Chebyshev Spectral Collocation with an Exact Operational Integration Matrix for Nonlinear Volterra–Hammerstein Equations

Abstract

This paper proposes a Chebyshev spectral collocation method for nonlinear Volterra–Hammerstein integral equations of the second kind with smooth kernels. We approximate the unknown solution using a finite Chebyshev expansion and place collocation points at the Chebyshev–Gauss–Lobatto nodes. Conventional methods apply separate grids for collocation and quadrature. In contrast, we discretise the integral operator using a Chebyshev operational integration matrix that is exact for polynomials of degree up to N. This construction allows collocation and integration to share a single node set, leaving the integrand’s interpolation error as the dominant remaining error source. We present a complete convergence analysis in Lω2(I) and L∞(I). Under a strict contraction assumption on the Volterra integral operator and for m-times continuously differentiable data, we establish ∥u−uN∥Lω2(I)≤CN−m and ∥u−uN∥L∞(I)≤CN12−m. Numerical experiments validate our theoretical findings. For the epidemiological SIR model, our scheme achieves an accuracy of 10−8 using only N=12 collocation points. A systematic comparison with a Gauss–Legendre quadrature-based Chebyshev collocation method clarifies the distinct advantages of each approach. The quadrature-based method yields higher accuracy on smooth problems by factors between 10 and 105. Meanwhile, the operational integration matrix reduces the wall-clock runtime per Newton iteration by up to a factor of 8 at N=80. Thus, this work delivers analytical and structural contributions: a self-contained convergence theory for the operational matrix approach and a unified node set for collocation and integration. A lower computational cost during repeated Newton iterations is also demonstrated.