OSFESOR Code – The Delay Differential Equation Tool “Improving Delay Differential Equations Solver”

After having reviewed the RETARD code, which was originally written by Hairer & Wanner in 1995 with the aim of solving delay differential equations (DDEs), a new arithmetic called OSFESOR code is presented in this paper. The OSFESOR (Optimal stepsize for each step of RETARD) code is a tool for automatic implementation of DDEs in Fortran 77. Consequently, by use of the OSFESOR code, it is possible during the run of a program to evaluate the solution of DDEs with optimal stepsize, H, for each step , to evaluate the accuracy of any result provided by the computer . In short, the use of OSFESOR code serves to validate the result provided by a computer, and to assure the user of the reliability of scientific computations.


1-Introduction
The simulation of phenomena represents virtually all of scientific computation. Yet just what is the simulation of phenomena? It is the development of a scientific code, which translates a mathematical model and employs numerical methods to resolve it. RETARD code which was written by Hairer & Wanner, which is modified in 1995, [1] was concerned with Runge-Kutta methods for a system of first order ordinary differential equations of the form: The basic strategy investigated, there involves the adaptation of Runge -Kutta method for the ordinary differential equation : With prescribed initial value, we assume that the reader is familiar with Runge-Kutta methods for (2), the concept of their order and internal stage -orders and their representation by corresponding tableaux of the form:  This code which is based on an explicit Runge-Kutta method for (2) of order (4) 5 due to Dormand & Prince is adapted to the delay differential equations of (1).
Thus, a specific Runge-Kutta method for solving a delay differential equation involves 1-A choice of Runge-Kutta tableau. 2-A choice of approximation method for interplant [2].

2-Interpolation:
Runge-Kutta method can be adapted for DDEs, such that we consider the fifth-order Dormand & Prince explicit RK method. There are several possible choices of denseoutput for RK method [3], [ are the fourth order continuous embedded formula .
The RK parameter are "formally explicit" if i j for a ij  = 0 and will be called "local" if ] for all i. A CRK triple allows one to obtain a formulate (dense -output) for the numerical solution of a DDEs. Now after having reviewed the RETARD code, which solving a system of first order DDEs(1),a new arithmetic called OSRESOR code is presented in this paper .

3-scientific computation of the optimal stepsize for each step in OSFESOR code :
Consider the method error e m and the error due to the propagation of round-off error, called the computation error e c here. It is well known that when the dis-cretizing step h decreases, e m decreases and that, on the contrary, when h increases, e m also increases. It has been shown that, when h increases, ec decreases, and that when h decreases, e c increases. This means that e m & e c act in the opposite way . Thus the best approximation of the solution that can be obtained on a computer corresponds to an optimal dis-cretizing step h.
The scientific computation of the optimal stepsize h for each step in OSFESOR code can be computed by the following outlines:

3.1-The Outlines:
The computation of optimal stepsize h requires three phases: *The optimal stepsize h is evaluated for each interval [x k ,x k +h k ] as follows Analytical solution: not available Sources: E.Hairer , G.Wanner &Norsett,solving ODEs 1,sipringer series in comp. Math.vol. 8(1980).(see [9]) Other information: This system of DDEs describes a disease model a reference solution is (see [9]) y 1 (40)=0.0912491205663460 y 2 (40)=0.0202995003350707 The results in the table below obtained with the use of RETARD & OSFESOR codes: The result of RETARD code:

6-Conclusions:
The main advantage of using OSFESOR code is to develop DDEs methods making it possible to estimate the optimal step or mesh at each computation step , so that the best possible informational solution can be obtained . Thus we feel that the one can now use tools, such as the OSFESOR code, for validating the results provided by the computer and should also forcelly demand that manufactures creating arithmetic units and compilers include tools for numerical validation.