Solomon A. Okunuga

3-Point Block Methods for Direct Integration of General Second-Order Ordinary Differential Equations

A Multistep collocation techniques is used in this paper to develop a 3-point explicit and implicit block methods, which are suitable for generating solutions of the general second-order ordinary differential equations of the form 𝑦=𝑓(𝑥,𝑦,𝑦),𝑦(𝑥0)=𝑎,𝑦(𝑥0)=𝑏. The derivation of both explicit and implicit block schemes is given for the purpose of comparison of results. The Stability and Convergence of the individual methods of the block schemes are investigated, and the methods are found to be 0-stable with good region of absolute stability. The 3-point block schemes derived are tested on standard mechanical problems, and it is shown that the implicit block methods are superior to the explicit ones in terms of accuracy.

A multi-point numerical integrator with trigonometric coefficients for initial value problems with periodic solutions

Based on the collocation technique, we introduced a unifying approach for deriving a family of multi-point numerical integrators with trigonometric coefficients for the numerical solution of periodic initial value problems. A practical 3-point numerical integrator was presented, whose coefficients are generalizations of classical linear multistep methods such that the coefficients are functions of an estimate of the angular frequency ω. The collocation technique yields a continuous method, from which the main and complementary methods are recovered and expressed as a block matrix finite difference formula that integrates a second-order differential equation over non-overlapping intervals without predictors. Some properties of the numerical integrator were investigated and presented. Numerical examples are given to illustrate the accuracy of the method.

Implementing a Type of Block Predictor-corrector Mode for Solving General Second Order Ordinary Differential Equations

The paper is geared towards implementing a type of block predictor-corrector mode capable of integratinggeneral second order ordinary differential equations using variable step size. This technique will be carried out on nonstiff problems. The mode which emanated from Milne’s estimate has many computation advantages such as changing and designing a suitable step size, correcting to convergence, error control/minimization with better accuracy compare to other methods with fixed step size. Moreover, the approach will adopt the estimates of the principal local truncation error on a pair of explicit (predictor) and implicit (corrector) Adams family which are implemented in P(CE) m mode. Numerical examples are given to examine the efficiency of the method and compared with subsisting methods.

Step Size Bounds for a Class of Multiderivative Explicit Runge–Kutta Methods

A class of 3-Stage Multiderivative Explicit Runge-Kutta Methods was developed for the solution of Initial Value Problems (IVPs) in Ordinary Differential Equations. In this work, we present the bounds on the step size required for the implementation of this family of methods. This bound is one of the parameter required in the design of program codes for solving IVPs. A comparison of the step size bound was made vis-a-vis the existing Explicit Runge-Kutta Methods using some standard problems. The computation shows that the family of methods competes well with the popular methods.