# forward difference

## forward difference

[¦fȯr·wərd ′dif·rəns]
(mathematics)
One of a series of quantities obtained from a function whose values are known at a series of equally spaced points by repeatedly applying the forward difference operator to these values; used in interpolation or numerical calculation and integration of functions.
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We also take the forward difference scheme to the left boundary condition and the lower boundary condition.
We call [f.sup.[DELTA]](t) the delta (or Hilger) derivative of f(t) and it turns out that [f.sup.[DELTA]] is the usual derivative if T = R and is the usual forward difference operator if T = Z.
The forward difference operator is denoted by [DELTA].
In the transient term, we used the forward difference for
With the help of this method, we use backward space difference provided that the wave speed c is positive and if c is negative, we have to ensure the stability by using forward difference. For the above equation, may results as, at a grid point ( ) discussed within the region shown in fig.1.
[DELTA] is the first-order forward difference operator; that is, [DELTA]u(k) = u(k + 1) - u(k).
In a series of papers Villatoro and Ramos  used the modified equation approach to analyze and improve the efficiency of an Euler forward difference method to solve linear and nonlinear differential equations.
where {p(n)}, [q(n)} are sequences of real numbers, {[sigma]a(n)} is a sequence of positive integers such that [sigma](n) > n + 1, ([tau](n)} is a nondecreasing sequence of nonnegative integers such that [tau](n) < n and [DELTA] is the forward difference operator defined by the equation [DELTA]x(n) = x(n + 1) - x(n).
The discrete-time analogue of the Lie derivative of a function ([phi](x) is its difference, which can be defined in two different ways, either as the forward difference
One can check that in these cases we have [f.sup.[DELTA]](t) = f'(t), [f.sup.[DELTA]](t) = [[DELTA].sub.[omega]] f(t) and [f.sup.[DELTA]](t) = [D.sub.q] f(t), respectively, where [[DELTA].sub.[omega]] f(t) = f (t + [omega]) - f(t) / [omega] is the forward difference operator with stepsize [omega] , and [D.sub.q] f(t) = f(qt) - f(t) / t(q - 1) (for t [not equal to] 0) is the q-difference operator .

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