Chebyshev spectral solution of nonlinear Volterra-Hammerstein integral equations

Abstract

In this paper, the Chebyshev spectral (CS) method for the approximate solution of nonlinear Volterra-Hammerstein integral equations Y(τ) = F(τ) + fτ0 K(τ, r)G(r, Y(r))dr, τ ∈ [0, T], is investigated. The method is applied to approximate the solution not to the equation in its original form, but rather to an equivalent equation z(t) = g(t, y(t)), t ∈ [-1, 1]. The function z is approximated by the Nth degree interpolating polynomial zN, with coefficients determined by discretizing g(t, y(t)) at the Chebyshev-Gauss Lobatto nodes. We then define the approximation to y to be of the form yN(t) = f(t) + f1-1 k(t, s)zN(s)ds, t ∈ [-1, 1], and establish that, under suitable conditions, limN→∞ yN(0 = y(t) uniformly in t. Finally, a numerical experiment for a nonlinear Volterra-Hammerstein integral equation is presented, which confirms the convergence, demonstrates the applicability and the accuracy of the Chebyshev spectral (CS) method.