A new method called parallel R-point explicit block method for solving a single equation of higher order ordinary differential equation directly using a constant step size is developed. This method calculates the numerical solution at R point simultaneously is parallel in nature. Computational advantages are presented by comparing the results obtained with the new method with that of the conventional 1-point method. The numerical results show that the new method reduces the total number of steps and execution time. The accuracy of the parallel block and the conventional 1-point methods is comparable particularly when finer step sizes are used.
Exponentially-fitted numerical methods are appealing because L-stability is guaranteed when solving initial value problems of the form y' = λy, y(a) = η, λ ∈ , Re(λ) < 0. Such numerical methods also yield the exact solution when solving the above-mentioned problem. Whilst rational methods have been well established in the past decades, most of them are not ‘completely’ exponentially-fitted. Recently, a class of one-step exponential-rational methods (ERMs) was discovered. Analyses showed that all ERMs are exponentially-fitted, hence implying L-stability. Several numerical experiments showed that ERMs are more accurate than existing rational methods in solving general initial value problem. However, ERMs have two weaknesses: every ERM is non-uniquely defined and may return complex values. Therefore, the purpose of this study was to modify the original ERMs so that these weaknesses will be overcome. This study discusses the generalizations of the modified ERMs and the theoretical analyses involved such as consistency, stability and convergence. Numerical experiments showed that the modified ERMs and the original ERMs are found to have comparable accuracy; hence modified ERMs are preferable to original ERMs.
Linear array of permutations is hard to be factorised. However, by using a starter set, the process of listing the permutations becomes easy. Once the starter sets are obtained, the circular and reverse of circular operations are easily employed to produce distinct permutations from each starter set. However, a problem arises when the equivalence starter sets generate similar permutations and, therefore, willneed to be discarded. In this paper, a new recursive strategy is proposed to generate starter sets that will not incur equivalence by circular operation. Computational advantages are presented that compare the results obtained by the new algorithm with those obtained using two other existing methods. The result indicates that the new algorithm is faster than the other two in time execution.