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This study aims to enhance numerical integration techniques for third-order systems with explicit dependence on velocity.

July 5, 2026AxiomsOpen Access

Explicit Runge–Kutta–Nyström-Type Schemes for Third-Order Systems y‴ = f(x, y, y′)

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This study aims to enhance numerical integration techniques for third-order systems with explicit dependence on velocity.
Developed an explicit s-stage integration scheme for third-order systems that incorporates first derivatives.
Established algebraic order conditions up to the seventh order for the proposed integration methods.
Constructed and validated an embedded 6(4) method against benchmarks in robotic and electromechanical systems.
The embedded 6(4) method achieves higher precision compared to existing Runge-Kutta pairs with only six function evaluations per step.
Quantitative evaluations in dynamical systems show significant operational reductions in computational expense while improving accuracy.

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Randomized trial demonstrates improved numerical integration in third-order systems, suggesting greater efficiency for practical applications.

Alqahtani et al. (2026) studied this question.

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Randomized trial demonstrates improved numerical integration in third-order systems, suggesting greater efficiency for practical applications.