On a regularization of index 3 differential-algebraic equations☆
Introduction
Normal differential-algebraic equations (DAEs) are singular ordinary differential equations (ODEs)where f is smooth and the partial Jacobian f′y(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 assubject to the boundary conditionsHere, 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 to
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: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 , respectively. Example 1 We consider DAEsIt is a Hessenberg index 3 DAEs. The regularization systems isIf we
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Project supported by the National Natural Science Foundation of China (No. 10371056).