TABLE OF CONTENTS This chapter is entirely dedicated to the presentation of several approximation methods of the solution of a given Cauchy problem. Although these methods are no longer used in their original form, they are still of interest in many effective numerical algorithms. In the first section we prove that a Cauchy problem has only analytic solutions whenever the right-hand side of the corresponding differential equation is an analytic function. This theorem is, on one hand, an approximation result (ensuring the possibility to develop any solution in power series), and on the other hand a sufficient condition for the regularity of solutions. In the next three sections we discuss: the method of successive approximations, the method of polygonal lines, known also as Euler explicit method, and Euler implicit method. The chapter ends with a set of exercises and problems.
In this section, using the so-called majorant series method proposed by Cauchy and improved by Lindel ff, we shall prove that, whenever f is analytic on 
, the unique solution x of the Cauchy problem CP(D) is also analytic on its domain. This result allows us, either to approximate the solution by a partial sum of the power series which defines it, or even to find the solution explicitly as a power series,
This method of solving of a Cauchy problem by means of power series is one of the oldest and effective. In order to illustrate it, let us analyze the...
