D. O. Awoyemi’s research while affiliated with University of Ilorin and other places

This work considers the direct solution of general third order ordinary differential equation. The method is derived by collocating and interpolating the approximate solution in power series. A single hybrid three-step method is developed. Taylor series is used to generate the independent solution at selected grid and off grid points. The order, zero stability and convergence of the method were established. The developed method is then applied to solve some initial value problems of third order ODEs. The numerical results of the method confirm the superiority of the new method over the existing method.

This paper considers the development of a 2-step four-point continuous hybrid method for the direct solution of initial value problem (IVPs) of second order ordinary differential equations using the method of interpolation of the power series approximate solution and collocation of the differential system to develop our scheme. Taylor's series approximation is used to analyze and implement n i y  1 1 i n    at   , j 0 1 2. n i x   The method is found to be consistent and zero-stable. Numerical results show a superior accuracy compared to existing methods.

In this paper, we derived a continuous linear multistep method (LMM) with step number k = 4 through collocation and interpolation techniques using power series as basis function for approximate solution. An order-seven scheme is developed which was used to solve the third-order initial value problems (IVPS) in ordinary differential equation without first reducing to a system first-order. Taylor’s series algorithm of the same order was developed to implement our method. The result obtained was compared favourably with existing methods. Key words: Continuous collocation, multistep methods, interpolation, third-order, power series.

In this paper we derived a continuous linear multistep method (LMM) with step number k = 5 through collocation and interpolation techniques using power series as basis function for approximate solution. An order nine p-stable scheme is developed which was used to solve the third order initial value problems in ordinary differential equation without first reducing to a system of first order equations. Taylor’s series algorithm of the same order was developed to implement our method. The result obtained compared favourably with existing methods.

A one-step implicit hybrid block solution method for initial value problems of general second order ordinary differential equations has been studied in this paper. The one-step method is augmented by the inclusion of off step points to enable the multistep procedure. This guaranteed zero stability as well as consistency of the resulting method. The convergence and weak stability properties of the new method have been established. Results from the new method compared with those obtained from existing methods show that the new method gives better accuracy.

In this paper we derive finite difference methods by a power series form of multistep collocation for the solution of the initial value problems for ordinary differential equations. By selection of points for both interpolation and collocation, many important classes of finite difference methods are produced including new ones which are generally more accurate (smaller error constants) than the Adams–Moulton Methods with adequate absolute stability intervals for nonstiff problem.

In this paper, an implicit one step method for the numerical solution of second order initial value problems of ordinary differential equations has been developed by collocation and interpolation technique. The introduction of an o step point guaranteed the zero stability and consistency of the method. The implicit method developed was implemented as a block which gave simultaneous solutions, as well as their rst derivatives, at both o step and the step point. A comparison of our method to the predictor-corrector method after solving some sample problems reveals that our method performs better.

The direct solution of general second order ordinary differential equations is considered in this paper. The method is based on the collocation and interpolation of the power se-ries approximate solution to generate a continuous linear multistep method. We modified the existing block method in order to accommodate the general nth order ordinary differen-tial equation. The method was found to be efficient when tested on second order ordinary differential equation.

We propose an implicit multi-step method for the solution of initial value problems (IVPs) of third order ordinary differential equations (ODE) which does not require reducing the ODE to first order before solving. The development of the method is based on collocation of the differential system and interpolation of the approximate solution at selected grid points. This generates a system of equations, which are then solved using Gaussian elimination method. Three predictors, each of order 5, are also proposed to calculate some starting values in the main method. Analysis of basic properties is considered to guarantee the accuracy of the method. The results for method of step length k = 5 when compared with that of step length k = 4 show a better level of accuracy.KEYWORD: Zero stable, third order IVPs, predictor method, step length.

Problem statement: The conventional methods of solving higher order differential equations have been by reducing them to systems of first order equations. This approach is cumbersome and increases computational time. Approach: To address this problem, a numerical algorithm for direct solution of 5th order initial value problems in ordinary differential equations (odes), using power series as basis function, is proposed in this research. Collocation of the differential system is taken at selected grid points to reduce the number of functions to be evaluated per iteration. A number of predictors and their derivatives having the same order of accuracy with the main method are proposed. Results: The approach yields a multiderivative method of order six. Numerical examples solved show increased efficiency of the method with increased number of iterations, converging to the theoretical solutions. Conclusion/Recommendations: The new mutiderivative method is efficient to solve linear and nonlinear fifth order odes without reduction to system of lower order equations.