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New applications of Picardʼs successive approximations

✍ Scribed by Janne Gröhn

Publisher
Elsevier Science
Year
2011
Tongue
French
Weight
134 KB
Volume
135
Category
Article
ISSN
0007-4497

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✦ Synopsis

The iterative method of successive approximations, originally introduced by Émile Picard in 1890, is a basic tool for proving the existence of solutions of initial value problems regarding ordinary first order differential equations. In the present paper, it is shown that this method can be modified to get estimates for the growth of solutions of linear differential equations of the type

with analytic coefficients. A short comparison to the growth results in the literature, obtained by means of different methods, is also given. It turns out that many known results can be proved by applying Picard's successive approximations in an effective way. Self-contained considerations are carried out in the complex plane and in the unit disc, and some remarks about solutions of real linear differential equations are made.

📜 SIMILAR VOLUMES

Picard's successive approximation for no
Picard's successive approximation for non-linear two-point boundary-value problems
✍ M Lal; D Moffatt 📂 Article 📅 1982 🏛 Elsevier Science 🌐 English ⚖ 255 KB

It is demonstrated that Picard's successive approximation provides a simple and efficient method for solving linear and non-linear two-point boundary-value problems. For problems, where intrinsic convergence is slow, the method can be readily modified to speed up convergence.