Index definitions for nonlinear IAEs and DAEs: new classifications and numerical treatments

TL;DR

This paper extends the concept of index for nonlinear differential and integral algebraic equations, addressing the challenges of defining and computing these indices in the nonlinear context and their implications for numerical methods.

Contribution

It introduces a generalized index classification for nonlinear IAEs and DAEs, highlighting the dependence on exact solutions and guiding numerical solution approaches.

Findings

Generalized index definitions for nonlinear IAEs and DAEs

Dependence of nonlinear index on exact solutions

Implications for numerical methods like Runge-Kutta

Abstract

The definition of index for differential algebraic equations (DAEs) or integral algebraic equations (IAEs) in the linear case (time variable) depends only on the coefficients of integrals or differential operators and the coefficients of the unknown functions. Is this possible for the nonlinear case? In this paper we answer this question. In this paper, we generalize the index notion for the nonlinear case. One of the difficulties for nonlinear case, is its dependence on the exact solution which motivates us to give an important warning to whom want to solve DAEs using numerical methods such as Runge-Kutta, multistep or collocation methods.