In recent times, numerical approximation of 3rd-order boundary value problems (BVPs) has attracted great attention due to its wide applications in solving problems arising from sciences and engineering. Hence, A higher-order block method is constructed for the direct solution of 3rd-order linear and non-linear BVPs. The approach of interpolation and collocation is adopted in the derivation. Power series approximate solution is interpolated at the points required to suitably handle both linear and non-linear third-order BVPs while the collocation was done at all the multiderivative points. The three sets of discrete schemes together with their first, and second derivatives formed the required higher-order block method (HBM) which is applied to standard third-order BVPs. The HBM is self-starting since it doesn’t need any separate predictor or starting values. The investigation of the convergence analysis of the HBM is completely examined and discussed. The improving tactics are fully considered and discussed which resulted in better performance of the HBM. Three numerical examples were presented to show the performance and the strength of the HBM over other numerical methods. The comparison of the HBM errors and other existing work in the literature was also shown in curves.
The aim of this paper is to compute the numerical solution of special second order delay differential equations directly by a four-step multi-hybrid block method. The methods were generated using collocation and interpolation approach by means of a combination of power series and exponential function at some selected grid and off-grid points. The developed schemes and its first derivatives was combined to form block methods to concurrently solve special second order delay differential equations directly without reducing it to the system of first order. The basic properties of the methods such as order, error constants, consistency and convergence were examined. The developed methods were applied to solve some second order delay differential equations, the methods also solve application problem in other to test for the efficiency and accuracy of the methods. The results are displays in the tables.
JEL classification numbers: 65L05; 65L06; 65L20.
Keywords: Multi-Hybrid Block Method; Special Second Order Delay Differential Equations; Convergence; Four-step Block Method; Order seventeen.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.