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].

Now, taking the

forward difference of (8), the following inequality holds:

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 [4447] 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 differenceOne 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] [32], and [D.sub.q] f(t) = f(qt) - f(t) / t(q - 1) (for t [not equal to] 0) is the q-difference operator [10].