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<p>This study introduces a novel computational technique for solving fractional-order Duffing equations by incorporating fractional-order Liouville–Caputo derivatives and two integral boundary conditions. This extension aims to more accurately capture the dynamics of systems exhibiting memory effects. For the first time, we apply the quasilinearization-Lerch matrix collocation (QLM-Lerch) technique to the fractional-order Duffing equation. This innovative method combines quasilinearization, which simplifies nonlinear differential equations, with the Lerch matrix collocation method, renowned for its efficiency and accuracy in addressing complex boundary conditions. The effectiveness of the QLM-Lerch technique is validated through a detailed residual error analysis across various test examples, demonstrating its superior accuracy and reliability. The results highlight the QLM-Lerch approach’s effectiveness in solving the Duffing equation, showcasing improved performance compared to existing methods in the literature.</p>

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Fractional-order forcing nonlinear Duffing equations under integral boundary conditions: QLM-Lerch matrix collocation methodology

  • 期刊论文

  • 发表日期：

    2025年9月12日

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  • Mohammad Izadi,
  • Shah Jahan,
  • Waleed Adel

摘要

This study introduces a novel computational technique for solving fractional-order Duffing equations by incorporating fractional-order Liouville–Caputo derivatives and two integral boundary conditions. This extension aims to more accurately capture the dynamics of systems exhibiting memory effects. For the first time, we apply the quasilinearization-Lerch matrix collocation (QLM-Lerch) technique to the fractional-order Duffing equation. This innovative method combines quasilinearization, which simplifies nonlinear differential equations, with the Lerch matrix collocation method, renowned for its efficiency and accuracy in addressing complex boundary conditions. The effectiveness of the QLM-Lerch technique is validated through a detailed residual error analysis across various test examples, demonstrating its superior accuracy and reliability. The results highlight the QLM-Lerch approach’s effectiveness in solving the Duffing equation, showcasing improved performance compared to existing methods in the literature.