Professor Yusuph Amuda Yahaya

Wednesday 19th May, 2021

The Federal University of Technology Minna,
Main Campus,
Gidan Kwanu,
Minna, Niger State

 It is obvious that to assess the history of numerical methods in the 20th century we must first recognize the legacies of the previous century on which it was built. Notable among them include Adams and Bashforth (1883), Runge (1895), Heun and Kutta-Butcher (1906).
The famous Adams-Bashforth method plays an essential role in modern software. Together with the Adams-Moulton method and the practical use of Taylor's series method, following the important and prophetic work of Adams and of Runge, the new century began with further contributions to what is now known in the 21 st century as the Runge-Kutta Method by Heun and Kutta in 1900. That is what is now recognized as the starting point for modern one-step methods.
See Butcher (2000) for details. Today, Mr. Vice-Chancellor Sir, as you have seen in earlier sections ofthis lecture, there are many variants of the Runge-Kutta Method in common use, but the most popular is the So-called PECE mode as in Section 4.5 of this lecture. The modern analysis of continuous multi-step methods is intimately bound up with the work ofDahlquist (1956 and 1959) stated in 4.6 of section 4 of this lecture. This body of works is in several parts, of which the first deals with the concepts of consistency, stability, and convergence discussed extensively in sections 3.0 and part of4.0.All our works on
linear multistep methods show that consistency and stability are together equivalent to convergence. See section 4.0 of this lecture.
Development of single step and hybrid block methods that circumvented the Dahlquist stability barrier theorem for the solution of first-order initial value problems of ordinary differential equations, and combines the advantages of the block method and allow continuous linear multistep method to address the setbacks of the predictor-corrector method were the major contributions I made in my earlier days as a numerical analyst. Although a flurry of activities by other authors followed. I want to state here that the continuous hybrid method combines certain characteristics of CLMMs with Runge-Kutta Methods with the flexibility of changing step length and evaluating at off-step points. See Section 5.0 of this lecture.
As an improvement on the limitation of both the single-step and multistep methods, Yahaya (2004) introduced the concept of CLMM which was earlier briefly discussed in Yusuph and Onumanyi (2002) in their paper titled: New Multiple FDMs through Multistep Collocation for problem 3.1.2 ofthis lecture.