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# Finite Differences∂2y∂x2≈1Δx2(y(x−Δx)− 2 y(x)+ y(x+Δx)) \frac{\partial^2…
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\[
\begin{align}
e_{t+} \, u_{\left[ n \right]} &= \ u_{\left[ n+1 \right]}\\ \\
e_{t-} \, u_{\left[ n \right]} &= \ u_{\left[ n-1 \right]}
\end{align}
\]
-
\[
\left.
\begin{align}
\delta_{t+} &:= \frac{1}{\Delta t}\ \left( e_{t+}-1 \right) \\ \\
\delta_{t-} &:= \frac{1}{\Delta t}\ \left( 1-e_{t-} \right) \\ \\
\delta_{t} &:= \frac{1}{2 \ \Delta t}\ \left( e_{t+}-e_{t-} \right)
\end{align}
\right\}
\ \approx \
\frac{d}{dt}
\]
\[
\begin{align}
\delta_{t+} \, u_{\left[ n \right]} &= \ \frac{1}{\Delta t}\ \left( u_{\left[ n+1 \right]} - u_{\left[ n \right]} \right) \\ \\
\delta_{t-} \, u_{\left[ n \right]} &= \ \frac{1}{\Delta t}\ \left( u_{\left[ n \right]} - u_{\left[ n-1 \right]} \right) \\ \\
\delta_{t} \; u_{\left[ n \right]} &= \ \frac{1}{2 \Delta t}\ \left( u_{\left[ n+1 \right]} - u_{\left[ n-1 \right]} \right)
\end{align}
\]
-
\[
\left.
\begin{align}
\mu_{t+} &:= \frac{1}{2}\ \left( e_{t+}+1 \right) \\ \\
\mu_{t-} &:= \frac{1}{2}\ \left( 1+e_{t-} \right) \\ \\
\mu_{t} &:= \frac{1}{2}\ \left( e_{t+}+e_{t-} \right)
\end{align}
\right\}
\ \approx \
1
\]
-
-
\[
\delta_{t t}
\ := \ \delta_{t+} \ \delta_{t-}
\ = \
\frac{1}{\Delta t^2}
\left( e_{t+} - 2 + e_{t-} \right)
\ \approx \
\frac{d^2}{dt^2}
\]
-
\[
\delta_{t t t t}
\ := \ \delta_{t t} \ \delta_{t t}
\ = \
\frac{1}{\Delta t^4}
\left( e_{t+}^2 - 4 e_{t+} + 6 - 4 e_{t-} + e_{t-}^2 \right)
\ \approx \
\frac{d^4}{dt^4}
\]
-
\[
\delta_{t-} \delta_{x x}
\ = \
\frac{1}{2 \ \Delta t \ \Delta x^2}
\left( e_{t+}^2 - 4 e_{t+} + 6 - 4 e_{t-} + e_{t-}^2 \right)
\ \approx \
\frac{d^4}{dt^4}
\]
-
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