# 顯式和隱式方法

${\displaystyle Y(t+\Delta t)=F(Y(t))\,}$

${\displaystyle G{\Big (}Y(t),Y(t+\Delta t){\Big )}=0\qquad (1)\,}$

${\displaystyle Y(t+\Delta t)=F(Y(t+\Delta t))+G(Y(t)),\,}$

## 用前向歐拉方法和後向歐拉方法的說明

${\displaystyle {\frac {dy}{dt}}=-y^{2},\ t\in [0,a]\quad \quad (2)}$

${\displaystyle \left({\frac {dy}{dt}}\right)_{k}\approx {\frac {y_{k+1}-y_{k}}{\Delta t}}=-y_{k}^{2}}$

${\displaystyle y_{k+1}=y_{k}-\Delta ty_{k}^{2}\quad \quad \quad (3)\,}$

${\displaystyle {\frac {y_{k+1}-y_{k}}{\Delta t}}=-y_{k+1}^{2}}$

${\displaystyle y_{k+1}+\Delta ty_{k+1}^{2}=y_{k}}$

${\displaystyle y_{k+1}={\frac {-1+{\sqrt {1+4\Delta ty_{k}}}}{2\Delta t}}.\quad \quad (4)}$

${\displaystyle {\frac {y_{k+1}-y_{k}}{\Delta t}}=-{\frac {1}{2}}y_{k+1}^{2}-{\frac {1}{2}}y_{k}^{2}}$

${\displaystyle y_{k+1}+{\frac {1}{2}}\Delta ty_{k+1}^{2}=y_{k}-{\frac {1}{2}}\Delta ty_{k}^{2}}$

${\displaystyle {\frac {dy}{dt}}=y-y^{2},\ t\in [0,a]\quad \quad (5)}$

${\displaystyle \left({\frac {dy}{dt}}\right)_{k}\approx y_{k+1}-y_{k}^{2},\ t\in [0,a]}$

${\displaystyle y_{k+1}={\frac {y_{k}(1-y_{k}\Delta t)}{1-\Delta t}}\quad \quad (6)}$

## 來源

1. ^ U.M. Ascher, S.J. Ruuth, R.J. Spiteri: Implicit-Explicit Runge-Kutta Methods for Time-Dependent Partial Differential Equations页面存档备份，存于互联网档案馆, Appl Numer Math, vol. 25(2-3), 1997
2. ^ L.Pareschi, G.Russo: Implicit-Explicit Runge-Kutta schemes for stiff systems of differential equations, Recent Trends in Numerical Analysis, Vol. 3, 269-289, 2000