A numerical verification method for solutions of initial value problems for ODEs using a linearized inverse operator

We propose a new verification method to enclose solutions for initial value problems of systems of first-order nonlinear ordinary differential equations (ODEs) using a linearized inverse operator. The proposed approach can verify the existence and local uniqueness of the exact solution independent of the choice of the approximation scheme, while the existing methods usually depend on the numerical scheme for the approximate solution. In contrast, most of the well-known verification methods to enclose solutions for nonlinear ODEs work only on the specified approximate solution. Namely, in the existing verification methods the numerical scheme for computing an approximate solution is essentially limited to the Taylor method . Therefore, one of our purposes is to develop a verification method that can obtain guaranteed error bounds independent of the approximation scheme. We will present numerical examples of the proposed verification method that obtain rigorous error bounds of the approximate solutions obtained by the Euler method or the second-order Runge-Kutta method.