Direct one-step block methods for solving general second order non-stiff ordinary differential equations

In this thesis, one-step block methods are developed for solving Initial Value Problems (IVPs) of general second order Ordinary Differential Equations (ODEs). These methods is used to solve the general second order ODEs using variable step size. The proposed methods will obtain the approximation solutions at two, three, four and five points simultaneously in a block. These methods will also solve the general second order ODEs directly. This approach is more efficient than the common technique in reducing the problems to a system of first order equations. These methods will be formulated in terms of multistep method but the implementation is equivalent to the one-step method i.e. Runge-Kutta method. Lagrange interpolation polynomial is applied in order to compute the coefficients of the developed block methods formulae by integrating the closest point in the interval to obtain the approximate solutions. The stability region of the proposed method has also been studied. The numerical results showed that as the number of point increased in the block, the total number of steps is reduced. In addition, at smaller tolerances, the execution times of the proposed methods were faster in the tested problems as the number of points increased. In all cases, the accuracy of the proposed methods gave acceptable accuracy within the given tolerances. In conclusion, the proposed direct one-step block methods in this thesis are suitable for solving the general second order ODEs directly.

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