A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses

\begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P

\end{equation}

as predictor and

\begin{equation} y_{n+1}-y_{n}=\frac{h}{2}(f_{n+1}-f_{n}) \tag C

\end{equation}

IF $(P)$ and $(C)$ are used in PECE mode on the vector problem

\begin{equation} \frac{du}{dt}=u

\end{equation}

\begin{equation} \frac{dv}{dt}=-10u-11v+cos(2\pi t)

\end{equation}

with $u(0)$,$v(0)$ given, find the largest constant $\gamma >0$ for which the scheme is stable in the sense of Von Neumann (Fourier series stability and frequency) whenever $0<\gamma<0$. Give full details of your argument.

=>

I haven't try very well because its really difficult question for me.

I was thinking

\begin{equation} y_{n+1}=y_{n}+hf_{n} \tag P

\end{equation}

as predictor and

\begin{equation} y_{n+1}=y_{n}+\frac{h}{2}(f_{n+1}-f_{n}) \tag C

\end{equation}

iam trying to get transition matrix but these condition

\begin{equation} \frac{du}{dt}=u

\end{equation}

\begin{equation} \frac{dv}{dt}=-10u-11v+cos(2\pi t)

\end{equation}

i don't know how and where to use.

please help me.