On a regularization of index 3 differential-algebraic equations

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Abstract

In Hanke [J. Math. Anal. Appl. 151 (1990) 236–253], boundary value problems for linear differential-algebraic equations (DAEs) with time-varying coefficients A(t)x′(t) + B(t)x(t) = q(t) tractable with index 2 are considered. In this paper, we consider index 3 DAEs in an analogous way. A transferable DAEs is obtained by the regularization and the convergence rate is derived.

Introduction

Normal differential-algebraic equations (DAEs) are singular ordinary differential equations (ODEs)f(x(t),x(t),t)=0,where f is smooth and the partial Jacobian fy(y, x, t) is everywhere singular but has constant rank. Such systems are of special interest in view of various applications, e.g., electrical networks, constrained mechanical systems of rigid bodies, control theory problems, singular perturbation and discretization of partial differential equations, etc. (cf. [1], [2], [3], [4]).

It is well known that differential-algebraic equations can be difficult to solve when they have a higher index, i.e., index greater than one (cf. [1]). A lot of researchers show that the DAEs being tractable with higher index k(⩾2) leads to ill posed problems in Tikhonov’s sense, and a straightforward discretization generally does not work well. In order to treat the ill posed problems, some regularization for the DAEs with indexes 2 and 3 are provided by [5], [6], [9].

In the present paper, we consider linear variable coefficients index 3 systems such asA(t)x(t)+B(t)x(t)=q(t),t[a,b]subject to the boundary conditionsDax(a)+Dbx(b)=γ.Here, x is a vector-valued real function, x(t)  Rm, A(t) and B(t) are m × m matrices depending continuously on t  [a,b].

Section snippets

Main results

Now we consider the DAEs (1.2). We assume that the null space N0(t) = ker(A(t)) is smooth, i.e., there exists a continuously differentiable matrix function Q0 projects Rm onto N(A(t)) for every t  [a, b]. If we set P0 = I  Q0, then (1.2) is equivalent toA(P0x)+(B-AP0)x=q.

For simplicity we omit the argument t here and in the following if no confusion can arise.

A matrix chain is constructed in [8] as follows:A0=A,B0=B-AP0,Ai=Ai-1+Bi-1Qi-1,Bi=Bi-1Pi-1-Ai(P0Pi)P0Pi-1,i=1,2,,where Qi(t) denotes a

Numerical examples

In this section, we give two numerical examples to illustrate our results obtained in Sections 2. Here we denote the exact solution of original systems (1.2) and the numerical solution of the regularization systems by x(t) = (x1(t), x2(t), x3(t))T and x˜(t)=(x˜1(t),x˜2(t),x˜3(t))T, respectively.

Example 1

We consider DAEs100010000x1x2x3+11tett+100t20x1x2x3=q1q2q3,t[a,b].It is a Hessenberg index 3 DAEs. The regularization systems is1+εε0εet1+ε(t+1)00εt20x1x20+11tett+100t20x1x2x3=q1q2q3,t[a,b].If we

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Project supported by the National Natural Science Foundation of China (No. 10371056).

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