition to the translational velocity also an angular velocity, has not, so far, been derived. This can apparently be explained by the fact that in the case of the rotational motion, the Langevin equation for the translational motion velocity vector must be supplemented by a corresponding equation for an angular velocity vector. The latter equation must contain, in addition to the systematic moment of reaction linearly dependent on the angular velocity of rotation itself, a random moment rapidly varying with time. Moreover, to ensure the compatibility of two differential vector equations within the system, additional relations which must be introduced, must connect not only the coefficients of the systematic reactions, but also the. random vectors varying rapidly with time.In [1],the Boltzmann's equation for a mixture of two gases was used to derive a Fokker-Planck equation for a translational motion of Brownian particles. The same method can be applied to the Brownian motion of spherical particles which have, in addition to the translational velocities, angular velocities of self-rotations. In this case there is no need to introduce additional relations connecting the random rapidly varying vectors.In the present paper we derive the Fokker-Planck equations for a new model of rotating spherical molecules which was used in [2].     

On the energy flux vector for bending vibrations of a plate : PMM vol. 40, n 6, 1976, pp. 1131–1135

An expression for the energy flux vector of plate bending vibrations is obtained in invariant form. The derivation of expressions for the transverse force, bending and twisting moments in an arbitrary orthogonal coordinate system and the derivation of an orthogonality type condition for normal waves being propagated in a thin elastic strip with free edges are considered as applications.In a number of cases it turns out to be useful to consider the energy flux vector in analyzing the vibrations in systems with distributed parameters. The expressions for the Umov-Poynting vector in electrodynamics and for the energy flux vector in acoustics are well-known. An analogous vector for the bending Vibrations of a plate was mentioned only in [1 – 3], This vector is used in [1] to prove a uniqueness theorem for a two-component acoustic model consisting of an ideal compressible fluid and elastic plates in contact with it. However, the expression for the energy flux in [1] (it was later cited in [2, 3] with a reference to [1]) is erroneous. An exact expression (within the framework of the applicability of the Kirchhoff equation) is found below for the energy flux vector of the bending vibrations of a plate and some applications of the formulas obtained are mentioned.     

Isothermic Surfaces in Spaces of Arbitrary Dimension: Integrability, Discretization, and Bäcklund Transformations—A Discrete Calapso Equation

A vector analog of the classical Calapso equation governing isothermic surfaces in R ^{n +2} is introduced. It is shown that this vector Calapso system admits a nonlocal) scalar Lax pair based on the classical Moutard equation. The analog of Darboux's Bäcklund transformation for isothermic surfaces in R³ is derived in a systematic manner and shown that it may be formulated in terms of the classical Moutard transformation acting on the scalar Lax pair. A permutability theorem for isothermic surfaces is set down that manifests itself in an explicit superposition principle for the vector Calapso system. This superposition principle in vectorial form is shown to constitute an integrable discretization of the vector Calapso system and, therefore, defines discrete isothermic surfaces in R ^{n +2}. The discrete Calapso equation is related to the discrete Korteweg–de Vries equation and discrete holomorphic functions. A matrix Lax pair based on Clifford algebras and a scalar Lax pair are derived for the discrete Calapso equation. A discrete Moutard-type transformation for the discrete Calapso equation is obtained, and it is shown that the discrete Calapso equation may be specialized to an integrable discrete version of the O( n +2) nonlinear σ-model.     

High Order Energy-Preserving Method of the "Good" Boussinesq Equation

The fourth order average vector field (AVF) method is applied to solve the "Good" Boussinesq equation. The semi-discrete system of the "good" Boussinesq equation obtained by the pseudo-spectral method in spatial variable, which is a classical finite dimensional Hamiltonian system, is discretized by the fourth order average vector field method. Thus, a new high order energy conservation scheme of the "good" Boussinesq equation is obtained. Numerical experiments confirm that the new high order scheme can preserve the discrete energy of the "good" Boussinesq equation exactly and simulate evolution of different solitary waves well.     

Tikhonov regularization for infeasible absolute value equations

We investigate the optimum correction of an absolute value equation by minimally changing the coefficient matrix and right-hand side vector using Tikhonov regularization. Solving this problem is equivalent to minimizing the sum of fractional quadratic and quadratic functions. The primary difficulty with this problem is its nonconvexity. Nonetheless, we show that a global optimal solution to this problem can be found by solving an equation on a closed interval using the subgradient method. Some examples are provided to illustrate the efficiency and validity of the proposed method.     

Convergence analysis of the Newton–Shamanskii method for a nonsymmetric algebraic Riccati equation

In this paper, we consider the nonsymmetric algebraic Riccati equation arising in transport theory. An important feature of this equation is that its minimal positive solution can be obtained via computing the minimal positive solution of a vector equation. We apply the Newton–Shamanskii method to solve the vector equation. Convergence analysis shows that the sequence of vectors generated by the Newton–Shamanskii method is monotonically increasing and converges to the minimal positive solution of the vector equation. Numerical experiments show that the Newton–Shamanskii method is feasible and effective, and outperforms the Newton method. Copyright © 2008 John Wiley & Sons, Ltd.     

A modified Newton method for solving non-symmetric algebraic Riccati equations arising in transport theory

Liang Bao ^{The non-symmetric algebraic Riccati equation arising in transporttheory can be rewritten as a vector equation and the minimalpositive solution of the non-symmetric algebraic Riccati equationcan be obtained by solving the vector equation. In this paper,we apply the modified Newton method to solve the vector equation.Some convergence results are presented. Numerical tests showthat the modified Newton method is feasible and effective, andoutperforms the Newton method.}     

Diffeomorphism,periodic KdV and C. Neumann system

After reviewing few known facts regarding KdV equation as an Euler equation in the coadjoint orbit of the Virasoro–Bott group we are able to establish an interesting connection between the projective vector field, a vector field leaving fixed a given projective connection, and the C. Neumann system using the idea of Knörrer and Moser. We show that certain quadratic function of a projective field satisfies C. Neumann system.     

对弦振动方程与薄膜振动方程的探讨

为了提高振弦式产品和振膜式产品的技术指标,基于微小横振动中张力不相等的概念,把张力角度增量与角度之比作为变量,提出了对弦振动和薄膜振动方程进行修正的方法,修正方程对产品研发有指导意义.     

Three dimensional elastodynamics of 2D quasicrystals: The derivation of the time-dependent fundamental solution

The time-dependent differential equations of elasticity for 2D quasicrystals with general structure of anisotropy (dodecagonal, octagonal, decagonal, pentagonal, hexagonal, triclinic) are considered in the paper. These equations are written in the form of a vector partial differential equation of the second order with symmetric matrix coefficients. The fundamental solution (matrix) is defined for this vector partial differential equation. A new method of the numerical computation of values of the fundamental solution is suggested. This method consists of the following: the Fourier transform with respect to space variables is applied to vector equation for the fundamental solution. The obtained vector ordinary differential equation has matrix coefficients depending on Fourier parameters. Using the matrix computations a solution of the vector ordinary differential equation is numerically computed. Finally, applying the inverse Fourier transform numerically we find the values of the fundamental solution. Computational examples confirm the robustness of the suggested method for 2D quasicrystals with arbitrary type of anisotropy.     

The King-Werner method for solving nonsymmetric algebraic Riccati equation

For the nonsymmetric algebraic Riccati equation arising from transport theory, we concern about solving its minimal positive solution. In [1], Lu transferred the equation into a vector form and pointed out that the minimal positive solution of the matrix equation could be obtained via computing that of the vector equation. In this paper, we use the King-Werner method to solve the minimal positive solution of the vector equation and give the convergence and error analysis of the method. Numerical tests show that the King-Werner method is feasible to determine the minimal positive solution of the vector equation.     

On Lie symmetry analysis,conservation laws and solitary waves to a longitudinal wave motion equation

Under investigation in this work is a longitudinal wave motion equation, which describes the solitary waves propagation with dispersion caused by transverse Poisson’s effect in a magneto-electro-elastic circular rod. The Lie symmetry method is employed to study its vector fields and optimal systems, respectively. Furthermore, the symmetry reductions and eight families of soliton wave solutions of the equation are obtained on the basis of the optimal systems, including hyperbolic-type and trigonometric-type solutions. Two of reduced equations are Painlevé-like equations. Finally, by virtue of conservation law multiplier, the complete set of local conservation laws of the equation for the arbitrary constant coefficients is well constructed with a detailed derivation.     

Asymptotic theory of the global error and some techniques of error estimation

Summary The error of the approximate solution obtained by discretising a functional equation can be shown under certain conditions to possess an asymptotic expansion in terms of some parameter which is usually a representative step-length. We consider the case of two-parameter expansions, which is particularly relevant to parabolic equations. We derive results for the existence of the expansion and for the application of the classical difference correction and of defect correction. The theory is illustrated by the discussion of a simple parabolic problem     

Algebraic classification of physical structures with zero. II. Topological aspects

Some topological analogs are obtained of the author’s previous results on the classification of physical structures. A topological-algebraic axiomatics is considered enabling us to replace the algebraic axiom corresponding to the main equation of a physical structure by a more natural axiom. A physical structure of rank different from (2, 2) is shown to be a pair of vector spaces with a nondegenerate bilinear form over a topological skew field. The obtained results are applied to the classification of physical structures of rank different from (2, 2). The structures of rank (2, 2) are also considered. To describe them, a topological group structure corresponding to the biform action is introduced on the set of measurements.     

On an integrable case of perturbed keplerian motion

A general solution of a differential vector equation of perturbed Keplerian motion is derived for the case when the position vector and perturbing acceleration vector are collinear. A variable change is employed, in which the new independent variable is expressed in terms of the initial values of the phase variables and time, using the elliptical Jacobi function. The two-point boundary value problem for the initial equation is reduced to the Cauchy problem, A parametric representation is obtained for the regularized trajectory of motion of a material point under the action of a central force.     

Acoustic scattering by a thin cylindrical screen with Dirichlet and impedance boundary conditions on opposite sides of the screen

A problem on scattering acoustic waves by a thin cylindrical screen is studied. In doing so, the Dirichlet condition is specified on one side of the screen, while the impedance boundary condition is specified on the other side of the screen. The solution of the problem is subject to the radiating condition at infinity and to the propagative Helmholtz equation. By using the potential theory the scattering problem is reduced to a system of singular integral equations with additional conditions. By regularization and subsequent transformations, this system is reduced to a vector Fredholm equation of the second kind and index zero. It is proved that the obtained vector Fredholm equation is uniquely solvable. Therefore the integral representation for a solution of the original scattering problem is obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)