A Two-step with First and Second Derivative Scheme for Numerical Solution of First-Order Problems in Dynamical Systems

One of the numerical techniques used to solve differential equations is the linear multistep method (LMM). A two-step second-derivative intra-point block numerical method of uniform order ten is proposed for solving dynamical systems in ordinary differential equations. The derived two-step method with multi-derivatives effectively addresses the challenges in solving nonlinear dynamical systems – exhibiting phenomena such as multiple steady states, oscillations, and chaos. The inclusion of second derivative in the block method makes sure more information about the ODE is used in generating the solution thereby improving the accuracy of the method. The method is A-stable, making it suitable for solving nonlinear dynamic systems in ordinary differential equations (ODEs). In addition, the method possesses a higher order of accuracy, and the associated error constants are very small. This block method generates numerical solutions that provide solution profiles and phase portraits for the problems considered under various situations of dynamical systems. The results generated from this method underscore its potential as a robust and versatile tool for solving a wide range of practical problems arising in real-life.