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Time Difference and Averaging Operators
The definitions of time difference operators in the distributed setting are nearly unchanged from those introduced in §2.2 and applied to time series. For a grid function
, the forward and backward shifts, and the identity operation ``1" are defined as
and are to be regarded as applying to the time series
at all values of the index
and
. Approximations to a partial time derivative are defined as
![$\displaystyle \delta_{t+} \triangleq \frac{1}{k}\left(e_{t+}-1\right)\approx \f...
...ngleq \frac{1}{2k}\left(e_{t+}-e_{t-}\right)\approx \frac{\partial}{\partial t}$](img341.png) |
(5.1) |
and averaging approximations to the identity operation as
![$\displaystyle \mu_{t+} \triangleq \frac{1}{2}\left(e_{t+}+1\right)\approx 1\qqu...
...x 1\qquad\mu_{t\cdot} \triangleq \frac{1}{2}\left(e_{t+}+e_{t-}\right)\approx 1$](img342.png) |
(5.2) |
A simple approximation to a second time derivative is, as before,
![$\displaystyle \delta_{tt}=\delta_{t+}\delta_{t-}\approx\frac{\partial^2}{\partial t^2}$](img343.png) |
(5.3) |
Next: Spatial Difference Operators
Up: Grid Functions and Difference
Previous: Grid Functions
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Stefan Bilbao
2006-11-15