## Asymptotic Estimation of the Trapezoidal Method for a Class of Neutral Differential Equation with Variable Delay

Zhang Gengen,1, Wang Wansheng3, Xiao Aiguo2

 基金资助: 国家自然科学基金.  11701110

 Fund supported: the NSFC.  11701110

Abstract

In this paper, we investigate the stability of the trapezoidal method for a class of neutral differential equation with variable delay and obtain the asymptotic estimation of numerical solution with the aid of a functional inequality. The asymptotic estimation is more accurate than asymptotic stability in describing the behaviours of the numerical solution, and gives the upper bound estimates of the numerical solution for the nonstable case.

Keywords： Neutral delay differential equation ; Trapezoidal method ; Asymptotic estimation ; Asymptotic stability

Zhang Gengen, Wang Wansheng, Xiao Aiguo. Asymptotic Estimation of the Trapezoidal Method for a Class of Neutral Differential Equation with Variable Delay. Acta Mathematica Scientia[J], 2019, 39(3): 560-569 doi:

## 2 离散格式

$$$y'(t ) = a(t)y(t)+b(t)y(\theta(t))+ c(t)y^\prime(\phi(t)), \ t\geq t_0,$$$

$$$y(t_{n+1}) -y(t_{n}) = \int\nolimits_{t_{n}}^{t_{n+1}} a(s)y(s){\rm d}s + \int\nolimits_{t_{n}}^{t_{n+1}} b(s)y(\theta(s)){\rm d}s + \int\nolimits_{t_{n}}^{t_{n+1}} c(s)y^\prime(\phi(s)){\rm d}s,$$$

$$$y_{n+1} = R_n y_n + h S_n(\beta _n y_{[\overline{ \theta }_n]} + \alpha_n y_{[\overline{ \theta }_n]+1}) + T_n (\hat{\beta_n} y_{[\overline{ \phi }_n]} + \hat{\alpha_n} y_{[\overline{ \phi }_n]+1}) ,$$$

$$$R_n = \frac{2+h a_n}{2-h a_{n+1}}, \quad S_n = \frac{2b_n}{2-h a_{n+1}}, \quad T_n = \frac{2c_n }{2-h a_{n+1}} .$$$

## 3 渐近估计

$$$|S_n |h (|\beta_n| \varrho_{[\overline{ \theta }_n]} + |\alpha_n | \varrho_{[\overline{ \theta }_n]+1}) + | T_n | (|\hat{\beta_n}| \varrho_{[\overline{ \phi }_n]} + |\hat{\alpha_n}| \varrho_{[\overline{ \phi }_n]+1}) \leq (1-|R_n|)\varrho_n,$$$

$$$\begin{array}{lll} \tilde{S}: = \sup\limits_{n\in Z^+}(\left| S_n\right|) < \infty, \; \; \tilde{R}: = \sup\limits_{n\in Z^+}{\left| R_n\right|}< 1, \; \; \tilde{T}: = \sup\limits_{n\in Z^+}(\left| T_n\right|) < \infty, \\ \eta: = \sup\limits_{n\in Z^+}(| \alpha_n|+| \beta_n|) <\infty, \quad \tilde{\eta}: = \sup\limits_{n\in Z^+}(| \hat{\alpha_n}|+|\hat{\beta_n} |)<\infty , \end{array}$$$

$$$\tilde{\gamma}: = \frac{h\tilde{S}\eta +\tilde{T} \tilde{\eta}}{1-\tilde{R}}.$$$

$$$\varphi(\tau(t)) = \nu \varphi(t), \quad \nu = \tau'(t_0), \ t\geq t_0.$$$

$$$\varrho_n = \left\{ \begin{array}{ll} (\varphi(t_0 +(n-k^\star )h) )^{-\log_{\nu} \tilde{\gamma}}, \quad &{\hbox{当}} \; \; \tilde{\gamma} \geq1, \\ (\varphi(t_0 +(n+k^\star )h) )^{-\log_{\nu} \tilde{\gamma}}, &{\hbox{当}} \; \; 0< \tilde{\gamma}< 1, \end{array} \right.$$$

令$\overline{ \tau }_n = (\tau_n- t_0)/h$.$\tilde{\gamma}\geq 1$时, $\varrho_n$是一个递增序列,则利用(3.4), (3.6)和(3.7)式得

$$$\left|z_{n^{\star}} \right|\leq B_m \left( \frac{\varrho_{\sigma_m}}{\varrho_{n^{\star}}} \prod\limits_{l = \sigma_m}^{n^{\star}-1}{\left| R_l \right|} + \frac{1}{\varrho_{n^{\star}}} \sum\limits_{i = \sigma_m}^{n^{\star}-1}(1-| R_i|)\varrho_i \prod\limits_{l = i+1}^{n^{\star}-1}{| R_l |} \right).$$$

$\tilde{\gamma} \geq 1$时, $\varrho_n$是一个递增序列且$\triangle \varrho_i\geq 0$,故$\left|z_{n^{\star}} \right|\leq B_m$.$0 < \tilde{\gamma} < 1$时, $\varrho_n$是一个递减序列且$\triangle \varrho_i\leq 0$, $\triangle \varrho_i\geq \triangle \varrho_{i-1}$,利用中值定理及$\varphi'(t)$单调递减性得

(ⅲ)假设$R_{n^{\star}-1} \neq 0$且存在$k \in [\sigma_m, n^\star-2] \cap Z^+$使得$R_k = 0$.定义$\sigma^\star : = \sup(k, k \in [\sigma_m, n^{\star}-2] \cap Z^+ $$R_n = 0 ),则 对方程(3.9)两边同乘 \prod\limits_{l = \sigma^{\star}+1}^{n}\frac{1}{R_l} ,且从 \sigma^{\star}+1$$ n^{\star}-1$求和得

$\sigma^\star$的定义知$R_{\sigma^\star} = 0$,则由情况(i)结论得到估计式

$\sigma_m \geq \nu^{-m}(\sigma_0- \frac{1+\nu}{1-\nu})$及相应的无穷积收敛.所以当$m \rightarrow \infty$时,序列${B_m}$一致有界,即(3.8)式成立.

$$$y_{n+1} = R_n y_n + (h S_n \beta_n + T_n \hat{\beta_n}) y_{\lfloor\overline{ \theta }_n\rfloor} + (h S_n \alpha_n + T_n \hat{\alpha_n}) y_{\lfloor\overline{ \theta }_n\rfloor+1}),$$$

$$$\begin{array}{lll} \tilde{R}: = \sup\limits_{n\in Z^+}{\left| R_n\right|}< 1, \; \; S_{\alpha}: = \sup\limits_{n\in Z^+}(| h S_n \beta_n + T_n \hat{\beta_n} |) < \infty, \\ S_{\beta}: = \sup\limits_{n\in Z^+}(| h S_n \alpha_n + T_n \hat{\alpha_n}|) < \infty, \end{array}$$$

$$$\gamma: = \frac{ S_{\alpha} +S_{\beta}}{1-\tilde{R}},$$$

$$$y_n = O\left((\varphi(n))^{-\log_{\nu}\gamma}\right), \quad {\hbox{当}}\ n\rightarrow \infty .$$$

$$$y_{n+1} = R_n y_n + (h S_n \beta_n + T_n \hat{\beta_n}) y_{[ \overline{\varepsilon}_n ]} + (h S_n \alpha_n + T_n \hat{\alpha_n}) y_{[ \overline{\varepsilon}_n ]+1},$$$

## 4 数值算例

$$$y'(t) = -10y(t)- 15y(\frac{t}{2})+ 0.6 y'(\frac{t}{2}), \; \; y(0) = 1, \; \; t \geq 0.$$$

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