Uniformly exponentially stable approximation for the transmission line with variable coefficients and its application

We analyze an ideal transmission line, which is defined by the telegraph equation with variable coefficients, from the perspectives of numerical analysis and control theory in this note. Because the spatially semi-discrete scheme of the original system is insufficient for discussing uniform exponential stability, we apply a similar transform to the continuous system and produce an intermediate system that may be easily analyzed. To begin, we discuss uniform exponential stability for the intermediate system using an so called average central-difference semi-discrete scheme and the direct Lyapunov function approach. The proof is the same as in the continuous case. The Trotter-Kato Theorem is used to demonstrate the stability and consistency of numerical approximation scheme. Finally, we propose a semi-discrete strategy for the original system through an inverse transform. All results on intermediate system are then translated into the original system. The numerical state reconstruction problem is addressed as an essential application of the main results. Furthermore, several numerical simulations are used to validate the effectiveness of the numerical approximating algorithms.     

STABILIZED NUMERICAL APPROXIMATIONS OF THE BACKWARD PROBLEM OF A PARABOLIC EQUATION

1　IntroductionLetΩ be a bounded domain in Rn and Ω be its boundary.ThenΣ =Ω× ( 0 ,1 ) is abounded domain in Rn+1 .We consider the following backwad problem of a prabolic equa-tion: u t= ni,j=1 xiaij( x) u xj -c( x) u,　　 ( x,t)∈Σ,( 1 )u| Ω× [0 ,1 ] =0 , ( 2 )u| t=1 =g( x) . ( 3 )　　 Where { aij( x) } are smooth functions given onΩ satisfyingaij( x) =aji( x) ,　　 1≤ i,j≤ n, ( 4)α0 ni=1ζ2i ≤ ni,j=1aij( x)ζiζj≤α1 ni=1ζ2i,　　 ζ∈ Rn,x∈Ω. ( 5)　　Where0 <α…     

Computing Stability of Differential Equations with Bounded Distributed Delays

This paper deals with the stability analysis of scalar delay integro-differential equations (DIDEs). We propose a numerical scheme for computing the stability determining characteristic roots of DIDEs which involves a linear multistep method as time integration scheme and a quadrature method based on Lagrange interpolation and a Gauss–Legendre quadrature rule. We investigate to which extent the proposed scheme preserves the stability properties of the original equation. We derive and prove a sufficient condition for (asymptotic) stability of a DIDE (with a constant kernel) which we call RHP-stability. Conditions are obtained under which the proposed scheme preserves RHP-stability. We compare the obtained results with corresponding ones using Newton–Cotes formulas. Results of numerical experiments on computing the stability of DIDEs with constant and nonconstant kernel functions are presented.     

Two-parameter bifurcation in a two-dimensional simplified Hodgkin–Huxley model

The dynamical behaviors of a two-dimensional simplified Hodgkin–Huxley model exposed to external electric fields are investigated based on the qualitative analysis and numerical simulation. A necessary and sufficient condition is given for the existence of the Hopf bifurcation. The stability of equilibrium points and limit cycles is also studied. Moreover, the canards and bifurcation are discussed in the simplified model and original model. The dynamical behaviors of the simplified model are consistent with the original model. It would be a great help to further investigations of the original model.     

A Dynamical Method for Solving the Obstacle Problem

In this paper, we consider the unilateral obstacle problem, trying to find the numerical solution and coincidence set. We construct an equivalent format of the original problem and propose a method with a second-order in time dissipative system for solving the equivalent format. Several numerical examples are given to illustrate the effectiveness and stability of the proposed algorithm. Convergence speed comparisons with existent numerical algorithm are also provided and our algorithm is fast.     

Limitations of frequency domain approximation for detecting chaos in fractional order systems

In this paper, we analytically study the influences of using frequency domain approximation in numerical simulations of fractional order systems. The number and location of equilibria, and also the stability of these points, are compared between the original system and its frequency based approximated counterpart. It is shown that the original system and its approximation are not necessarily equivalent according to the number, location and stability of the fixed points. This problem can cause erroneous results in special cases. For instance, to prove the existence of chaos in fractional order systems, numerical simulations have been largely based on frequency domain approximations, but in this paper we show that this method is not always reliable for detecting chaos. This approximation can numerically demonstrate chaos in the non-chaotic fractional order systems, or eliminate chaotic behavior from a chaotic fractional order system.     

Stabilization of Interior-Point Methods for Linear Programming

The paper studies numerical stability problems arising in the application of interior-point methods to primal degenerate linear programs. A stabilization procedure based on Gaussian elimination is proposed and it is shown that it stabilizes all path following methods, original and modified Dikin's method, Karmarkar's method, etc.     

On Mayfield’s stability proof for the discretized transparent boundary condition for the parabolic equation

Mayfield’s numerical implementation of transparent boundary condition for the Schrödinger-type parabolic equations is revisited. An inaccuracy in the original proof of the conditional stability for the resulting scheme is pointed out. The highly unusual and impressive original result is reestablished and a new proof is presented. Some further remarks and estimates on the instability which occurs when the Mayfield condition is violated are given.     

一类含有稳定参数的Adams型隐式方法及其新算法

§1.引言 数值积分Stiff常微分方程初值问题,其积分过程的稳定性相当重要.用传统的数值方法,如Adams方法等,为保证计算稳定性,积分步长受到相当的限制.在stiff常微分方程初值问题的数值解法中,Gear方法是目前最通用的方法之一.但是,当阶p大     

The High Order Exponentially Fitted Nonequidistant Extrapolation Methods for Stiff Systems

A class of exponentially fitted nonequidistant extrapolation methods based on an L-stable linear single-step formula are studied. Theorems about the extrapolation coefficients are given. These methods keep good numerical stability "quasi-L-stability", while raising the accuracy of the original formula.