Falkner-Type Numerical Schemes for the Solution of Second Order Initial Value Problems in Ordinary Differential Equations

Abstract

Many physical phenomena in thermodynamics, electromagnetism, and classical mechanics are modelled using initial value problems of high-order differential equations. Since exact analytical solutions are usually impossible to find, researchers must rely on fast, accurate numerical methods instead. This paper utilises the collocation and interpolation technique to develop a new class of sixth-order Falkner-type schemes. Implementing these schemes as a block method allows high-order differential equations to be solved directly. This approach completely skips the standard, computationally heavy step of breaking down a high-order equation into a large system of first-order equations, which minimises processing time and avoids built-up calculation errors. The core properties of these new schemes—namely consistency, stability, and convergence—are thoroughly analysed to prove theoretical validity. Numerical experiments are conducted to verify the method in practice, with the results presented in clear data tables. These numerical results show outstanding precision, yielding absolute errors as low as 10⁻²⁷. The outcomes prove that the proposed block method is a highly effective, fast, and practical tool for solving these specific physical problems.