ON TWO CONTINUOUS COLLOCATION METHODS FOR SOLVING GENERAL SECOND ORDER INITIAL VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS

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).