Abstract:
In this thesis two zero stable Linear Multistep methods (LMM) with
continuous coetficients for solving general second order initial value
problems of ordinary differential equations which does not require that the
equation be reduced to a system of first order equation are considered. The
approach is based on c:ollocation of the differential systems arising from the
basis function at the grid points x = xn+i' 0 s i ~ k and interpolation of the
approximate solution at the selected grid points x = Xn+i' for 1 < i < 3 and
2 <i <4 for the step numbers k = 3 and 4 respectively.
Some predictors and their first derivatives are proposed to calculate
Yn+k and Y~+k for k = 3, 4. The use of Taylor series expansion is employed
for the calculation of Yn+i for i = 1,2. Evaluation of each method and its
predictors at x = Xn+k gives particular discrete schemes as special cases of the
methods and their predictors respectively. The new 4-step method was
analysed and found to be consistent and zero stable, hence convergent.
The new method was tested on some general second order initial
value problems of ordinary differential equations The result showed that the
method converges as h decreases. The new results were compared with the
exact and the earlier result of Awoyemi (1999), it was found that the new
result improved over that of Awoyerni (1999).
