Cauchy systems for fredholm integral equations with parameter imbedding

Consider the family of Fredholm integral equations u(t,gamma) = g(t) + gamm a integral(0)(1) k(t,gamma)u(y,gamma)dy, where gamma is sufficiently small to guarantee a solution, and the Cauchy system u(gamma)(t, gamma) = integra l(0)(1) K(t,y,gamma)u(y,gamma)dy, K-gamma(t,y,gamma) = integral(0)(1) K(t,y ',gamma)K(y',y,gamma)dy', 0 less than or equal to t, y less than or equal t o 1, 0 less than or equal to y, u(t,0) = g(t), K(t,gamma,0) = k(t,y), 0 les s than or equal to t, y less than or equal to 1. The equivalence between th e family of Fredholm integral equations and the Cauchy system is demonstrat ed. The numerical method is illustrated with an example. (C) 2000 Elsevier Science Inc. All rights reserved.