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-...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Elsevier B.V.
2026
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| Online Access: | http://psasir.upm.edu.my/id/eprint/119992/ http://psasir.upm.edu.my/id/eprint/119992/1/119992.pdf |
| _version_ | 1848868093600727040 |
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| author | Lee, K.C. Hashim, I. Mohd Aris, M.N. Senu, N. |
| author_facet | Lee, K.C. Hashim, I. Mohd Aris, M.N. Senu, N. |
| author_sort | Lee, K.C. |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-15T14:46:55Z |
| format | Article |
| id | upm-119992 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-15T14:46:55Z |
| publishDate | 2026 |
| publisher | Elsevier B.V. |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-1199922025-09-22T07:04:55Z http://psasir.upm.edu.my/id/eprint/119992/ A novel two-derivative multistep collocation method with fitting-techniques with application to Duffing problem Lee, K.C. Hashim, I. Mohd Aris, M.N. Senu, N. 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. Elsevier B.V. 2026-01 Article PeerReviewed text en cc_by_4 http://psasir.upm.edu.my/id/eprint/119992/1/119992.pdf Lee, K.C. and Hashim, I. and Mohd Aris, M.N. and Senu, N. (2026) A novel two-derivative multistep collocation method with fitting-techniques with application to Duffing problem. Mathematics and Computers in Simulation, 239. pp. 420-441. ISSN 0378-4754 https://linkinghub.elsevier.com/retrieve/pii/S0378475425002174 10.1016/j.matcom.2025.05.024 |
| spellingShingle | Lee, K.C. Hashim, I. Mohd Aris, M.N. Senu, N. A novel two-derivative multistep collocation method with fitting-techniques with application to Duffing problem |
| title | A novel two-derivative multistep collocation method with fitting-techniques with application to Duffing problem |
| title_full | A novel two-derivative multistep collocation method with fitting-techniques with application to Duffing problem |
| title_fullStr | A novel two-derivative multistep collocation method with fitting-techniques with application to Duffing problem |
| title_full_unstemmed | A novel two-derivative multistep collocation method with fitting-techniques with application to Duffing problem |
| title_short | A novel two-derivative multistep collocation method with fitting-techniques with application to Duffing problem |
| title_sort | novel two-derivative multistep collocation method with fitting-techniques with application to duffing problem |
| url | http://psasir.upm.edu.my/id/eprint/119992/ http://psasir.upm.edu.my/id/eprint/119992/ http://psasir.upm.edu.my/id/eprint/119992/ http://psasir.upm.edu.my/id/eprint/119992/1/119992.pdf |