# 时滞微分方程

${\displaystyle {\frac {d}{dt}}x(t)=f(t,x(t),x_{t}),}$

## 例子

• 连续时滞微分方程
${\displaystyle {\frac {d}{dt}}x(t)=f\left(t,x(t),\int _{-\infty }^{0}x(t+\tau )d\mu (\tau )\right)}$
• 离散时滞微分方程
${\displaystyle {\frac {d}{dt}}x(t)=f(t,x(t),x(t-\tau _{1}),\ldots ,x(t-\tau _{n}))}$ for ${\displaystyle \tau _{1}>\ldots >\tau _{n}\geq 0}$.
• 离散时滞线性方程
${\displaystyle {\frac {d}{dt}}x(t)=A_{0}x(t)+A_{1}x(t-\tau _{1})+\ldots +A_{m}x(t-\tau _{m})}$

${\displaystyle {\frac {d}{dt}}x(t)=ax(t)+bx(\lambda t),}$

## 时滞微分方程求解

${\displaystyle {\frac {d}{dt}}x_{t}=f(x(t),x(t-\tau ))}$

${\displaystyle {\frac {d}{dt}}\psi (t)=f(\psi (t),\phi (t-\tau ))}$,

${\displaystyle \psi (0)=\phi (0)}$. 这样就可以利用前面区间的解作为非齐次项一步步求得整个区间上的解. 在实际的计算中, 初值问题通常采用数值计算.

### 例子

${\displaystyle x(t)=a\int _{s=0}^{t}\phi (t-\tau )\,dt+C}$,

## 简化为常微分方程(ODE)

• 例 1 考虑方程
${\displaystyle {\frac {d}{dt}}x(t)=f\left(t,x(t),\int _{-\infty }^{0}x(t+\tau )e^{\lambda \tau }d\tau \right).}$

${\displaystyle {\frac {d}{dt}}x(t)=f(t,x,y),\quad {\frac {d}{dt}}y(t)=x-\lambda y.}$
• 例 2 方程
${\displaystyle {\frac {d}{dt}}x(t)=f\left(t,x(t),\int _{-\infty }^{0}x(t+\tau )\cos(\alpha \tau +\beta )d\tau \right)}$

${\displaystyle {\frac {d}{dt}}x(t)=f(t,x,y),\quad {\frac {d}{dt}}y(t)=\cos(\beta )x+\alpha z,\quad {\frac {d}{dt}}z(t)=\sin(\beta )x-\alpha y,}$

${\displaystyle y=\int _{-\infty }^{0}x(t+\tau )\cos(\alpha \tau +\beta )d\tau ,\quad z=\int _{-\infty }^{0}x(t+\tau )\sin(\alpha \tau +\beta )d\tau .}$

## 特征方程

${\displaystyle {\frac {d}{dt}}x(t)=A_{0}x(t)+A_{1}x(t-\tau _{1})+\ldots +A_{m}x(t-\tau _{m})}$

${\displaystyle det(-\lambda I+A_{0}+A_{1}e^{-\tau _{1}\lambda }+\ldots +A_{m}e^{-\tau _{m}\lambda })=0}$.

${\displaystyle {\frac {d}{dt}}x(t)=-x(t-1).}$

${\displaystyle -\lambda -e^{-\lambda }=0.\,}$

${\displaystyle \lambda =W_{K}(-1)}$,

## 注释

1. ^ Michiels, Niculescu, 2007 Chapter 1
2. ^ Jarlebring 2008 Chapter 2