Abstract
The paper explored the applicability of the Dzhumabaev parametrization method to Cauchy problems with nonlinear integrodifferential operator equation arising in applied mathematics. The proposed method reformulated the problem under consideration as an equivalent parametric multipoint problem. Subsequently, linearization of the problem and stepwise solution of a sequence of linear approximations was used to solve the parametric problem. Particular emphasis was placed on solving the nonlinear special Cauchy problem for Fredholm integrodifferential equations. A novel strategy was employed to determine solutions of the original problem through this approach. The effectiveness of the proposed approach was demonstrated by numerical examples.