G-stability one-leg hybrid methods for solving DAEs

P Agarwal,Fatma M Yousry,Iman H Ibrahim

doi:10.1186/s13662-019-2019-2

Abstract

This paper introduces the solution of differential algebraic equations using two hybrid classes and their twin one-leg with improved stability properties. Physical systems of interest in control theory are sometimes described by systems of differential algebraic equations (DAEs) and ordinary differential equations (ODEs) which are zero index DAEs. The study of the first hybrid class includes the order of convergence, A(α)-stability, stability regions, and G-stability for its one-leg twin in two cases: for step (k=1) and steps (k=2). For the second class, G-stability of its one-leg twin is studied in two cases: for steps (k=2) and steps (k=3). Test problems are introduced with different step size at different end points.

Highlights

The one-leg twin of the first class is derived and its G-stability is discussed in Sect
Consider the initial value problems of the form f x (t); x(t); t = 0; x(t0) – a = 0, t ∈ [t0; T], (1)where a ∈ Rm is a consistent initial value for (1) and the function f : Rm × Rm × [t0; T] → Rm is assumed to be sufficiently smooth
Some systems can be reduced to an ordinary differential equations (ODEs) system, which are zero index differential algebraic equations (DAEs), and can be solved by numerical ODE methods after reduction

Summary

Introduction

The one-leg twin of the first class is derived and its G-stability is discussed in Sect. 4. The one-leg twin of the second class [14] is derived and its G-stability is discussed in Sect.

Results

Conclusion