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<p>A central challenge in kernel-based approximation theory is the trade-off between approximation accuracy and numerical stability. Hybrid radial kernels (HRKs) have emerged as an effective strategy to mitigate this conflict. In this paper, we propose a meshless collocation scheme utilizing HRKs for the stable and accurate numerical solution of <i>d</i>-dimensional nonlinear Volterra integral equations (VIEs). The proposed method constructs a hybrid basis by blending distinct radial kernels, with the shape parameters and weight coefficients optimized via a modified Particle Swarm Optimization (PSO) algorithm. To ensure the theoretical foundations of the scheme, a rigorous convergence analysis is provided. Numerical experiments across various dimensional cases demonstrate that the HRK-PSO approach effectively circumvents the ill-conditioning typically associated with small shape parameters in standard RBF methods. Comparative results show that the proposed technique outperforms established benchmarks, such as Haar wavelet and HS-SVD methods, in terms of both precision and numerical robustness.</p>

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A PSO-optimized hybrid radial kernel method for the stable and accurate solution of Volterra integral equations

  • 期刊论文

  • 发表日期：

    2026年4月21日

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  • Tahereh Akbari,
  • Mohsen Esmaeilbeigi,
  • Davoud Moazami

摘要

A central challenge in kernel-based approximation theory is the trade-off between approximation accuracy and numerical stability. Hybrid radial kernels (HRKs) have emerged as an effective strategy to mitigate this conflict. In this paper, we propose a meshless collocation scheme utilizing HRKs for the stable and accurate numerical solution of d-dimensional nonlinear Volterra integral equations (VIEs). The proposed method constructs a hybrid basis by blending distinct radial kernels, with the shape parameters and weight coefficients optimized via a modified Particle Swarm Optimization (PSO) algorithm. To ensure the theoretical foundations of the scheme, a rigorous convergence analysis is provided. Numerical experiments across various dimensional cases demonstrate that the HRK-PSO approach effectively circumvents the ill-conditioning typically associated with small shape parameters in standard RBF methods. Comparative results show that the proposed technique outperforms established benchmarks, such as Haar wavelet and HS-SVD methods, in terms of both precision and numerical robustness.