A Class of Adams-like Implicit Collocation Methods of Higher Orders for the solutions of Initial Value Problems

The focus of this research work is the derivation of a class of Adams-like collocation multistep methods of orders not exceeding p=9. Numerical quadrature rule is used to derive steps k= 3,...,8 of the Adams methods. Convergence of each formula derived is established in this paper. As a numerical experiment, the step six pair of the Adams method so derived was used as predictor-corrector pair to solve a non-stiff initial value problem. The absolute errors show an accuracy of o(h7).

We seek the values of the coefficients β j so that the corresponding formula has the maximum possible degree of precision.(Note here that the corresponding numerical integration rule would use points from outside the range of integration).A ready example in literature is the popular two-step order two explicit method given as The Adams Moulton methods are solely due to John Couch Adams, like the Adams Bashfor th methods.The name of Forest Ray Moulton became associated with the method because he realised that they could be used in tandem with the Adams Bashforth methods as a predictor-corrector pair (Moulton 1926).The twostep order three implicit method is given as.

Theorem due Dahlquist
A linear multistep 0 method is said to be convergent if it is consistent and zero stable.
A linear multistep method is consistent if it satisfies the conditions bellow i. ii.
A linear multistep method is said to be zero stable if the roots of the characteristic polynomial l(r) = if | r | ≤1 and each root is distinct.

Order of accuracy of Linear Multistep methods
Equation ( 2) is said to have order of accuracy p if p is the largest positive integer such that for any sufficiently smooth curve in the rectangular domain D of the first order initial value problem (Equation 1).There exist constants k and h 0 such that for |T n | ≤ kh p for 0<h<, for any k+1 points (on the solution curve.We can thus deduce that the method (equation 2) is of order of accuracy p if and only if in this case is the error constant.

Highest Attainable Order of Stable Multistep Methods
It is a natural task to investigate the stability of the multistep methods with highest possible order.Counting the order conditions shows that for order p the parameters of a linear multistep method have to satisfy p+1 linear equations.As 2k+1 free parameters are involved, (without loss of generality one can assume α k =1), this suggests that 2k is the highest attainable order.According to Dahlquist (1956), "The order p of a stable linear k -step method satisfies the following conditions:

Predictor -Corrector Methods
The idea behind this is that an Adams-Bash forth formulae is used to estimate the value of y(x k+1 ) and this is then used in place of y k+1 on the right-hand side of an Adams-Moulton formula in order to obtain an improved estimate for y k+1 .The Adams Bashforth method is thus used to predict the value y k+1 and the Adams Moulton formula is the corrector which may be applied iteratively until convergence.

Methodology
For the k-step implicit Adams method, we seek that is We require thus we have the following equations to solve: We seek the values of the coefficients β j so that the corresponding formula has the maximum possible degree of precision.
To confirm this methodology, we show for step k = 3 for the Adams-Moulton scheme as follows:

CONCLUSION
The proposed methods have been shown to be accurate and efficient.They provide option for continuous output.Further improved accuracy can be achieved by block hybrid methods for numerical derivatives combined with self-starting block hybrid methods for the direct integration of.

Table 1 : Order, error constants and convergence of implicit Adams methods k-step Order Error constants Convergence property
The table below shows the absolute errors which is of at least order O(h 6 )