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Chebyshev–Picard iteration methods for solving delay differential equations

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  • Zhou, Quan
  • Wang, Yinkun
  • Liu, Yicheng

Abstract

In this paper, we propose an effective Chebyshev–Picard iteration (CPI) method for solving delay differential equations with a constant delay. This approach adopts the Chebyshev series to represent the solution and improves the accuracy of the solution by successive Picard iterations. The CPI method is implemented in a matrix–vector form efficiently without matrix inversion. We also present a multi-interval CPI method for solving long-term simulation problems. Further, the convergence of the CPI method is analyzed by evaluating the eigenvalues of the coefficient matrices of the iteration. Several numerical experiments including both the linear and nonlinear systems with delay effects are presented to demonstrate the high accuracy and efficiency of the CPI method by comparison with the classic methods.

Suggested Citation

  • Zhou, Quan & Wang, Yinkun & Liu, Yicheng, 2024. "Chebyshev–Picard iteration methods for solving delay differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 217(C), pages 1-20.
    • Handle: RePEc:eee:matcom:v:217:y:2024:i:c:p:1-20
    DOI: 10.1016/j.matcom.2023.09.023
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    References listed on IDEAS

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    1. Tafakkori–Bafghi, M. & Loghmani, G.B. & Heydari, M., 2022. "Numerical solution of two-point nonlinear boundary value problems via Legendre–Picard iteration method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 199(C), pages 133-159.
    2. Yan, Xiaoqiang & Zhang, Chengjian, 2019. "Solving nonlinear functional–differential and functional equations with constant delay via block boundary value methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 166(C), pages 21-32.
    3. Hongliang Liu & Aiguo Xiao & Lihong Su, 2013. "Convergence of Variational Iteration Method for Second-Order Delay Differential Equations," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-9, February.
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