Nonlinear Theory and Its Applications, IEICE
Online ISSN : 2185-4106
Special section on Recent Progress in Verified Numerical Computations
An algorithm of identifying parameters satisfying a sufficient condition of Plum's Newton-Kantorovich like existence theorem for nonlinear operator equations
Kouta SekineAkitoshi TakayasuShin'ichi Oishi
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2014 Volume 5 Issue 1 Pages 64-79

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Abstract

This paper presents an algorithm of identifying parameters satisfying a sufficient condition of Plum's Newton-Kantorovich like theorem. Plum's theorem yields a numerical existence test of solutions for nonlinear partial differential equations. The sufficient condition of Plum's theorem is given by the nonemptiness of a region defined by one dimensional nonlinear inequalities. The aim of this paper is to develop a systematic method of constructing an inner inclusion of this region. If $\underline\rho \in {\mathbb R}^+$ is the minimum included in this region, $\underline\rho$ gives the minimum of the error bounds. Moreover, if $\overline\rho \in {\mathbb R}^+$ is the maximum included in this region, then $\overline\rho$ gives the maximum radius of a ball in which the exact solution is unique. In this paper, an algorithm is developed for finding $\rho_e$ and $\rho_u$ such that they belong to this region and become close approximations of $\underline\rho$ and $\overline\rho$, respectively. Finally, to illustrate features of Plum's theorem with our proposed algorithm, some numerical results compared with results by Plum's theorem with the Newton method are presented. In addition to this, Plum's theorem with our algorithm is also compared with Newton-Kantorovich's theorem. One of the most important facts found in this paper is that, for some examples, $\rho_e$ become smaller than error bounds obtained by Newton-Kantorovich's theorem. Moreover, also for these examples, $\rho_u$ become greater than regions indicating uniqueness of the exact solution derived by Newton-Kantorovich's theorem. This implies that Plum's theorem can be seen as a modification of Newton-Kantorovich's theorem.

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