Showing papers on "Matrix difference equation published in 2014"
01 May 2014
TL;DR: In this paper, the periodicity, the boundedness and the global stability of the positive solutions of the following nonlinear difference equation were studied, and it was shown that the positive solution of the difference equation has bounded periodicity and boundedness.
Abstract: In this article, we study the periodicity, the boundedness and the global stab ility of the positive solutions of the following nonlinear difference equation
80 citations
01 Aug 2014
TL;DR: Overall, this article is based on diagonalizing symmetric matrices, replacing sums by integrals, and computing Fourier coefficient approximations with a variety of boundary conditions.
Abstract: When the heat equation and wave equation are approximated by ut = −Ku and utt = −Ku (discrete in space), the solution operators involve e −Kt , √ K ,c os( √ Kt), and sinc( √ Kt). We compute these four matrices and find accurate approximations with a va- riety of boundary conditions. The second difference matrix K is Toeplitz (shift-invariant) for Dirichlet boundary conditions, but we show why eKt also has a Hankel (anti-shift- invariant) part. Any symmetric choice of the four corner entries of K leads to Toeplitz plus Hankel in all functions f (K). Overall, this article is based on diagonalizing symmetric matrices, replacing sums by integrals, and computing Fourier coefficients.
36 citations
TL;DR: A class of different form difference scheme for solving the Riesz space fractional reaction dispersion equation based on the parameter spline function and improved matrix transform method is developed.
22 citations
TL;DR: The Chen–Lee–Liu (C–L–L) equation is investigated by the Darboux transformation (DT) method and a specific construction of the N-fold DT for C–L-L equation is derived in a simple way.
22 citations
TL;DR: In this article, the periodicity, the boundedness and the global stability of the positive solutions of the following nonlinear difference equation were studied, and it was shown that the positive solution of the difference equation has bounded periodicity and boundedness.
Abstract: In this article, we study the periodicity, the boundedness and the global stab ility of the positive solutions of the following nonlinear difference equation
20 citations
TL;DR: In this paper, a reduction technique for the generalized Riccati difference equation arising in optimal control and optimal filtering is proposed, which is based on a study on the generalized discrete algebraic RICCati equation.
Abstract: This paper proposes a reduction technique for the generalized Riccati difference equation arising in optimal control and optimal filtering. This technique relies on a study on the generalized discrete algebraic Riccati equation. In particular, an analysis on the eigenstructure of the corresponding extended symplectic pencil enables to identify a subspace in which all the solutions of the generalized discrete algebraic Riccati equation are coincident. This subspace is the key to derive a decomposition technique for the generalized Riccati difference equation. This decomposition isolates a “nilpotent” part, which converges to a steady-state solution in a finite number of steps, from another part that can be computed by iterating a reduced-order generalized Riccati difference equation.
16 citations
TL;DR: In this article, an analytical representation for the solution of the neutron point kinetics equation, free of stiffness and assuming that reactivity is a continuous or sectionally continuous function of time, is presented.
14 citations
01 Jan 2014
14 citations
TL;DR: This paper shows a method to find a linear transformation that reduces the number of variables to represent a given incompletely specified index generation function and first generates the difference matrix, and then finds a minimal set of variables using a covering table.
Abstract: This paper shows a method to find a linear transformation that reduces the number of variables to represent a given incompletely specified index generation function. It first generates the difference matrix, and then finds a minimal set of variables using a covering table. Linear transformations are used to modify the covering table to produce a smaller solution. Reduction of the difference matrix is also considered. key words: minimal cover, linear transformation, functional decomposition, incompletely specified function, logic minimization
13 citations
TL;DR: In this paper, a conservative Crank-Nicolson finite difference scheme for the initial-boundary value problem of generalized Rosenau-KdV equation is proposed, which shows a discrete analogue of the main conservation law associated to the equation.
Abstract: A conservative Crank-Nicolson finite difference scheme for the initial-boundary
value problem of generalized Rosenau-KdV equation is proposed. The difference scheme
shows a discrete analogue of the main conservation law associated to the equation. On the other
hand the scheme is implicit and stable with second order convergence. Numerical experiments
verify the theoretical results.
12 citations
TL;DR: In this paper, a nonlocal boundary value problem for a third-order hyperbolic equation with variable coefficients is considered in the one and multidimensional cases, and a priori estimates for the nonlocal problem are obtained in the differential and difference formulations.
Abstract: A nonlocal boundary value problem for a third-order hyperbolic equation with variable coefficients is considered in the one- and multidimensional cases. A priori estimates for the nonlocal problem are obtained in the differential and difference formulations. The estimates imply the stability of the solution with respect to the initial data and the right-hand side on a layer and the convergence of the difference solution to the solution of the differential problem.
Posted Content•
TL;DR: In this article, the solvability of the Sylvester-like matrix equation through an auxiliary standard (or generalized) generalized sylvester matrix equation is discussed and the closed-form solutions can be found by using previous results.
Abstract: Many applications in applied mathematics and control theory give rise to the unique solution of a Sylvester-like matrix equation associated with an underlying structured matrix operator $f$. In this paper, we will discuss the solvability of the Sylvester-like matrix equation through an auxiliary standard (or generalized) Sylvester matrix equation. We also show that when this Sylvester-like matrix equation is uniquely solvable, the closed-form solutions can be found by using previous result. In addition, with the aid of the Kronecker product some useful results of the solvability of this matrix equation are provided.
11 Dec 2014
TL;DR: In this paper, the forbidden set and the global behavior of solutions of a fourth-order difference equation are discussed, and an explicit formula for the solutions of the solutions is introduced.
Abstract: We determine the forbidden set, introduce an explicit formula for the solutions, and discuss the global behavior of solutions of a fourth order difference equation.
Patent•
10 Dec 2014
TL;DR: In this paper, a sorting method based on an isomorphic sequence was proposed to solve the problem in detection of pulse groups which are practically submersed into a large quantity of pulse streams.
Abstract: The invention provides a pulse sorting method based on an isomorphic sequence, and aims to solve the problem in detection of pulse groups which are practically submersed into a large quantity of pulse streams. The method comprises the following steps S1, acquiring a pulse repetition time interval sequence through first-order backward difference of the arrival time sequence of pulse streams, and quantizing the value of each element of the pulse repetition time interval sequence; S2, sieving the repetition substrings of the pulse repetition time interval sequence by using a suffix array and a maximum public prefix; S3, deleting shorter sub-strings from sub-strings having inclusion relations, and combining and jointing sub-strings having overlapping relations; S4, constructing a pulse stream arrival time difference matrix specific to the remaining pulse streams; S5, extracting a positive real number sequence of each line of the difference matrix to construct a pile of arrays, and sequencing the arrays to obtain a plurality of subsets; S6, searching for the sum of the subsets and a maximum common subsequence, and determining the position of a target pulse; S7, performing harmonic wave verification and pulse loss verification.
TL;DR: In this article, the authors studied the five-wave classical scattering matrix for nonlinear and dispersive Hamiltonian equations with a nonlinearity of the type u∂u/∂x.
Abstract: We study the five-wave classical scattering matrix for nonlinear and dispersive Hamiltonian equations with a nonlinearity of the type u∂u/∂x. Our aim is to find the most general nontrivial form of the dispersion relation ω(k) for which the five-wave interaction scattering matrix is identically zero on the resonance manifold. As could be expected, the matrix in one dimension is zero for the Korteweg-de Vries equation, the Benjamin-Ono equation, and the intermediate long-wave equation. In two dimensions, we find a new equation that satisfies our requirement.
TL;DR: In this article, a method of finding asymptotic formulas of solutions of a second-order difference equation with coefficients and, where is a real unbounded sequence and and are some perturbations.
Abstract: In this paper we present a method of finding asymptotic formulas of solutions of a second-order difference equation with coefficients and , where is a real unbounded sequence and and are some perturbations.
TL;DR: In this paper, the nonlocal boundary value problem for a hyperbolic-elliptic equation in a Hilbert space is considered and the stability estimate for the solution of the given problem is obtained.
Abstract: The nonlocal boundary value problem for a hyperbolic–elliptic equation in a Hilbert space is considered. The stability estimate for the solution of the given problem is obtained. The first and second orders of difference schemes approximately solving this boundary value problem are presented. The stability estimates for the solution of these difference schemes are established. The theoretical statements for the solution of these difference schemes are supported by the results of numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.
TL;DR: In this paper, one general q-exponential operator identity is established by solving one simple q-difference equation and some multilinear generating functions for H n are discussed.
Abstract: A new method for solving the 1D Poisson equation is presented using the finite difference method. This method is based on the exact formulation of the inverse of the tridiagonal matrix associated with the Laplacian. This is the first time that the inverse of this remarkable matrix is determined directly and exactly. Thus, solving 1D Poisson equation becomes very accurate and extremely fast. This method is a very important tool for physics and engineering where the Poisson equation appears very often in the description of certain phenomena.
TL;DR: In this article, a new approach is presented for obtaining the solutions to Yakubovich-conjugate quaternion matrix equation based on the real representation of quaternions.
Abstract: A new approach is presented for obtaining the solutions to Yakubovich--conjugate quaternion matrix equation based on the real representation of a quaternion matrix. Compared to the existing results, there are no requirements on the coefficient matrix . The closed form solution is established and the equivalent form of solution is given for this Yakubovich--conjugate quaternion matrix equation. Moreover, the existence of solution to complex conjugate matrix equation is also characterized and the solution is derived in an explicit form by means of real representation of a complex matrix. Actually, Yakubovich-conjugate matrix equation over complex field is a special case of Yakubovich--conjugate quaternion matrix equation . Numerical example shows the effectiveness of the proposed results.
15 May 2014
TL;DR: In this paper, a generalization of the improved Zakharov equation for the "almost" 2-D water waves at the surface of deep water is presented, which is very suitable for analytic study as well as for numerical simulation.
Abstract: In the paper [1] authors applied canonical transformation to water wave equation not only to remove cubic nonlinear terms but to simplify drastically fourth order terms in Hamiltonian. After the transformation well-known but cumbersome Zakharov equation is drastically simplified and can be written in X-space in compact way. This new equation is very suitable for analytic study as well as for numerical simulation At the same time one of the important issues concerning this system is the question of its integrability. The first part of the work is devoted to numerical and analytical study of the integrability of the equation obtained in [1]. In the second part we present generalization of the improved Zakharov equation for the "almost" 2-D water waves at the surface of deep water. When considering waves slightly inhomogeneous in transverse direction, one can think in the spirit of Kadomtsev-Petviashvili equation for Korteveg-de-Vries equation taking into account weak transverse diffraction. Equation can be written instead of classical variables η(x,y,t) and ψ(x,y,t) in terms of canonical normal variable b(x,y,t). This equation is very suitable for robust numerical simulation. Due to specific structure of nonlinearity in the Hamiltonian the equation can be effectively solved on the computer. It was applied for simulation of sea surface waving including freak waves appearing.
TL;DR: In this paper, the problem of condensate origination for an auxiliary field in the Kardar-Parisi-Zhang equation and its matrix generalization was studied, and it was shown that the problem can be solved exactly in the large-N limit.
Abstract: We study the problem of the condensate (stochastic average) origination for an auxiliary field in the Kardar-Parisi-Zhang equation and its matrix generalization. We cannot reliably conclude that there is a condensate for the Kardar-Parisi-Zhang equation in the framework of the one-loop approximation improved by the renormalization group method. The matrix generalization of the Kardar-Parisi-Zhang equation permits a positive answer to the question of whether there is a nonzero condensate, and the problem can be solved exactly in the large-N limit.
TL;DR: In this paper, the Schur and doubling methods are generalized to the case of unilateral quadratic matrix equations, and the efficiency of the algorithms proposed to solve the UQM equation is demonstrated by way of examples.
Abstract: The Schur and doubling methods, which are usually used to solve the algebraic Riccati equation, are generalized to the case of unilateral quadratic matrix equations. The efficiency of the algorithms proposed to solve the unilateral quadratic matrix equation is demonstrated by way of examples. The algorithms are compared with well-known ones. It is shown that the solutions of the unilateral quadratic matrix equation can be used to update model parameters.
TL;DR: It is shown that the unique prime period-2 solution of the equation above is globally asymptotically stable.
Abstract: We study the following nonautonomous difference equation: , , where is a period-2 sequence and the initial values . We show that the unique prime period-2 solution of the equation above is globally asymptotically stable.
TL;DR: In this paper, the generalized discrete Karm-Yakubovich-transpose matrix equation XAX T B = CY is considered and the generalized Leverrier algorithm is proposed.
Abstract: This paper is concerned with the complete parametric solutions to the generalized discrete Yakubovich-transpose matrix equation XAX T B = CY. which is related with several types of matrix equations in control theory. One of the parametric solutions has a neat and elegant form in terms of the Krylov matrix, a block Hankel matrix and an observability matrix. In addition, the special case of the generalized discrete Yakubovich-transpose matrix equation, which is called the Karm-Yakubovich-transpose matrix equation, is considered. The explicit solutions to the Karm-Yakubovich-transpose matrix equation are also presented by the so-called generalized Leverrier algorithm. At the end of the paper, two examples are given to show the efficiency of the proposed algorithm.
TL;DR: A fixed point theorem in partially ordered sets is proved and the existence of a unique Hermitian positive definite solution for the matrix equation is derived.
Abstract: The nonlinear matrix equation, with is investigated. A fixed point theorem in partially ordered sets is proved. And then, by means of this fixed point theorem, the existence of a unique Hermitian positive definite solution for the matrix equation is derived. Some properties of the unique Hermitian positive definite solution are obtained. A residual bound of an approximate solution to the equation is evaluated. The theoretical results are illustrated by numerical examples.
TL;DR: A new iterative scheme has been constructed for finding minimal solution of a rational matrix equation of the form X + A*X −1 A = I and the convergence of the method has been studied and tested via numerical experiments.
Abstract: A new iterative scheme has been constructed for finding minimal solution of a rational matrix equation of the form X + A*X−1A = I. The new method is inversion-free per computing step. The convergence of the method has been studied and tested via numerical experiments.
01 Jan 2014
TL;DR: In this article, the authors developed easily verifiable tests that can be applied to determine whether a higher-order autonomous difference equation has a p-periodic solution, based on the theory of circulant matrices.
Abstract: In this paper we develop easily verifiable tests that we can apply to determine whether or not a higher order autonomous difference equation has a p-periodic solution. One of the main tools in our investigations is a transformation, recently introduced by the authors, which formulates a given higher order difference equation as a first order recursion. The second important tool is the theory of circulant matrices. The periodicity conditions are formulated in terms of the coefficients of the higher order equation, along with examples showing that they have nontrivial applications.
TL;DR: A direction-of-arrival (DOA) estimation algorithm is presented based on covariance differencing and sparse signal recovery, in which the desired signal is embedded in noise with unknown covariance, to eliminate the noise component by forming the difference of original and transformed covariance matrix.
TL;DR: In this paper, an approximate discrete-time method for the resolution of the matrix differential equation of Riccati in the general case of varying-time (dependant of time) matrices is presented.
Abstract: The Riccati Matrix Differential Equation (RMDE) is an interesting equation in different fields of science and engineering practice. In fact, that the arithmetic solution for this matrix differential equation in the general case of varying-time matrices is very difficult to find. The literature offers the solution of this differential equation in the case of dependant constant matrices (i.e. invariant-time matrices). The present approach is an approximate discrete-time method for the resolution of the matrix differential equation of Riccati in the general case of varying-time (dependant of time) matrices; the method in fact, is a discretisation of the exact matrix solution, that evaluates for any so small step of time, and which is function of the solution of the preceding step of time and the constitute equation matrices. The proposed algorithm is verified, for a controlled structure under Modified El-Centro earthquake by a comparison with the same uncontrolled structure, which constitutes by a two Degrees Of Freedom (2DOF) system. The results of this comparison offer good differences between the controlled and the uncontrolled systems.