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

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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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Acknowledgements

The authors sincerely thank two anonymous reviewers for their valuable comments.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Zhaoxiang Li & Feng Zhang

  2. Department of Mathematics, Texas A &M University, College Station, TX, 77843, USA

    Jianxin Zhou

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  1. Zhaoxiang Li
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  2. Feng Zhang
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Correspondence to Jianxin Zhou.

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Supported in part by the NNSF of China (No. 12271366, 11871043, 12171322), the NSF of Shanghai, China (No. 21ZR1447200, 22ZR1445500) and the Science and Technology Innovation Plan of Shanghai (No. 20JC1414200).

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Li, Z., Zhang, F. & Zhou, J. Partial Newton-Correction Method for Multiple Fixed Points of Semi-linear Differential Operators by Legendre–Gauss–Lobatto Pseudospectral Method. J Sci Comput 97, 32 (2023). https://doi.org/10.1007/s10915-023-02341-z

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  • DOI: https://doi.org/10.1007/s10915-023-02341-z

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