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Journal of Scientific Computing

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01-11-2023

Partial Newton-Correction Method for Multiple Fixed Points of Semi-linear Differential Operators by Legendre–Gauss–Lobatto Pseudospectral Method

Authors: Zhaoxiang Li, Feng Zhang, Jianxin Zhou

Published in: Journal of Scientific Computing | Issue 2/2023

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Abstract

Inspired by several numerical methods for finding multiple solutions, a partial Newton-correction method (PNCM) is proposed to find multiple fixed points of semi-linear differential operators. First a new augmented singular transform is developed to form a barrier so that an algorithm search outside the subspace generated by previously found fixed points cannot pass the barrier and penetrate into the inside to reach an old fixed point. Thus a fixed point found by an algorithm must be new. Its mathematical validations are established. A flow chart of PNCM is presented. Then a more accurate Legendre–Gauss–Lobatto pseudospectral scheme is constructed and convertes a semi-linear fixed point problem into a linear partial differential equation and an algebraic equation. It greatly simplifies the computation. Finally numerical results are presented to show the effectiveness of these approaches.

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Title: Partial Newton-Correction Method for Multiple Fixed Points of Semi-linear Differential Operators by Legendre–Gauss–Lobatto Pseudospectral Method
Authors: Zhaoxiang Li
Feng Zhang
Jianxin Zhou
Publication date: 01-11-2023
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 2/2023
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-023-02341-z

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