A novel two-derivative multistep collocation method with fitting-techniques with application to Duffing problem

K.C. Lee, I. Hashim, M.N. Mohd Aris and N. Senu

Mathematics and Computers in Simulation (MATCOM), 2026, vol. 239, issue C, 420-441

Abstract: The general k-step fifth-order two-derivative linear multistep collocation method (TDLMM5) using collocation technique with Gegenbauer polynomial as basis function is derived for direct integrating second-order ordinary differential equation in the form u″(t)=f(t,u(t)) with periodic solution. Fifth-order two-derivative linear multistep method with various collocation points and off-set points is developed using collocation and interpolation approach. Order, stability, consistency and convergence of TDLMM5 are analyzed and discussed. Then, trigonometrically-fitting technique is adapted into TDLMM5 by setting u(t) as the linear combination of the functions {sin(λt),cos(λt)},λ∈R and turn the coefficients of TDLMM5 into frequency-dependent. Numerical experiment is conducted to verify the proposed method is superior compared to other existing methods in the literature with similar order. Additionally, the trigonometrically-fitted TDLMM5, denoted as TFTDLMM5, is applied to the well-known damped and driven oscillator problem, known as the Duffing problem. The outcome demonstrates that the proposed method is still successful in modeling this real-world application problem.

Keywords: Two-derivative linear multistep methods; Gegenbauer polynomial; Collocation; Second-order ordinary differential equations; Stability; Consistency; Trigonometrically-fitting technique (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:239:y:2026:i:c:p:420-441

DOI: 10.1016/j.matcom.2025.05.024

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