In Section 4, the numerical treatment with the corresponding difference format is presented; then the numerical solutions of constant coefficient DAEs are presented by Padee approximation and the implicit Runge-Kutta method
. In Section 5, some numerical examples and error estimates are proposed.
The ODE45 consist 6 stage pair of embedded runge-Kutta method
of order 4 and 5.
With these models we can verify the performance of the LDU decomposition with respect to the previous implicit BT method and the classic explicit Runge-Kutta method
of 4th order (exRK).
The dimensionless shock dynamic equations (6) are solved using the fourth-order Runge-Kutta method
. When dimensionless peak pulse acceleration [beta][[??].sub.0] = 0.1, frequency parameter ratio of system [[lambda].sub.1] = 10, mass ratio of system [[lambda].sub.2] = 0.01, and dimensionless pulse duration [[tau].sub.0] = 0.5 are defined, respectively, the effects of angle on dimensionless response acceleration-time history of critical components are given in Figure 2.
In this work, a fourth-order Runge-Kutta method
was used to obtain the discrete solution .
The details of calculations using the fourth-order Runge-Kutta method
satisfy the following relationship:
For comparison purposes, the Fehlberg fourth-fifth order Runge-Kutta method
with degree four interpolant (RKF45) [42, 43] build-in in Maple CAS software was used to obtain the exact solution of the pollution problems.
Hayashi  handled small and vanishing delay by proposing three algorithms of iterative scheme which are extrapolation, special interpolant, and iteration procedure with the adaptation of continuous Runge-Kutta method
, while Neves and Thompson  handled small and vanishing delay by restricting the step size to be smaller and using extrapolation, respectively.
In recent years, many different methods and different basis functions have been used to estimate the solution of the system of integral equations, such as Adomian decomposition method [1, 2], Taylor's expansion method [3, 4], homotopy perturbation method [5, 6], projection method and Nystrom method , Spline collocation method [ 8], Runge-Kutta method
, sinc method , Tau method , block-pulse functions, hat basis functions method , and operational matrices [13,14].
(12) for the middle segment of threaded connection were solved separate numerically by Runge-Kutta method
. (It was realized by using the suite of mathematical programs Maple-9).
Numerical experiments are carried out to investigate controlled systems by using Fourth-order Runge-Kutta method
with time step 0.01.
The real-time virtual model represents the ECSTR described by (1) and (2) implemented in Delphi, using the 4th order Runge-Kutta method
. The application works with/without a controller KRGN 90 real or virtual (Pollak, 2004) and (Remias, 2004) or the controller UDC 3000/3300 Honeywell (Kuzma, 2006) and (Rybar, 2005).