Derivation of Four-Point Integrator for the Solutions of Second-order Oscillatory Problems
DOI:
https://doi.org/10.62054/ijdm/0104.01Keywords:
A-stability, Four-point, Integrator, Oscillation, Second-orderAbstract
The research introduces a new four-point integrator (FPI) for solving second-order differential equations characterized by oscillatory behavior. The proposed FPI is developed through a continuous scheme within a linear multistep framework, utilizing two off-step points to enhance efficiency. Unlike conventional methods that reduce higher-order equations to first-order systems, often resulting in loss of inherent characteristics, the FPI maintains essential equation properties. Through this approach, the FPI enables a self-starting, block-by-block algorithmic implementation that does not require predictor values, simplifying computation. The integrator’s properties are rigorously analyzed, confirming its consistency, zero-stability, and convergence, all of which contribute to its reliability in handling oscillatory problems. Additionally, stability analysis, including the region of absolute stability, demonstrates the A-stability of the integrator. Numerical experiments conducted on some second-order problems reveal the FPI’s computational accuracy, effectively comparing with established solvers such as ode45, ode15s and other methods. These results highlight the potential of the FPI as a viable alternative for directly solving oscillatory differential equations with damping characteristics, providing computational efficiency and accuracy.
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