Chebyshev spectral solution of nonlinear Volterra-Hammerstein integral equations

[摘要] In this paper, the Chebyshev spectral (CS) method for the approximate solution of nonlinear Volterra-Hammerstein integral equations Y(tau = F(tau) + integral(0)(t)au K(tau, r)G(r, Y(r)) dr, tau epsilon [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 epsilon [-1, 1]. The function z is approximated by the Nth degree interpolating polynomial z(N), 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 y(N)(t) = f(t) + integral(-1)(1) (k) over cap(t,s)z(N)(s)ds, t epsilon [-1,1], and establish that, under suitable conditions, lim(N-->infinity) y(N)(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.

[发布日期] 1996-12-17

[关键词] spectral Chebyshev projection;Volterra-Hammerstein;Chebyshev-Gauss rule