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Analysis of backward differentiation formula for nonlinear differentialalgebraic equations with 2 delays
SpringerPlus volume 5, Article number: 1013 (2016)
Abstract
This paper is concerned with the backward differential formula or BDF methods for a class of nonlinear 2delay 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.
Background
For a system of differentialalgebraic equations (DDAEs) (Brenan et al. 1996),
F and y are vector valued and \(\partial F/\partial y'\) may be singular. In some cases, time delays appear in variables of unknown functions so that the differentialalgebraic equations (DAEs) are converted to delay differentialalgebraic equations (DDAEs) (Ascher and Petzold 1995),
where F and y are vector valued, \(\tau >0\) is a constant, \(\partial F/\partial y'\) may be singular. If \(y'(t\tau )\) does not vanish, it is actually called neutral delay differentialalgebraic equations (NDDAEs), otherwise it is called delay differentialalgebraic equations (DDADs). In 1995, authors in Ascher and Petzold (1995) discussed the convergence of BDF methods and Runge–Kutta methods solving initialvalue differentialalgebraic equations of retarded and neutral types, corresponding to the structure of Hessenberg forms; in 1997, authors Zhu and Petzold (1997) considered the asymptotic stability of linear constant coefficient differentialalgebraic equations and obtained numerical results on θmethods, Runge–Kutta methods and linear multistep methods to these systems. In 1998, Zhu and Petzold (1998) got further results on stability of Hessenberg DDAEs of retarded or neutral type. In 2005, stability of Rosenbrock methods for neutral delay differentialalgebraic equatuons was discussed in Zhao and Xu (2005). Earlier, authors of Torelli (1989), Mechee et al. (2013) were interested in the numerical treatments on delay differential equations which are delay differentialalgebraic equations with \(\partial F/\partial y'\) nonsingular. Authors in Fan et al. (2013), Liu et al. (2014) gave criteria for stability of neutral delay differentialalgebraic 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 differentialalgebraic 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 twostep BDF methods are stable and asymptotically stable.
Asymptotic behavior of 2delay differentialalgebraic equations
Now we consider the following nonlinear system of delay differentialalgebraic equations,
According to Ascher and Petzold (1995) the assumption that \(\varphi _{v}\) 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 \([2\tau , 0]\). So in fact, we will investigate (1) and (2) by following nonlinear system of delay differentialalgebraic equations,
and its perturbed equations
From results of Torelli (1989), we hope the estimations on \(u(t)\tilde{u}(t)\) and \(v(t)\tilde{v}(t)\) satisfy
In practice, the following definition is to be considered.
Definition 1
Liu et al. (2014) System (1)–(2) is said to be stable, if the follow inequalities are satisfied,
where M > 0 is a constant,
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 2step 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 2step BDF methods
Firstly, the 2step 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 kstep BDF, which has order p = k,
this can be written in scaled form where \(\alpha _{0}=1\),
here we apply 2step BDF, the formula can be written as
For the initial value problem of the ordinary differential equations
The 2setp BDF methods can be written as:
where \(x_{n}\sim x(t_{n})\), \(h>0\) is the step size. To solve (5)–(8) and (9)–(12) by (18)–(19), we get
The perturbations of (20)–(21) are
If the step size is \(h>0\) and \(t_{n}=nh\) and the numerical approximations are \(u_{n}\approx u(t_{n})\), it should be note that \(t_{i}\tau\) may not be a grid point \(t_{j}\) for any j. Then a function interpolation is needed so that
here \(0<\delta _{u}, \delta _{v}<1\), 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 \(u_{n+im}, v_{n+jm}\) are on grid points or obtained by interpolations.
The stability of 2step BDF methods
Let \(u_{\tau }=u(t\tau ), v_{\tau }=v(t\tau )\). We require that f, \(\varphi\) in (5), (6), (9), and (10) satisfy the following Lipschitz conditions (1)–(4):

(1)
$$\langle f(t,u,u_{\tau },v,v_{\tau })f(t,\tilde{u},u_{\tau },v,v_{\tau }), u\tilde{u}\rangle \le \sigma (t)\Vert u\tilde{u}\Vert ^{2},$$

(2)
$$\begin{aligned}&\Vert f(t,u,u_{\tau },v,v_{\tau })f(t,u,\tilde{u}_{\tau },v,v_{\tau })\Vert \le \gamma _{1}(t)\Vert u_{\tau }\tilde{u}_{\tau }\Vert ,\\&\Vert f(t,u,u_{\tau },v,v_{\tau })f(t,u,u_{\tau },\tilde{v},v_{\tau })\Vert \le \gamma _{2}(t)\Vert v\tilde{v}\Vert ,\\&\Vert f(t,u,u_{\tau },v,v_{\tau })f(t,u,u_{\tau },v,\tilde{v}_{\tau })\Vert \le \gamma _{3}(t)\Vert v_{\tau }\tilde{v}_{\tau }\Vert , \end{aligned}$$

(3)
\(\varphi _{v}\) is nonsingular, so that for g(u, v) in (3), there exist \(L>0\), \(K>0\), such that
$$\begin{aligned}&\Vert g(u, v)g(\tilde{u}, v)\Vert \le L\Vert u\tilde{u}\Vert ,\quad \Vert g(u, v)g(u, \tilde{v})\Vert \le K\Vert v\tilde{v}\Vert ,\\&\sigma (t)<0,\quad \frac{1}{2}\sigma _{1}(t)+\gamma _{1}(t)+(L+K)\gamma _{2}(t)+(L+K)\gamma _{3}(t)\le \sigma (t),\quad t>0, \end{aligned}$$where \(\sigma _{1}(t)\) is an increasing function described in the following Theorem 1.
Note: \(\sigma _(t)<0\) means the right side of function in condition (1) is negative, examples in the last section show the situation exists.

(4)
The Frechet derivatives of g(u, v) with regard to u, v, \(\frac{\partial g}{\partial u}\), \(\frac{\partial g}{\partial v}\) exist in the product space \(\mathbb {R}^{d}\times \mathbb {R}^{d}\), \(\frac{\partial g}{\partial v}\) is continuous, \((\frac{\partial g}{\partial u})^{1}\) exists, and
$$\begin{aligned} \sup _{u,v\in \mathbb {R}^{d}}\left\ \left( \frac{\partial g}{\partial v}\right) ^{1}\left( \frac{\partial g}{\partial u}\right) \right\ =L<\infty , \end{aligned}$$where \(u=(u_{1},u_{2},\ldots ,u_{d})^{T}\), \(v=(v_{1},v_{2},\ldots ,v_{d})^{T}\), \(\langle u,v\rangle =\sum ^{d}_{i=1}u_{i}v_{i}\), \(\Vert u\Vert ^{2}=\langle u,u\rangle\),
$$\left\ \frac{\partial g}{\partial u}\right\ =\sup _{\omega \in \mathbb {R}^{d}, \Vert \omega \Vert =1}\left\ \left( \frac{\partial g}{\partial u}\right) \omega \right\ .$$
Here \(\sigma (t), \sigma _{1}(t), \gamma _{i}(t),i=1,2,3, t>0\) are increasing functions defined on time. The Frechet derivatives are described as follows, If
\(f_{j}(x)\;(j=1,2,\ldots ,m)\) has firstorder 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 \(\Phi\), \(\tilde{\Phi }\), and each step \(h>0\), the solution sequences \(\{u_{n}, v_{n}\}\), \(\{\tilde{u}_{n}, \tilde{v}_{n}\}\) for (5)–(8) and (9)–(12) in which \(f, \varphi\) 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 2step BDF methods are stable for DDAEs if \(f, \varphi\) satisfy conditions (1)–(4) and
Note: it seems more natural if \(\Vert f(t,u,u_{\tau },v,v_{\tau })f(t,\tilde{u},u_{\tau },v,v_{\tau }) \Vert \le \sigma _{1}(t)\Vert u\tilde{u}\Vert\) is true, 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.
Proof
Let \(\bar{V}_{n}=u_{n}\tilde{u}_{n}\). Substituted into (20) and (24),
An inner product of (28) with \(\bar{V}_{n+1}=u_{n+1}\tilde{u}_{n+1},\)
apply Schwartz theorem and condition (1)–(2), we obtain
Assume that \(\Vert \bar{V}_{n+2}\Vert \ne 0\) (otherwise no perturbations), note (3) and condition (3), (4), we conclude
(29) divided by \(\Vert \bar{V}_{n+2}\Vert\), and note the consistency of the initial value function, we get
where \(\omega (t_{n+2})=\gamma _{1}(t_{n+2})+K\gamma _{2}(t_{n+2})+L\gamma _{3}(t_{n+2}), n=0, 1, 2, \ldots , \quad \bar{V}_{0}, \bar{V}_{1}\) are two initial values for 2step BDF methods where \(\Vert V_{0}\Vert \le \max \limits _{2\tau \le t\le 0}\Vert \Phi (t)\tilde{\Phi }(t)\Vert\), \(\bar{V}_{1}\) is evaluated by using Implicit Euler method
and conditions (1)–(3), by a simple induction, we get
and with the condition of this theorem, yields
hence, as \(n=0\),
with condition (3) and (33) and the incretion of \(\sigma (t), \sigma _{1}(t), \gamma _{i}(t),i=1,2,3\), we get
as \(n=1\), we evaluate \(\Vert \bar{V}_{3}\Vert\) in (31) in terms of \(\Vert \bar{V}_{2}\Vert , \Vert \bar{V}_{2}\bar{V}_{1}\Vert\) in the following.
then, from condition (1) and (34)
substitute (37) into (31), take n = 2, also note the incretion of \(\sigma (t), \sigma _{1}(t), \gamma _{i}(t),i=1,2,3\), we get the estimation of \(\Vert \bar{V}_{3}\Vert\) in the following,
note condition (3), (33) and (36), \(\bar{V}_{3m}, \bar{V}_{32m}\) are initial valued functions with (33) or (36) satisfied too, so we get
similarly, when n = 2, 3, 4,…, by iteration,
applying mathematical induction, we conclude it is true for all \(n\ge 0\). As for \(\Vert v_{n}\tilde{v}_{n}\Vert\), just see (28)
\(\square\)
The asymptotic stability of 2step BDF methods
Now we give the following definition.
Definition 3
The delay differentialalgebraic equations (5)–(8) are asymptotically stable if and only if for every consistent initial value functions \(\Phi (t)\), \(\tilde{\Phi }(t)\), solutions \(\{u(t), v(t)\}\), \(\{\tilde{u}(t), \tilde{v}(t)\}\) satisfy
Theorem 2
If \(f, \varphi\) satisfy conditions (1)–(4) and the following (\(3'\))
Then the 2step BDF methods are asymptotically stable for DDAEs Here \(\sigma (t), \sigma _{1}(t), \gamma _{i}(t),i=1,2,3, t\ge 0\) are increasing functions. Note: The system is stable if \(q=1\) while q strictly less than 1 is required for asymptotic stability.
Proof
Let \(V_{n}=\Vert u_{n}\tilde{u}_{n}\Vert\), from (31) (35) (38), we have
Let \(0\le n\le 2m1\) in the above inequality, we get
Note condition (3′), there is \(0<p<1\) such that
Therefore, when \(0 \le n\le 2m1\)
For the case \(n=2m\)
As indicated above,
For the case \(2rm\le n \le 2(r+1)m1\), it can be shown by induction that
When \(r\rightarrow \infty ,\quad n\rightarrow \infty\)
Thus,
\(\square\)
Numerical examples
First, we give an example for Theorem 1.
Example 1
Let u(t), \(v(t)\in \mathbb {R}\), \(f:\mathbb {R}\times \mathbb {R}\times \mathbb {R}\times \mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\), \(g:\mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\).
where \(\sigma (t)\), \(\sigma _{1}(t)\), \(P_{1}(t)\), \(P_{2}(t)\), \(P_{3}(t)\) are polynomials of t, \(u(t\tau )=0\). Condition (1)–(4) say if,
then (40)–(41) is stable. In fact,

(1)
$$\begin{aligned}&\langle f(t,u,u_{\tau },v,v_{\tau })f(t,\tilde{u},u_{\tau },v,v_{\tau }),u\tilde{u}\rangle =\sigma (t)u\tilde{u}^{2},\\&f(t,u,u_{\tau },v,v_{\tau })f(t,\tilde{u},\tilde{u}_{\tau },\tilde{v},\tilde{v}_{\tau })\le \sigma _{1}(t)u\tilde{u} \end{aligned}$$

(2)
$$\begin{aligned}&f(t,u,u_{\tau },v,v_{\tau })f(t,u,\tilde{u}_{\tau },v,v_{\tau })=P_{1}(t)\cdot f_{1}(u_{\tau })f_{1}(\tilde{u}_{\tau })\le L_{1}\gamma _{1}(t)u_{\tau }\tilde{u}_{\tau },\\&f(t,u,u_{\tau },v,v_{\tau })f(t,u,u_{\tau },\tilde{v},v_{\tau })=P_{2}(t)\cdot f_{2}(v)f_{2}(\tilde{v})\le L_{2}\gamma _{2}(t)v\tilde{v},\\&f(t,u,u_{\tau },v,v_{\tau })f(t,u,u_{\tau },v,\tilde{v}_{\tau })=P_{3}(t)\cdot f_{3}(v_{\tau })f_{3}(\tilde{v}_{\tau })\le L_{3}\gamma _{3}(t)v_{\tau }\tilde{v}_{\tau }. \end{aligned}$$
Let \(\tilde{u}=u+h\), \(\tilde{v}=v+k\), using Taylor’s formula,
where \(\tilde{g}(u,v)=g(u+\theta h,v+\theta k)\), \(0<\theta <1\).
If \(g(u,v)=g(\tilde{u},\tilde{v})=0\), (46) results in
If (42) is true, then
together with (43), (44), (45), (47), we know that all the conditions of Theorem (16) are satisfied. For example,
and checking the note from Theorem 1, we just take
It can be easily verified by Lagrange mean value theorem that the Lipschitz constant,
Let
then
we get \(L=1\) and
Conditions (1)–(4) are satisfied in the above example. Example 2 and Example 3 show the stable results, while Example 4 shows unstable results.
Example 2
Let \(x(t), y(t)\in \mathfrak {R}\), \(f: \mathfrak {R}\times \mathfrak {R}\times \mathfrak {R}\times \mathfrak {R}\times \mathfrak {R}\rightarrow \mathfrak {R}\), \(\varphi : \mathfrak {R}\times \mathfrak {R}\times \mathfrak {R}\rightarrow \mathfrak {R}\).
Take \(\tau =1\), then \(\sigma (t)=(2+\frac{1}{2}e^{2t})<0, \quad L=\frac{1}{2},\quad K=\frac{1}{2}\), \(\sigma _{1}(t)=2\)
Therefore,
The above results show that all the stability conditions are satisfied, so 2step 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 with different step sizes. Table 2 shows numerical results made by implicit Euler method and 2step BDF method at some time. It can be seen by comparison that BDF method converges much faster than implicit Euler method.
Example 3
Let \(x(t),\quad y(t)\in \mathfrak {R}\quad f: \mathfrak {R}\times \mathfrak {R}\times \mathfrak {R}\times \mathfrak {R}\times \mathfrak {R}\rightarrow \mathfrak {R},\quad g: \mathfrak {R}\times \mathfrak {R}\rightarrow \mathfrak {R}\)
Here we take \(\tau =2\), its initial functions are
Obviously,
Then it can be found by a simple computation and without losing generality, by taking supremum values
they satisfy
The above results show that all the stability conditions are satisfied, so 2step 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.
Example 4
Here \(u(t)=(x_{1}(t), x_{2}(t), x_{3}(t))^{T}, v(t)=y(t)\). By simple calculation, we get,
For the initial data
The solution is
Obviously, this solution is not stable. In fact, we could not find any \(\sigma (t)\) satisfies
Thus conditions of Theorem 1 are not valid.
Conclusions and notes
While investigating nonlinear 2delayed differentialalgebraic equations, we get two sufficient conditions for the stability and asymptotic stability of 2step 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 differentialalgebraic equations with multidelays 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 \(\Vert f(t,u,u_{\tau },v,v_{\tau })f(t,\tilde{u},u_{\tau },v,v_{\tau }) \Vert \le \sigma _{1}(t)\Vert u\tilde{u}\Vert\), but we find results can also be true and the proofs are analogous.
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Acknowledgements
The research is supported by the scientific Computing key Laboratory of Shanghai University and the Shanghai Natural Science Foundation, No. 15ZR1431200.
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The author declares that he has no competing interests.
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Sun, L. Analysis of backward differentiation formula for nonlinear differentialalgebraic equations with 2 delays. SpringerPlus 5, 1013 (2016). https://doi.org/10.1186/s400640162422z
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DOI: https://doi.org/10.1186/s400640162422z