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Finite Difference Schemes (FDSs) aim to solve differential equations by means of finite differences. For example, as discussed in §H.2, if denotes the displacement in meters of a vibrating string at time seconds and position meters, we may approximate the first- and second-order partial derivatives by

$\displaystyle {\dot y}(t,x)$	$\displaystyle \isdef$	$\displaystyle \frac{\partial}{\partial t}y(t,x) \;\approx\; \frac{y(t,x)-y(t-T,x)}{T}$
$\displaystyle y'(t,x)$	$\displaystyle \isdef$	$\displaystyle \frac{\partial}{\partial x}y(t,x) \;\approx\; \frac{y(t,x)-y(t,x-X)}{X}$
$\displaystyle {\ddot y}(t,x)$	$\displaystyle \isdef$	$\displaystyle \frac{\partial^2}{\partial t^2} y(t,x) \;\approx\; \frac{y(t+T,x) - 2 y(t,x) + y(t-T,x) }{T^2}$
$\displaystyle y''(t,x)$	$\displaystyle \isdef$	$\displaystyle \frac{\partial^2}{\partial x^2} y(t,x) \;\approx\; \frac{y(t,x+X) - 2 y(t,x) + y(t,x-X) }{X^2} \protect$	(N.1)

where

denotes the time sampling interval and

denotes the spatial sampling interval. Other types of finite-difference schemes were derived in Appendix L (§L.3), including a look at frequency-domain properties. These finite-difference approximations to the partial derivatives may be used to compute solutions of differential equations on a discrete grid:

$\begin{displaymath} \begin{array}{rclcl} x &\to& x_m&=& mX\nonumber \\ t &\to& t_n&=& nT\nonumber \end{array}\end{displaymath}$