Adaptive residual‑controlled overlapping Taylor continuation framework for nonlinear delay differential equations

Çevik M., Baykuş Savaşaneril N.

INTERNATIONAL JOURNAL OF ADVANCES IN ENGINEERING SCIENCES AND APPLIED MATHEMATICS, cilt.18, sa.5, ss.47-64, 2026 (ESCI)

Özet

Delay differential equations (DDEs) model hereditary effects and delayed feedback in scientific and engineering systems, but their numerical solution becomes challenging when delays vary with time, derivatives are delayed, or nonlinear dynamics persist over extended intervals. This study introduces the adaptive residual-controlled overlapping Taylor (ARCOT) method, which combines high-order local polynomial reconstruction, residual-based acceptance and rejection, adaptive interval regulation, and overlapping continuation. The governing-equation residual is used to regulate successive approximation intervals, while delayed states are evaluated through available polynomial representations. The method is assessed using five benchmarks involving proportional delays, nonlinear time-dependent delays, higher-order equations with delayed derivatives, and nonlinear delayed-feedback systems, including the Mackey–Glass equation. Numerical experiments demonstrate systematic convergence under interval refinement, small errors against analytical or independently computed reference solutions, and sampled residuals satisfying prescribed tolerances. Comparisons with RK4 and DOP853 quantify computational cost and accuracy across the investigated configurations. Extended Mackey–Glass simulations show bounded numerical trajectories and close agreement with a refined DOP853 reference in finite-time solutions, post-transient statistics, and delay portraits. Although computational efficiency varies across problems and the results do not establish general long-time stability or chaotic attractor preservation, the findings demonstrate that ARCOT provides an accurate and adaptable numerical framework for the investigated classes of nonlinear DDEs.