In this research work, we present the derivation and implementation of an order six block integrator for the solution of first-order ordinary differential equations using interpolation and collocation procedures. The approximate solution used in this work is a combination of power series and exponential function. We further investigate the properties of the block integrator and found it to be zero-stable, consistent and convergent. The block integrator is further tested on some real-life numerical problems and found to be computationally reliable.

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We proposed a continuous blocks method for the solution of second order initial value problems with constant step size in this paper. The method was developed by interpolation and collocation of power series approximate solution to generate a continuous linear multistep method;this is evaluated for the independent solution to give a continuous block method which is evaluated at selected grid point to give discrete block method. The basic properties of the method were investigated and was found to be zero stable, consistent and convergent. The efficiency of the method was tested on some numerical examples and found to give better approximation than the existing methods.

In this article, a variable step size strategy is adopted in formulating a new variable step hybrid block method (VSHBM) for the solution of the Kepler problem, which is known to be a rigid and stiff differential equation. To derive the VSHBM, the step size ratio r is left the same, halved, or doubled in order to optimize the total number of steps, minimize the number of formulae stored in the code, and ensure that the method is zero-stable. The method is formulated by integrating the Lagrange polynomial with limits of integration selected at special points. The article further analyzed the stability, order, consistency, and convergence properties of the VSHBM. The stability regions of the VSHBM at different values of the step size ratios were also plotted and plots showed that the method is fit for solving the Kepler problem. The results generated were then compared with some existing methods, including the MATLAB inbuilt stiff solver (ode 15 s), with respect to total number of failure steps, total number of steps, total function calls, maximum error, and computation time.

Integration of a larger stiff system of initial value problems emerging from chemical kinetics models requires a method that is both efficient and accurate, with a large absolute stability region. To determine the solutions of the stiff chemical kinetics ordinary differential equations that help in explaining chemically reactive flows, a numerical integration methodology known as the 3-point variable step block hybrid method has been devised. An appropriate time step is automatically chosen to give accurate results. To check the efficiency of the new method, the numerical integration of a few renowned stiff chemical problems is evaluated such as Belousov–Zhabotinskii reaction and Hires, which are widely used in numerical studies. The results generated are then compared with the MATLAB stiff solver, ode15s.