Asian Journal of Pure and Applied Mathematics

Converting ordinary differential equations (ODEs) into a recurrence relation provides a rapid discrete-domain iteration technique for simulation and computer modelling of complex natural systems. The aim is to derive a simple recursive method that could be applied to solve first to fourth-order ODEs in any science discipline. A recursion is simple to implement as an algorithm in a computational loop. Recurrence solutions are derived for first, third and fourth-order ODEs that are routinely encountered in many science disciplines. In this work, the central difference method is adopted to generate recursive formulas that approximate derivatives of the ODE. Use of the central difference method is shown to be less error prone and has speed and computational advantages over simpler Euler and more complex Runge-Kutta methods. Stability of computed results that depend on initial conditions are addressed, including solutions that show cyclic or chaotic behaviour. Systems containing time delays or forcing functions with unstable or chaotic solutions may be solvable by a recurrence approximation method. A central difference recursion is often overlooked in algorithm development, nevertheless its simplicity increases speed and reduces coding effort when evaluating ODEs and dynamic processes. One application of the central difference method is provided to show the dynamics of a dual mass-spring system requiring a solution to a second order ODE. Since non-linear terms are avoided with a central difference technique, its importance to scientific modelling is understated.