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

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.