A new type of approximating sequence for the solution of the Cauchy problem consisting of piecewise linear functions

This paper continues the study of a new successive approximation method for solving the Cauchy problem. The motivation for considering such a method consists in the fact that in practice real difficulties arise in the problem of computing integrals from the respective recurrence relations. We give below an original method to avoid these difficulties by defining a similar recurrence for solving the Cauchy problem, so that the obtaining approximating sequence consists in functions which are piecewise linear. We prove the (uniform) convergence and finally, a numerical example is given.