Solving the Duffing Equation via Hybrid Fourier-Block-Pulse Basis Functions

Articles in Press

Document Type : Research article

Authors

Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, P. O. Box 35195-363, Semnan, Iran

Abstract

In this paper, a novel numerical method based on hybrid Fourier and block-pulse functions (HFBPFs) is proposed for solving the nonlinear Duffing equation. Although hybrid Fourier-block-pulse bases have been previously applied to linear integral equations and calculus of variations problems, their application to the Duffing equation---a benchmark nonlinear oscillator with cubic stiffness---has not been reported in the literature. The proposed method combines the local behavior of block-pulse functions with the spectral accuracy of Fourier series within each subinterval. The operational matrix of integration and the product operational matrix are derived for the hybrid basis, and the Duffing equation is reduced to a system of nonlinear algebraic equations. Numerical examples demonstrate that the HFBPFs method achieves approximately three orders of magnitude improvement in accuracy compared with pure block-pulse functions, while avoiding the Gibbs phenomenon that affects pure Fourier approximations for non-smooth solutions. The method offers dual tunability through the number of subintervals and the number of Fourier components per subinterval, providing flexibility in balancing accuracy and computational cost.

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Main Subjects

Analytical and Numerical Solutions for Nonlinear Equations

Articles in Press, Accepted Manuscript
Available Online from 06 October 2026

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History

  • Receive Date: 01 August 2026
  • Accept Date: 04 October 2026
  • Publish Date: 06 October 2026

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