Chebyshev spectral solution of nonlinear Volterra-Hammerstein integral equations

In this paper, the Chebyshev spectral (CS) method for the approximate solution of nonlinear Volterra-Hammerstein integral equations (τ)=F(τ)+∫oτK(τr)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 Labatto nodes. We then define the approximation to y to be of the form yN(t)=f(t)+∫-11K(t,s)zN(s)ds,τ∈[-1,1] and establish that, under suitable conditions, limN→∞ yN(t) = 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.