Quasilinearization and Numerical Solution of Nonlinear Volterra Integro-differential Equations

When we use the projection methods in order to obtain the approximation solution of nonlinear equations, we always have some difficulties such as solving nonlinear algebraic systems. The method of generalized quasilinearization when is applied to the nonlinear integro-differential equations of Volterra type, gives two sequences of linear integro-differential equations with solutions monotonically and quadratically convergent to the solution of nonlinear equation. In this paper we employ step-by-step collocation method to solve the linear equations numerically and then approximate the solution of the nonlinear equation. In this manner we do not encounter solving nonlinear algebraic systems. Error analysis of the method is performed and to show the accuracy of the method some numerical examples are proposed.