Order Matrix Differential Equations Research Articles

Liouville-Green formulas for the fourth order matrix differential operators

In this paper, Liouville-Green analytical approximations for the solutions of the fourth order matrix differential equation are constructed. Using the Gronwall’s inequality, we have also determined the error bounds for the Liouville-Green approximations.

The Existence and Uniqueness of Solution for Fractional Order Matrix Differential Equation

The Existence and Uniqueness of Solution for Fractional Order Matrix Differential Equation

The semi-analytical method for damping of tubular transition layer damping structure

To solve the limited vibration consumption of the traditional tubular damping structure (TTDS), the tubular transition layer damping structure (TTLDS) is proposed; Based on viscoelastic materials and theories of thin cylindrical shells, the governing equation, the first order matrix differential equation describing vibration of TTLDS under harmonic excitation, is derived by considering the interaction between all layers and the dissipation caused by the shear deformation for transition layer and damping layer. By using the extended homogeneous capacity precision integration method to solve the control equation, a semi-analytical method for studying the vibration and damping characteristics of TTLDS is given. By way of comparison, the correctness of the method provided in paper is verified. At last, the influence of thickness, material and location of transition layer on damping effect is analyzed. The results show that the change for the thickness or material of the transition layer can make the structural damping effect change greatly, while the change for location of the transition layer plays only a few roles on the structural damping effect.

Optimal control in teleoperation systems with time delay: A singular perturbation approach

The main goal of controller design in teleoperation systems is to achieve optimal performance, transparency and stability in presence of factors such as time delay in communication channel and modeling uncertainties. The teleoperation systems usually have complex dynamic. Consequently, differential equation solution of optimal control problem is difficult and complex for them. This paper presents a novel method for designing optimal controller based on singular perturbation framework for these systems. Firstly, we use the Taylor expansion to model the time delay, with considering time delay term; we derive a singular perturbation formulation for the teleoperation system. Using singular perturbation model and Chang decoupling transformation, singularly perturbed differential equations of optimal control problem is decomposed into the reduced order slow and fast differential equations. A formula is obtained that produces the solution of original differential equations of optimal control problem in terms of solutions of the slow and fast reduced order matrix differential equations. The reduced-order differential equations decrease the complexity of the optimal control problem for teleoperation systems. The simulations verify the effectiveness of the proposed control method and excellent performances tracking with high speed and small control signal.

The Method of Matrix Differential Equation to Calculate the Natural Frequencies of Variable Cross-section Beams on the Non-uniform Elastic Foundations

The tradition methods to calculate the natural frequencies of beam on the elastic foundation were based on high order differential equation by defection. This paper derived the general equation based on the one order matrix differential equation represented by the mixed variable (a vector formed by deflection, angle of rotation, bending moment and shearing force) to calculate the natural frequencies of the variable cross-section beams on the non-uniform elastic foundations. To calculate the natural frequencies of this kind of beams, we divide the beams into several sections, calculate the transfer matrices of each section, multiply the matrices in proper order and obtain the overall transfer matrix of the mixed variable from one end to the other end of the variable cross-section beam on non-uniform elastic foundations. According to the conditions of boundary at both ends of the beam, we can calculate out the natural frequencies in MATLAB easily.

Oscillation results for second order matrix differential equations with damping

By using the positive linear functional, including the general means and Riccati technique, some new oscillation criteria are established for the second order matrix differential equations (r(t)P(t)?(X(t))K(X'(t)))' + p(t)R(t)?(X(t))K(X'(t)) + Q(t)F(X'(t))G(X(t)) = 0,t ? t0 > 0. The results improve and generalize those given in some previous papers.

Closed form solution for the equations of motion for constrained linear mechanical systems and generalizations: An algebraic approach

In this paper, a mathematical methodology is presented for the determination of the solution of motion for linear constrained mechanical systems applicable also to systems with singular coefficients. For mathematical completeness and also to incorporate some other interesting cases, the methodology is formulated for a general class of higher order matrix differential equations. Thus, describing the system in an autoregressive moving average (ARMA) form, the closed form solution is derived in terms of the finite and infinite Jordan pairs of the system׳s polynomial matrix. The notion of inconsistent initial conditions is considered and an explicit formula for the homogeneous system is given. In this respect, the methodology discussed in the present note provides an alternative view to the problem of computation of the response of complex multi-body systems. Two interesting examples are provided and applications of the equation to such systems are illustrated.

Solution of differential equation for the Euler-Bernoulli beam

The paper presents the solution of a fourth order differential equation with various coefficients occurring in the vibration problem of the Euler-Bernoulli beam. The concern- ing equation is written as a first order matrix differential equation. To solve the equation, the power series method is proposed.

Power series solution of first order matrix differential equations

In the paper, the solution of second order differential equations with various coefficients is presented. The concerning equations are written as first order matrix differ- ential equations and solved with the use of the power series method. Examples of applica- tion of the proposed method to the equations occurring in the technical problems are presented.

On Chebyshev matrix polynomials, matrix differential equations and their properties

The main aim of this paper is to define and study of the Chebyshev matrix polynomials. An explicit representation, a three-term matrix recurrence relations for the Chebyshev matrix polynomials are given. These polynomials appear as finite series solutions of second order matrix differential equations. The expansion Chebyshev matrix polynomials in series of Hermite matrix polynomials and the Christoffel formula of summation are established.

On an analytical representation for the solution of the neutron point kinetics equation free of stiffness

In this work, we report on 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. To this end, we cast the point kinetics equation in a first order linear differential equation. Next, we split the corresponding matrix into a diagonal matrix plus a matrix that contains the remaining terms. Expanding the neutron density and the delayed neutron precursor concentrations in a truncated series allows one to construct a recursive system in form of a first order matrix differential equation with source. The initialization of the recursion procedure is of diagonal form and has no source but satisfies the initial conditions. The remaining equations are subject to null initial conditions and include the time-dependent diagonal elements together with the off-diagonal elements as a source term. The solution is obtained in an analytical representation which may be evaluated for any time value, because it is free of stiffness. We present numerical simulations and comparisons against results from the literature for a constant, a step, a ramp, a quadratic, and a sine shaped reactivity function.

MMaMS 2014 Dynamic Model of a Mobile Robot

The contribution presents the dynamic model of a differential two-wheeled mobile robot in the joint frame, i.e. in the plane of angular position of both the right-hand side wheel and the left-hand side one. The aim is to create an appropriate MIMO plant model for the purpose of the mobile robot motion control synthesis. Based on the Lagrangian formalism, the complete description of the second order matrix differential equation (the dynamic model of the mobile robot) is provided, including the influence of inertial coupling forces. A pair of wheel driving torques represents the input of given model. Additionally, a decoupled dynamic model for the robust motion control implementation is presented.

A Convergence Theorem Associated With a Pair of Second Order Differential Equations

We consider the second order matrix differential equation   0, 0 M x        . Where M is a second-order matrix differential operator and  is a vector having two components. In this paper we prove a convergence theorem for the vector function 1 2 ( ) ( ) ( ) f x f x f x        which is continuous in 0 x    and of bounded variation in 0 x    , when ( ) p x and ( ) q x tend to  as x tend to  .

Liouville–Green approximation for Bleustein–Gulyaev waves in functionally graded materials

The present work aims at studying Bleustein–Gulyaev (BG) waves in a specific class of functionally graded materials. The inhomogeneity under question requires that the elastic stiffness, the piezoelectric term and the dielectric term all vary proportionally to the same inhomogeneity function. However, the density is allowed here to vary in a different manner. As a result we are able to analyse BG waves for media with variable bulk Shear Horizontal (SH) wave velocity profiles. We make use of the Liouville–Green approximation for second order Matrix Differential Equations, with explicit computable error bounds. Our analysis is limited to a certain integrability condition and to the numerical evaluation of a series of improper integrals. We present dispersion curves for BG waves for the metalized boundary condition for different inhomogeneity profiles. The results show that variations in the SH bulk wave velocities have a profound effect on the BG dispersion curves for the metalized boundary condition.

An approximate quantum Cramér–Rao bound based on skew information

A closed-form expression for Wigner-Yanase skew information in mixed-state quantum systems is derived. It is shown that limit values of the mixing coefficients exist such that Wigner-Yanase information is equal to Helstrom information. The latter constitutes an upper bound for the classical expected Fisher information, hence the inverse Wigner-Yanase information provides an approximate lower bound to the variance of an unbiased estimator of the parameter of interest. The advantage of approximating Helstrom's sharp bound lies in the fact that Wigner-Yanase information is straightforward to compute, while it is often very difficult to obtain a feasible expression for Helstrom information. In fact, the latter requires the solution of an implicit second order matrix differential equation, while the former requires just scalar differentiation.

A novel matrix method for coupled vibration and damping effect analyses of liquid-filled circular cylindrical shells with partially constrained layer damping under harmonic excitation

A novel matrix method is further developed for a liquid-filled circular cylindrical shell with partially constrained layer damping (CLD), which consists of treating liquid domain with Bessel function approach, and shell domain with transfer matrix equation based on a new set of first order matrix differential equation. In order to indicate its advantage to the finite element method (FEM), free vibration analysis on such an empty shell under clamped–clamped boundary are carried out by using the present method together with FEM. Meanwhile, coincident result is yielded for the liquid-filled shell by adopting the present method with by other transfer matrix method. Finally, a series of valuable numerical results are obtained by this method.

On differential equations for orthogonal polynomials on the unit circle

In this paper we characterize sequences of orthogonal polynomials on the unit circle whose corresponding Carathéodory function satisfies a Riccati differential equation with polynomial coefficients, in terms of second order matrix differential equations. In the semi-classical case, a characterization in terms of second order linear differential equations with polynomial coefficients is deduced.

Oscillation of nonlinear second order matrix differential equations

Abstract In this paper, sufficient conditions are obtained for oscillation of all nontrivial, prepared, symmetric solutions of a class of nonlinear second order matrix differential equations of the form $$(P(t)Y')' + Q(t)F(Y) = 0, t \geqslant 0,$$ and $$Y'' + Q(t)F(Y) = 0, t \geqslant 0.$$

Matrix differential equations and inverse preconditioners

In this article, we propose to model the inverse of a given matrix as the state of a proper first order matrix differential equation. The inverse can correspond to a finite value of the independent variable or can be reached as a steady state. In both cases we derive corresponding dynamical systems and establish stability and convergence results. The application of a numerical time marching scheme is then proposed to compute an approximation of the inverse. The study of the underlying schemes can be done by using tools of numerical analysis instead of linear algebra techniques only. With our approach, we recover some known schemes but also introduce new ones. We derive in addition a masked dynamical system for computing sparse inverse approximations. Finally we give numerical results that illustrate the validity of our approach.

Matrix differential equations and inverse preconditioners

In this article, we propose to model the inverse of a given matrix as the state of a proper first order matrix differential equation. The inverse can correspond to a finite value of the independent variable or can be reached as a steady state. In both cases we derive corresponding dynamical systems and establish stability and convergence results. The application of a numerical time marching scheme is then proposed to compute an approximation of the inverse. The study of the underlying schemes can be done by using tools of numerical analysis instead of linear algebra techniques only. With our approach, we recover some known schemes but also introduce new ones. We derive in addition a masked dynamical system for computing sparse inverse approximations. Finally we give numerical results that illustrate the validity of our approach.