After simplification, we get the truncation error
In this paper, a relative truncation error
[[epsilon].sub.t] is employed to generate an adaptive truncation scheme, which determines the accuracy of the H-LU factors.
The local truncation error
at [x.sub.n+k] of the method is defined to be expression L[y([x.sub.n]); h], when y(x) is the theoretical solution of the initial value problem
Ifa(x), f (x) [subset] [C.sup.1] ([bar.[OMEGA]]), then for the truncation error
[R.sub.i] we have
In this section, stability and local truncation error
of the nonstandard scheme (10) are examined.
We omit the ROC of the TOMs and BVM8 because their errors are mainly due to round-off errors rather than to truncation errors
. Figure 1 also shows the efficiency curves of these methods.
Although the abovementioned processing method can well maintain the system accuracy, the whole system still shows some truncation error
In this paper, two approximations to variable-order Caputo fractional derivatives were developed, and the analysis for the truncation errors
of new formulas was made.
For instance, for an arbitrary term [LAMBDA] = f ([tau], [delta], [bar.x]), the first [tau] derivative of [LAMBDA] with a second-order truncation error
centered difference can be obtained from
The central difference scheme has a truncation error
of order (Eq.), its results are less accurate for large coefficient of highest order derivatives.
and [[eta].sub.ORA,E] is the truncation error
resulting from truncating the infinite series in (10) at n = N.
However, this scheme is conditionally consistent, and the truncation error
depends on the ratio of the time stepsize and the square of the space stepsize.
In addition,  also computed the truncation error
bounds for truncating the infinite summation series in the final expressions.
Its truncation error
is more symmetric, but its quantization accuracy is not improved distinctly.