Analysis of backward differentiation formula for nonlinear differential-algebraic equations with 2 delays

This paper is concerned with the backward differential formula or BDF methods for a class of nonlinear 2-delay differential algebraic equations. We obtain two sufficient conditions under which the methods are stable and asymptotically stable. At last, examples show that our methods are true.

of Torelli (1989), Mechee et al. (2013) were interested in the numerical treatments on delay differential equations which are delay differential-algebraic equations with ∂F /∂y ′ nonsingular. Authors in Fan et al. (2013), Liu et al. (2014) gave criteria for stability of neutral delay differential-algebraic equations geometrically and obtained stable regions over which numerical methods could be used effectively. Among these results, there are few achievements on nonlinear systems. In fact, the solution of nonlinear system depends on a nonlinear manifold of a product space and on consistent initial valuedvectors over a space of continuous functions so that research on nonlinear DDAEs is more complicated and still remains investigated.
Authors in Kuang and Cong (2005), Ascher and Petzold (1998) denote that numerical approaches for the solution of differential-algebraic equations (DAEs) can be divided roughly into two classes. One is direct discretizations of the given system, the other is involving a reformulation, combined with a discretization. Practically all the winning methods have stiff decay. For initial value DAEs which are cumbersome and especially for DAEs whose underlying ODEs are stiff, the backward differentiation formulae (BDF) and Radau collocation methods are the overall methods of choice.
In this paper, we investigate a class of nonlinear DDAE system, and show the conditions under which two-step BDF methods are stable and asymptotically stable.

Asymptotic behavior of 2-delay differential-algebraic equations
Now we consider the following nonlinear system of delay differential-algebraic equations, According to Ascher and Petzold (1995) the assumption that ϕ v is nonsingular allows one to solve the constraint equations (2) for v(t) using the implicit theorem, yielding by substituting (3) into (1) we obtain the DODE Thus, the DDAEs (1) and (2) are stable if the DODE (4) is stable. Note that if all the delay terms are present in this retarded DODE, then the initial conditions need to be defined for t on [−2τ , 0]. So in fact, we will investigate (1) and (2) by following nonlinear system of delay differential-algebraic equations,

and its perturbed equations
From results of Torelli (1989), we hope the estimations on u(t) −ũ(t) and v(t) −ṽ(t) satisfy In practice, the following definition is to be considered. To study the stability of DDAE (1)-(2), we can investigate equations (5)-(8) and the perturbations (9)-(12). Ascher and Petzold (1995) showed that under some conditions the analytical solutions of the system is stable and asymptotically stable. In the next section, we will discuss the stability behavior of 2-step BDF methods for a class of the system based on the assumption that the analytical solution exists uniquely and stable.

The stability and asymptotic stability of 2-step BDF methods
Firstly, the 2-step BDF methods are introduced as follows.

Backward differentiation formula
For the differential equation the Backward Differentiation Formula or BDF methods are derived by differentiating the polynomial which interpolates past values of y, each step is h, and setting the derivative at t n to f (t n , y n ). This yields the k-step BDF, which has order p = k, this can be written in scaled form where α 0 = 1, here we apply 2-step BDF, the formula can be written as For the initial value problem of the ordinary differential equations The 2-setp BDF methods can be written as: where x n ∼ x(t n ), h > 0 is the step size. To solve (5)-(8) and (9)-(12) by (18)-(19), we get The perturbations of (20)-(21) are u n =φ 1 (t n ),ṽ n =ψ 1 (t n ), −m ≤ n ≤ 0, (27) u n =φ 2 (t n ),ṽ n =ψ 2 (t n ), −2m ≤ n ≤ −m, (mh = τ , m ≥ 1).
If the step size is h > 0 and t n = nh and the numerical approximations are u n ≈ u(t n ), it should be note that t i − τ may not be a grid point t j for any j. Then a function interpolation is needed so that here 0 < δ u , δ v < 1, the convergence order of interpolation is 2 and the local truncation error of the method is 3, then the convergence order of the iteration by BDF method is two (Kuang and Cong 2005). For simplicity, we just consider u n+i−m , v n+j−m are on grid points or obtained by interpolations.
(4) The Frechet derivatives of g (u, v) with regard to u, v, ∂g ∂u , ∂g ∂v exist in the product Here |σ (t)|, σ 1 (t), γ i (t), i = 1, 2, 3, t > 0 are increasing functions defined on time. The Frechet derivatives are described as follows, If f j (x) (j = 1, 2, . . . , m) has first-order continuous partial derivative at x = x 0 , then the Frechet derivative F ′ (x) can be expressed by the following matrix: Definition 2 A numerical method for solving DDAEs is called stable, if for every consistent initial value functions , ˜ , and each step h > 0, the solution sequences {u n , v n }, {ũ n ,ṽ n } for (5)- (8) and (9)-(12) in which f , ϕ satisfy conditions (1)-(4), satisfy for some M > 0. Now the sufficient condition with which the DDAEs are stable is as follows.
Theorem 1 The 2-step BDF methods are stable for DDAEs if f , ϕ satisfy conditions (1)-(4) and but we find proofs are analogous with this condition but only cumbersome and results are true without this assumption throughout the discussion in this paper.

Numerical examples
First, we give an example for Theorem 1.
Take τ = 1, then σ (t) = (−2 + 1 2 e −2t ) < 0, L = 1 2 , K = 1 2 , σ 1 (t) = 2 Therefore, The above results show that all the stability conditions are satisfied, so 2-step BDF methods for the system are stable and asymptotically stable. This can be seen in the following graph (Fig. 1). Table 1 lists errors between numerical solutions and the exact solutions actually .
Here we take τ = 2, its initial functions are Obviously, Then it can be found by a simple computation and without losing generality, by taking supremum values 10 6 10 −6 9.21 × 10 −12 3.86 × 10 −11 they satisfy The above results show that all the stability conditions are satisfied, so 2-step BDF methods for the system are stable and asymptotically stable. The simple illustrations are shown in the following graph (Fig. 2) and we can check it by errors of the solutions listed in the following Table 3.

For the initial data
The solution is Obviously, this solution is not stable. In fact, we could not find any σ (t) satisfies Thus conditions of Theorem 1 are not valid.

Conclusions and notes
While investigating nonlinear 2-delayed differential-algebraic equations, we get two sufficient conditions for the stability and asymptotic stability of 2-step BDF methods and think about how to check the conditions with some example. Although it is quite an early stage, the discussion is a useful enlightenment for differential-algebraic equations with multi-delays in the future. Note the Lipschitz conditions play a key role in this research. Apparently the second inequality in condition (1) seems more nature with the form �f (t, u, u τ , v, v τ ) − f (t,ũ, u τ , v, v τ )� ≤ σ 1 (t)�u −ũ�, but we find results can also be true and the proofs are analogous.