A Pseudospectral Method for Hammerstein Equations

This paper considers Hammerstein equations 1 w x y t sf t q k t, s g s, y s ds, tg y1, 1 , . . . . . Hy 1 and presents a pseudospectral method for solving them numerically. The method is applied to approximate the solution not to the equation in its original form, but rather to an equivalent equation z t.sg t, y t... The function z is approximated by a polynomial z N of degree N, with coefficients determined by collocation at the Legendre]Gauss]Lobatto nodes. We define the desire approximation to y by yN t.sf t.qH1 k t, s.zN s. ds, tgwy1, 1x, y1 and establish that, under suitable conditions, limNª` yN t.sy t., uniformly in t. Advantages of the proposed method, compared with existing methods in the literature, are discussed. An illustrative example is included to show the accuracy of the proposed method