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

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.