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Journal of Scientific Computing

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12-04-2015

Petviashvilli’s Method for the Dirichlet Problem

Authors: D. Olson, S. Shukla, G. Simpson, D. Spirn

Published in: Journal of Scientific Computing | Issue 1/2016

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Abstract

We examine Petviashvilli’s method for solving the equation \( \phi - \Delta \phi = |\phi |^{p-1} \phi \) on a bounded domain \(\Omega \subset \mathbb {R}^d\) with Dirichlet boundary conditions. We prove a local convergence result, using spectral analysis, akin to the result for the problem on \(\mathbb {R}\) by Pelinovsky and Stepanyants in [16]. We also prove a global convergence result by generating a suite of nonlinear inequalities for the iteration sequence, and we show that the sequence has a natural energy that decreases along the sequence.

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We use the definition \({{\mathrm{cn}}}= {{\mathrm{cn}}}(x;m)\) rather than \({{\mathrm{cn}}}= {{\mathrm{cn}}}(x; k^2)\).

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Title: Petviashvilli’s Method for the Dirichlet Problem
Authors: D. Olson
S. Shukla
G. Simpson
D. Spirn
Publication date: 12-04-2015
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2016
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-015-0023-6

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