Abstract

Initial value problems of the third order featuring explicit dependence on velocity, denoted as y‴=f(x,y,y′), emerge regularly across applications such as electromechanical networks, structural mechanics, and robotic trajectory control. Despite their practical prevalence, these differential equations remain insufficiently addressed by standard numerical integration techniques. Orthodox Runge–Kutta–Nyström (RKN) schemes are fundamentally formulated for differential equations lacking the first derivative, specifically y‴=f(x,y). Due to this algorithmic constraint, researchers frequently resort to computationally demanding first-order system reductions or rely upon standard Runge–Kutta methods. The present study resolves this methodological gap by defining an explicit s-stage integration architecture that natively incorporates the first derivative within the internal stage evaluations. Such structural modifications require the deployment of a supplementary coefficient matrix, denoted as D, to formulate the corresponding order theory. The complete set of algebraic order conditions is systematically established up to the seventh order, accompanied by a generic mathematical framework for generating schemes of arbitrary order. Based on this analytical foundation, an embedded 6(4) method is constructed. This specific pair achieves strict error tolerances utilizing merely six function evaluations per integration step, representing a substantial operational reduction compared to the eight computations strictly required by equivalent Runge–Kutta pairs. Direct numerical integration of the native third-order system prevents the dimensionality increase from reducing to first-order systems. Performance validation of the numerical solver involves two representative physical benchmarks: a coupled robotic appendage subjected to platform excitation and an electromechanical actuator array regulated by transient control inputs. Both dynamical systems exhibit severe velocity-dependent dissipation mechanisms and nonlinear external forcing. Quantitative numerical evaluations confirm that the constructed 6(4) pair yields higher precision and demands less computational expenditure than prevailing RK and RKN integrators. The analytical and empirical findings establish that derivative-capable Nyström integration algorithms furnish mathematically rigorous and computationally efficient numerical solutions for velocity-coupled third-order dynamics.

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