Application of finite difference method of lines on the heat equation

In this article, we apply the method of lines (MOL) for solving the heat equation. The use of MOL yields a system of first–order differential equations with initial value. The solution of this system could be obtained in the form of exponential matrix function. Two approaches could be applied on this problem. The first approach is approximation of the exponential matrix by Taylor expansion, Padé and limit approximations. Using this approach leads to create various explicit and implicit finite difference methods with different stability region and order of accuracy up to six for space and superlinear convergence for time variables. Also, the second approach is a direct method which computes the exponential matrix by applying its eigenvalues and eigenvectors analytically. The direct approach has been applied on one, two and three‐dimensional heat equations with Dirichlet, Neumann, Robin and periodic boundary conditions.     

A fourth‐order compact finite difference scheme for solving an N‐carrier system with Neumann boundary conditions

In this study, we develop a fourth‐order compact finite difference scheme for solving a model of energy exchanges in a generalized N‐carrier system with heat sources and Neumann boundary conditions, which extends the concept of the well‐known parabolic two‐step model for microheat transfer. By using the matrix analysis, the compact finite difference numerical scheme is shown to be unconditionally stable. The accuracy of the solution obtained by the scheme is tested by a numerical example. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Dynamics and interpretation of some integrable systems via matrix orthogonal polynomials

In this work we characterize a high-order Toda lattice in terms of a family of matrix polynomials orthogonal with respect to a complex matrix measure. In order to study the solution of this dynamical system, we give explicit expressions for the Weyl function, generalized Markov function, and we also obtain, under some conditions, a representation of the vector of linear functionals associated with this system. We show that the orthogonality is embedded in these structure and governs the high-order Toda lattice. We also present a Lax-type theorem for the point spectrum of the Jacobi operator associated with a Toda-type lattice.     

The canonical structure of a pencil of degenerate matrix functions

In this paper we study global properties of a pencil of identically degenerate matrix functions with a compact domain of definition. Matrix functions are assumed to have a constant rank and all roots of the characteristic equation of the matrix pencil are assumed to have a constant multiplicity at each point in the domain of definition. We obtain sufficient conditions for the smooth orthogonal similarity of matrix functions to the upper triangular form and sufficient conditions for the smooth equivalence of the pencil of matrix functions to its canonical form. We illustrate the obtained results with simple examples.     

Estimate on the Pathwise Lyapunov Exponent of Linear Stochastic Differential Equations with Constant Coefficients

In this article, we study the problem of estimating the pathwise Lyapunov exponent for linear stochastic systems with multiplicative noise and constant coefficients. We present a Lyapunov type matrix inequality that is closely related to this problem, and show under what conditions we can solve the matrix inequality. From this we can deduce an upper bound for the Lyapunov exponent. In the converse direction, it is shown that a necessary condition for the stochastic system to be pathwise asymptotically stable can be formulated in terms of controllability properties of the matrices involved.     

On approximated ILU and UGS preconditioning methods for linearized discretized steady incompressible Navier-Stokes equations

When the artificial compressibility method in conjunction with high-order upwind compact finite difference schemes is employed to discretize the steady-state incompressible Navier-Stokes equations, in each pseudo-time step we need to solve a structured system of linear equations approximately by, for example, a Krylov subspace method such as the preconditioned GMRES. In this paper, based on the special structure and concrete property of the linear system we construct a structured preconditioner for its coefficient matrix and estimate eigenvalue bounds of the correspondingly preconditioned matrix. Numerical examples are given to illustrate the effectiveness of the proposed preconditioning methods.     

Analysis of a set of nonlinear dynamics trajectories: Stability of difference equations

For a set of difference equations generated by discretization of the set of differential equations with Hukuhara derivative a principle of comparison with matrix Lyapunov function is specified and sufficient stability conditions of certain type are established. The analysis is carried out in terms of a matrix Lyapunov function of special structure. For an essentially nonlinear multiconnected switched difference system, conditions are obtained providing the asymptotic stability of its zero solution for any switching law. An example is presented to demonstrate efficiency of the proposed approaches.     

对角占优矩阵奇异-非奇异的充分必要判据

本文研究对角占优矩阵奇异-非奇异的充分必要条件.基于Taussky定理,本文得出,可约对角占优矩阵的奇异性由其独立Frobenius块的奇异性决定,从而将这一问题化为不可约对角占优矩阵的奇异-非奇异性问题;运用Taussky定理研究奇异不可约对角占优矩阵的相似性和酉相似性,获得这类矩阵元素辐角间的关系;并与Taussky定理给出的这类矩阵元素模之间的关系结合在一起,研究不可约对角占优矩阵奇异的充分必要条件;最后给出不可约对角占优矩阵奇异-非奇异性的判定方法.     

Perturbation and robust stability of autonomous singular linear matrix difference equations

In the perturbation theory of linear matrix difference equations, it is well known that the theory of finite and infinite elementary divisors of regular matrix pencils is complicated by the fact that arbitrarily small perturbations of the pencil can cause them to disappear. In this paper, the perturbation theory of complex Weierstrass canonical form for regular matrix pencils is investigated. By using matrix pencil theory and the Weierstrass canonical form of the pencil we obtain bounds for the finite elementary divisors of a perturbed pencil. Moreover we study robust stability of a class of linear matrix difference equations (of first and higher order) whose coefficients are square constant matrices.     

Polynomial stability of semigroups generated by operator matrices

In this paper, we study the stability properties of strongly continuous semigroups generated by block operator matrices. We consider triangular and full operator matrices whose diagonal operator blocks generate polynomially stable semigroups. As our main results, we present conditions under which also the semigroup generated by the operator matrix is polynomially stable. The theoretical results are used to derive conditions for the polynomial stability of a system consisting of a two-dimensional and a one-dimensional damped wave equation.     

Eight expressions for generalized inverses of a bordered matrix

A bordered matrix is a two-by-two partitioned matrix with its lower-right corner equal to a null matrix. In this article, we present eight partitioned matrices consisting of the Moore–Penrose inverses of submatrices in a bordered matrix, and give necessary and sufficient conditions for the eight partitioned matrices to be generalized inverses of the bordered matrix.     

Ultrametric and Tree Potential

In this article we study which infinite matrices are potential matrices. We tackle this problem in the ultrametric framework by studying infinite tree matrices and ultrametric matrices. For each tree matrix, we show the existence of an associated symmetric random walk and study its Green potential. We provide a representation theorem for harmonic functions that includes simple expressions for any increasing harmonic function and the Martin kernel. For ultrametric matrices, we supply probabilistic conditions to study its potential properties when immersed in its minimal tree matrix extension. C. Dellacherie thanks support from Nucleus Millennium P04-069-F for his visit to CMM-DIM at Santiago. The research of S. Martinez is supported by Nucleus Millennium Information and Randomness P04-069-F and by the BASAL CONICYT Program. The research of J. San Martin is supported by FONDAP and by the BASAL CONICYT Program.     

Stability of difference schemes for two-dimensional parabolic equations with non-local boundary conditions

The stability of difference schemes for one-dimensional and two-dimensional parabolic equations, subject to non-local (Bitsadze-Samarskii type) boundary conditions is dealt with. To analyze the stability of difference schemes, the structure of the spectrum of the matrix that defines the linear system of difference equations for a respective stationary problem is studied. Depending on the values of parameters in non-local conditions, this matrix can have one zero, one negative or complex eigenvalues. The stepwise stability is proved and the domain of stability of difference schemes is found.     

Multigrid solution of high order discretisation for three-dimensional biharmonic equation with Dirichlet boundary conditions of second kind

In this paper, we propose two compact finite difference approximations for three-dimensional biharmonic equation with Dirichlet boundary conditions of second kind. In these methods there is no need to define special formulas near the boundaries and boundary conditions are incorporated with these techniques. The unknown solution and its second derivatives are carried as unknowns at grid points. We derive second-order and fourth-order approximations on a 27 point compact stencil. Classical iteration methods such as Gauss–Seidel and SOR for solving the linear system arising from the second-order and fourth-order discretisation suffer from slow convergence. In order to overcome this problem we use multigrid method which exhibit grid-independent convergence and solve the linear system of equations in small amount of computer time. The fourth-order finite difference approximations are used to solve several test problems and produce high accurate numerical solutions.     

A Taylor polynomial approach for solving the most general linear Fredholm integro‐differential‐difference equations

In this study, a matrix method is developed to solve approximately the most general higher order linear Fredholm integro‐differential‐difference equations with variable coefficients under the mixed conditions in terms of Taylor polynomials. This technique reduces the problem into the linear algebraic system. The method is valid for any combination of differential, difference and integral equations. An initial value problem and a boundary value problem are also presented to illustrate the accuracy and efficiency of the method. Copyright © 2012 John Wiley & Sons, Ltd.     

Symmetric modified finite volume element methods for self-adjoint elliptic and parabolic problems

The finite volume element method is a discretization technique for partial differential equations, but in general case the coefficient matrix of its linear system is not symmetric, even for the self-adjoint continuous problem. In this paper we develop a kind of symmetric modified finite volume element methods both for general self-adjoint elliptic and for parabolic problems on general discretization, their coefficient matrix are symmetric. We give the optimal order energy norm error estimates. We also prove that the difference between the solutions of the finite volume element method and symmetric modified finite volume element method is a high order term.     

The structure of the spectrum of a system of difference equations

In this paper, we investigate the structure of the discrete spectrum of the system of non-selfadjoint difference equations of first order using the uniqueness theorems of analytic functions. We also obtained the sufficient conditions on coefficients of this system under which its discrete spectrum is finite.     

On asymptotic equivalence of perturbed linear systems of differential and difference equations

Standard results on asymptotic integration of systems of linear differential equations give sufficient conditions which imply that a system is strongly asymptotically equivalent to its principal diagonal part. These involve certain dichotomy conditions on the diagonal part as well as growth conditions on the off-diagonal perturbation terms. Here, we study perturbations with a triangularly-induced structure and see that growth conditions can be substantially weakened. In addition, we give results for not necessarily triangular perturbations which in some sense “interpolate” between the classical theorems of Levinson and Hartman-Wintner. Some analogous results for systems of linear difference equations are also given.     

The necessary and sufficient conditions for the existence of periodic solutions of a type of neutral functional differential equations

In this paper, using Fourier series, we study the problem of the existence of periodic solutions of a type of periodic neutral differential difference system. Some necessary and sufficient conditions for the existence of periodic solutions of a type of neutral functional equation system are obtained, and at the same time, we present a method with formula shows how to find the periodic solutions.     

退化时滞微分系统的通解

本文用D-逆阵给出退化滞后系统的基础解、常数交易公式与通解,确定了这类退化系统的基本理论．它将被广泛地应用于各类退化控制系统．