The Grunwald-Letnikov variable-, fractional-order

backward difference with step h >0 (GL-VFOBD-h) of function x(*) with an order function v: Z [right arrow] [R.sub.+] [union] {0} started at [k.sub.0] = 0 is defined as a finite sum

In the next definition, the Grunwald-Letnikov fractional-order

backward difference (GL-FOBD) is generalized to the Grunwald-Letnikov variable-, fractional-order

backward difference (GL-VFOBD) in part (a) and to the Grunwald-Letnikov variable-, fractional-order

backward difference with initialization (GL-VFOBDwl) in part (b).

Figure 4 shows the comparison results of angular acceleration's error calculated by the

backward difference and the SGF.

Backward Difference Formula (BDF) Runge-Kutta Method (RK4) and Differential Transform Method (DTM).

For the lower boundary, where i = M, substitute the approximate unknown value of the right boundary by the

backward difference approximation to [partial derivative]C/[partial derivative]x = 0.

[8] have introduced onepoint

backward difference methods for solving higher-order ODEs.

In this paper, we proposed a dynamic compensation method based on a digital filter with

backward difference design.

Another possibility is to define it as the

backward differencewhere [DELTA] f (x) = f (x + 1) - f (x) and [nabla] (x) = f (x) - f (x - 1) denote the forward and

backward difference operators, respectively.

In [10], Wong established the following discrete Opial type inequality about the

backward difference operator:

grid we introduce the forward and

backward difference quotients with respect to x

For stabilization of the calculations the first equation of (2.1) is approximated by using

backward differences formula: