Numerical integration of positive linear differential-algebraic systems

In the simulation of dynamical processes in economy, social sciences, biology or chemistry, the analyzed values often represent non-negative quantities like the amount of goods or individuals or the density of a chemical or biological species. Such systems are typically described by positive ordinary differential equations (ODEs) that have a non-negative solution for every non-negative initial value. Besides positivity, these processes often are subject to algebraic constraints that result from conservation laws, limitation of resources, or balance conditions and thus the models are differential-algebraic equations (DAEs). In this work, we present conditions under which both these properties, the positivity as well as the algebraic constraints, are preserved in the numerical simulation by Runge–Kutta or multistep discretization methods. Using a decomposition approach, we separate the dynamic and the algebraic equations of a given linear, positive DAE to give positivity preserving conditions for each part separately. For the dynamic part, we generalize the results for positive ODEs to DAEs using the solution representation via Drazin inverses. For the algebraic part, we use the consistency conditions of the discretization method to derive conditions under which this part of the approximation overestimates the exact solution and thus is non-negative. We analyze these conditions for some common Runge–Kutta and multistep methods and observe that for index-1 systems and stiffly accurate Runge–Kutta methods, positivity is conditionally preserved under similar conditions as for ODEs. For higher index problems, however, none of the common methods is suitable.     

Three-layer finite-difference method for linear partial differential-algebraic systems

We consider a boundary value problem for a linear partial differential-algebraic system with a special structure of the matrix pencil, which permits one to split the system by an appropriate transformation into a system of ordinary differential equations, a hyperbolic system, and a linear algebraic system. For the numerical solution of such problems, we use a three-layer method. We prove the theorem on the stability and convergence of the suggested numerical method. The results of numerical experiments are presented as well.     

On data-dependence of exponential stability and stability radii for linear time-varying differential-algebraic systems

This paper is addressed to some questions concerning the exponential stability and its robustness measure for linear time-varying differential-algebraic systems of index 1. First, the Bohl exponent theory that is well known for ordinary differential equations is extended to differential-algebraic equations. Then, it is investigated that how the Bohl exponent and the stability radii with respect to dynamic perturbations for a differential-algebraic system depend on the system data. The paper can be considered as a continued and complementary part to a recent paper on stability radii for time-varying differential-algebraic equations [N.H. Du, V.H. Linh, Stability radii for linear time-varying differential-algebraic equations with respect to dynamic perturbations, J. Differential Equations 230 (2006) 579-599].     

On the continuous dependence of solutions of linear systems of differential-algebraic equations on a parameter

We consider linear systems of ordinary differential equations with identically degenerate matrix multiplying the derivative of the unknown vector function. The matrices specifying the system are assumed to depend on a parameter. We obtain criteria for the continuous dependence of the solutions of the system on the parameter and the asymptotic equivalence of solutions of the original and perturbed systems.     

Transformation of high order linear differential-algebraic systems to first order

We study general nonsquare linear systems of differential-algebraic systems of arbitrary order. We analyze the classical procedure of turning the system into a first order system and demonstrate that this approach may lead to different solvability results and smoothness requirements. We present several examples that demonstrate this phenomenon and then derive existence and uniqueness results for differential-algebraic systems of arbitrary order and index. We use these results to identify exactly those variables for which the order reduction to first order does not lead to extra smoothness requirements and demonstrate the effects of this new formulation with a numerical example.Dedicated to Richard S. Varga on the occasion of his 77th birthday.     

Index for linear systems of differential-algebraic equations with partial derivatives

We investigate time-varying linear differential algebraic equations with partial derivatives. We introduce concept of insolubility index as the least possible order of the differential operator which transforms the initial system into a structural form with separated the “algebraic” and “differential” subsystems. The approach does not assume the existence of differential indexes with respect to independent variables.     

On a property of the reducibility exponent of linear differential systems

We establish that the reducibility exponent (Differentsial’nye Uravneniya, 2007, vol. 43, no. 2, pp. 191–202) of each linear system

$$\dot x = A(t)x, x \in \mathbb{R}^n , t \geqslant 0$$

, with piecewise continuous bounded coefficient matrix A does not belong to the set of values of σ for which the perturbed system (1_A+Q) with an arbitrary piecewise continuous perturbation Q satisfying the condition \(\overline {\lim } _{t \to + \infty } t^{ - 1} \ln \left\| {Q(t)} \right\| \leqslant - \sigma \) is reducible to the original system (1 _A ) by some Lyapunov transformation.

     

On the Hamiltonian property of linear dynamical systems in Hilbert space

Mathematical Notes - Conditions for the operator differential equation $$dot x = Ax$$ possessing a quadratic first integral (1/2)(Bx, x) to be Hamiltonian are obtained. In the finite-dimensional...     

On a least-squares collocation method for linear differential-algebraic equations

Summary The aim of this note is to extend some results on least-squares collocation methods and to prove the convergence of a least-squares collocation method applied to linear differential-algebraic equations. Some numerical examples are presented.     

On the uniqueness of the solution of nonlinear differential-algebraic systems

We consider the Cauchy problem for a system of nonlinear ordinary differential equations unsolved for the derivative of the unknown vector function and identically degenerate in the domain. We prove a theorem on the coincidence of two smooth solutions of the considered problem. We show that, under some additional assumptions, the above-mentioned problem cannot have classical solutions with less smoothness. We obtain conditions under which the problem has a fixed finite number of solutions.     

Convergence of general linear methods on differential-algebraic systems of index 3

Many numerical methods used to solve ordinary differential equations or differential-algebraic equations can be written as general linear methods. The purpose of this paper is to extend the known convergence results for Runge-Kutta and linear multistep methods to a large class of new promising numerical schemes. The theoretical results are illustrated by some numerical experiments.     

On convergence conditions of waveform relaxation methods for linear differential-algebraic equations

For linear constant-coefficient differential-algebraic equations, we study the waveform relaxation methods without demanding the boundedness of the solutions based on infinite time interval. In particular, we derive explicit expression and obtain asymptotic convergence rate of this class of iteration schemes under weaker assumptions, which may have wider and more useful application extent. Numerical simulations demonstrate the validity of the theory.     

On the general nonlinear self-adjoint spectral problem for differential-algebraic systems

We consider a general self-adjoint spectral problem, nonlinear with respect to the spectral parameter, for linear differential-algebraic systems of equations. Under some assumptions, we present a method for reducing such a problem to a general self-adjoint nonlinear spectral problem for a system of differential equations. In turn, this permits one to pass to a problem for a Hamiltonian system of ordinary differential equations. In particular, in this way, one can obtain a method for computing the number of eigenvalues of the original problem lying in a given range of the spectral parameter.     

Asymptotic property for linear integro-differential systems

We study the asymptotic property of linear integro-differential systems by means of the resolvent matrices and the useful equivalent system of Rao and Srinivas in [M.R.M. Rao, P. Srinivas, Asymptotic behavior of solutions of Volterra integro-differential equations, Proc. Amer. Math. Soc. 94 (1985) 55–60. [11]].     

Asymptotic property of linear Volterra difference systems

By using the resolvent matrix and the comparison principle, we investigate the asymptotic behavior of linear Volterra difference systems.