Chebyshev technique in the numerical solution of Delay Differential Equations

We investigate the stability of a numerical solution of the delay differential equation by using explicit Runge-kutta method, when the delay has been approximated by using Chebyshev polynomial instead of using Lagrange and Hirmate interpellation polynomials.


Introduction
The model problem Delay Differential Equation which we considered, has the forms is referred to as the "log".In general, the delay is a function of both t and the solution ) (t y . In our research, we use various types of Runge-kutta methods to find a numerical solution for equation (1).Previous work on numerical methods for delay differential equations has been done by paul [2,3]

Basic Explicit Runge-kutta methods
The s-stages Runge-kutta formula for computing the numerical solution eq.(1) at t n +h n is defined by where a ij ,b i and c i are the coefficients of the Runge-Kutta formula and y n is the associated approximation of y(t n ).The Runge-Kutta formula is usually represented by the following

Definition:
A Runge-kutta process is said to be explicit if a ij =0 for ji, and semi-explicit if a ij =0 for j>i, otherwise,the process is said to be implicit.In this paper ,we are interested only in the second and fourth order Runge-Kutta formulas, which have the following forms and tables 0 0 0

Solution of DDE using explicit Runge-kutta method
To find a numerical solution for for equation.(1),we can classify the problem into two types:-1.The problem with no delay term (i.e.,d(t,y(t))=0) which is a first order ordinary differential equations, and the analytic solution can be obtained directly.
2. The problem with constant delay.3. the problem with variable delay.
In case ( 2) and (3), we need to evaluate y(t-d(t,y(t)) and since Runge-Kutta methods produce an approximate results at mesh points only, and the approximate solution between the mesh points is a matter of interpolation.Besides the other good qualities o f this method, Lagrange or Hermite interpolation, between mesh points provides a numerical solution just accurate When using an interpolation scheme to evaluate the delay term, one must aware of the following:-1.The number of solution values to be retained at any one time.2. Selection of the points for interpolation and which interpolating scheme need to be used and at which order [1,2] Suppose we have progressed through the i-th point in our mesh, t i .In order to take the next step, we must compute The computation is requiring a value for y(u(t j )) and since u(t i ) will in general be an on-mesh point, some approximation must be needed.
We first find a positive integer j such that . Since we are dealing with retarded problem i j  ,a sufficient past data need to be available to construct an approximate interpolation for y, which is then be used to evaluate u(t i ) and this value is used to help approximating i f It is possible with interpolation schemes using both function and derivative values that an extrapolation can be done.
This occurs if j=i, since in that case, f i is not known and hence the last full set of data is available only at 1  i t .Such situation was infrequent but never the less did occur.
Once j is found we use the past data at mesh points 1 3 ,   j j t t and t j for a fourpoint interpolation scheme and the data at 1 2 ,   j j t t and t j for a three point interpolation scheme.Since the problems were solved using a variety of step sizes h the interpolation routines had to be written to accommodate this scheme

Step size control:[6]
The step size has chosen as small as necessary to get an accurate approximation.On the other hand it is chosen as big as possible to reach the end of the internal in a few steps as possible.For this reason, and to maintain the stability we can use the following adaptive procedure: Assume that we are given required the error tolerance (), and that local truncation error is estimated, by E then:a-If E, then reject the computed solution.For choosing the next step size, we have to take account of the possibility of a point of jump discontinuity in the kth derivative of the solution, where kp+1.hence, we register the point t * =t n +h as a possible point of discontinuity and then we choose the next step size as 2 h .Using the step size 2 h ,the solution and its derivative value at t n we calculate the next approximation of solution and LTE estimate E and repeat the test in (a).b-If E then we accept me the next approximation to the solution and let the next step mesh point to be t n =t n +h.For choosing the next step size, we make the following tests:i-If iiii-If (t n +h) T where T is a point where the solution is required, then we take next step size to be h=T-t n .using the step size h ,the solution and its derivative values, we calculate the next approximation and repeat the test in (a).

Conclusions:
From the above results, we can notice that the fourth order Runge-Kutta method gave better results than the second order.Also, the results obtained by using Chebyshev polynomials are better than those obtained by Lagrange or Hermite interpolation.
step size, otherwise keep the Same step size and go to (iii)

Table :
Chebyshev technique in the numerical solution of Delay Differential Equations..