multistepexamples.tex

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\section*{Examples of Multi-Step Schemes for Solving Ordinary Differential Equations (ODEs)}

\subsection*{1. Adams–Bashforth Second-Order Method (Explicit)}
\[
y_{n+1} = y_n + \frac{h}{2} \left[ 3f(t_n, y_n) - f(t_{n-1}, y_{n-1}) \right]
\]
\begin{itemize}
    \item \textbf{Order}: 2
    \item \textbf{Type}: Explicit
    \item \textbf{Steps}: 2
\end{itemize}

\subsection*{2. Adams–Bashforth Fourth-Order Method (Explicit)}
\[
y_{n+1} = y_n + \frac{h}{24} \left[ 55f(t_n, y_n) - 59f(t_{n-1}, y_{n-1}) + 37f(t_{n-2}, y_{n-2}) - 9f(t_{n-3}, y_{n-3}) \right]
\]
\begin{itemize}
    \item \textbf{Order}: 4
    \item \textbf{Type}: Explicit
    \item \textbf{Steps}: 4
\end{itemize}

\subsection*{3. Adams–Moulton Second-Order Method (Implicit, Trapezoidal Rule)}
\[
y_{n+1} = y_n + \frac{h}{2} \left[ f(t_{n+1}, y_{n+1}) + f(t_n, y_n) \right]
\]
\begin{itemize}
    \item \textbf{Order}: 2
    \item \textbf{Type}: Implicit
    \item \textbf{Steps}: 1
\end{itemize}

\subsection*{4. Adams–Moulton Third-Order Method (Implicit)}
\[
y_{n+1} = y_n + \frac{h}{12} \left[ 5f(t_{n+1}, y_{n+1}) + 8f(t_n, y_n) - f(t_{n-1}, y_{n-1}) \right]
\]
\begin{itemize}
    \item \textbf{Order}: 3
    \item \textbf{Type}: Implicit
    \item \textbf{Steps}: 2
\end{itemize}

\subsection*{5. Backward Differentiation Formula (BDF2) (Implicit)}
\[
y_{n+1} = \frac{4}{3}y_n - \frac{1}{3}y_{n-1} + \frac{2h}{3}f(t_{n+1}, y_{n+1})
\]
\begin{itemize}
    \item \textbf{Order}: 2
    \item \textbf{Type}: Implicit
    \item \textbf{Steps}: 2
\end{itemize}

\subsection*{6. Nyström Explicit Midpoint Method (Explicit)}
\[
y_{n+1} = y_{n-1} + 2h \cdot f(t_n, y_n)
\]
\begin{itemize}
    \item \textbf{Order}: 2
    \item \textbf{Type}: Explicit
    \item \textbf{Steps}: 2
\end{itemize}

\subsection*{7. Milne's Fourth-Order Predictor (Explicit)}
\[
y_{n+1} = y_{n-3} + \frac{4h}{3} \left[ 2f(t_n, y_n) - f(t_{n-1}, y_{n-1}) + 2f(t_{n-2}, y_{n-2}) \right]
\]
\begin{itemize}
    \item \textbf{Order}: 4
    \item \textbf{Type}: Explicit
    \item \textbf{Steps}: 4
\end{itemize}

\subsection*{8. Simpson's Fourth-Order Corrector (Implicit)}
\[
y_{n+1} = y_{n-1} + \frac{h}{3} \left[ f(t_{n+1}, y_{n+1}) + 4f(t_n, y_n) + f(t_{n-1}, y_{n-1}) \right]
\]
\begin{itemize}
    \item \textbf{Order}: 4
    \item \textbf{Type}: Implicit
    \item \textbf{Steps}: 2
\end{itemize}

\section*{Summary}
These methods span explicit and implicit families (Adams–Bashforth, Adams–Moulton, BDF, and others) and vary in order and step count.

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