Rational methods for solving first order ordinary differential equation

In this study, two classes of rational methods of second to fourth order of accuracy are proposed. The formulation of the methods are based on two distinct rational functions that are proposed in thesis, where the first class of methods are derived based on rational function with denominator of degree one, as the degree of the numerator increases. Meanwhile, the second class uses a rational function with the numerator of degree one, as the degree of its denominator increases. The derivation and implementation techniques are adapted from an existing study mentioned in the thesis. The concept of the closest points of approximation is applied on the Taylor series expansion in the derivation of the methods to increase the accuracy of the proposed methods. The stability regions of the proposed rational methods are illustrated. The second order methods from the first class is found to be A-stable, while third and fourth order methods are found to be absolutely stable. On the other hand, the methods from the second class are all A-stable. Besides that, the algorithm for the proposed methods are developed with constant step size strategy, in which the strategy to compute the starting values by an existing methods is also included. Both classes of methods are tested in solving initial value problems of different nature which are singular, stiff and singular perturbation. Based on the numerical results, it is observed that the proposed methods are capable to give comparable or more accurate solutions compared to some of the existing methods in solving the tested problems. The application of closest points of approximation concept have shown the capability of the proposed methods in solving problem with integer singular point compared to the existing rational multistep methods. Nevertheless, as the proposed methods are compared to the existing methods which apply self-starting mechanism in its formula, it is found that the accuracy of the proposed methods is comparable or outperformed by the existing methods. In terms of efficiency, the proposed methods require comparable or lesser time of execution compared to the existing methods of the same order. Besides that, the proposed methods also require lesser number of total function evaluation compared to the existing methods, except for the second order methods, where the number is found to be similar to the existing methods. In conclusion, the proposed methods are suitable in solving problems with singularity, stiff and singularly perturbed problems.

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