If we apply the half-explicit

Euler method which was proposed in [17] to the test DAE (3.16), then we obtain the DAE stability function

To estimate the depth of the deep structures, the

Euler method is applied the RTE-TMI map.

[11,12] studied the Vaserstein bound for

Euler method and they proved an O([h.sup.(2/3-epsilon])]) for a one-dimensional diffusion process where h is the step-size and then they generalize the result to SDEs of any dimension with 0(h[square root of log(1/h))] bound when the coefficients are time-homogeneous.

As we all know, there are many solutions corresponding to the second-order ODEs, including the

Euler method, the improved

Euler method and the Runge-Kutta method et al.

The results provided by the presented method will be compared with those by the modified

Euler method, Newmark method, widely-used fourth-order RK method, and the exact results, respectively.

The direct application of the method to such pathological cases has been compared with the classical

Euler method, showing that singularity and ambiguity drawbacks do not affect the proposed solution.

When applying the multiplicative

Euler method to the mth subsystem, we obtain a single step at a time:

The numerical properties of impulsive differential equations attracted attentions of scholars since Ran et al.'s work in [7], which showed that the explicit

Euler method is stable for impulsive differential equations, while the implicit

Euler method is not.

(a) The exponential

Euler method (explicit), denoted by expEuler:

Here [theta] = 0 defines the explicit

Euler method, [theta] = 1 defines the backward

Euler method and [theta] =1/2 gives the symmetric (or trapezoidal)

Euler method [13].

For a single small time step, this approach is much easier to implement and more efficient than the implicit method; at the same time, the stress is return-mapped effectively onto the subsequent yield surface by enforcing the consistency condition, and the problem of drift in the explicit forward

Euler method is overcome.