Inverse norm estimation of perturbed Laplace operators and corresponding eigenvalue problems

Kouta Sekine; Kazuaki Tanaka; Shin'ichi Oishi

doi:10.1016/j.camwa.2021.12.002

Inverse norm estimation of perturbed Laplace operators and corresponding eigenvalue problems

Kouta Sekine^*, Kazuaki Tanaka, Shin'ichi Oishi

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

In numerical existence proofs for solutions of the semi-linear elliptic system, evaluating the norm of the inverse of a perturbed Laplace operator plays an important role. We reveal an eigenvalue problem to design a method for verifying the invertibility of the operator and evaluating the norm of its inverse based on Liu's method and the Temple-Lehmann-Goerisch method. We apply the inverse-norm's estimation to the Dirichlet boundary value problem of the Lotka-Volterra system with diffusion terms and confirm the efficacy of our method.

Original language	English
Pages (from-to)	18-26
Number of pages	9
Journal	Computers and Mathematics with Applications
Volume	106
DOIs	https://doi.org/10.1016/j.camwa.2021.12.002
Publication status	Published - 2022 Jan 15

Keywords

Computer-assisted proofs
Eigenvalue evaluation
Norm of inverse operators
Rigorous numerical computations
System of partial differential equations

ASJC Scopus subject areas

Modelling and Simulation
Computational Theory and Mathematics
Computational Mathematics

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10.1016/j.camwa.2021.12.002

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title = "Inverse norm estimation of perturbed Laplace operators and corresponding eigenvalue problems",

abstract = "In numerical existence proofs for solutions of the semi-linear elliptic system, evaluating the norm of the inverse of a perturbed Laplace operator plays an important role. We reveal an eigenvalue problem to design a method for verifying the invertibility of the operator and evaluating the norm of its inverse based on Liu's method and the Temple-Lehmann-Goerisch method. We apply the inverse-norm's estimation to the Dirichlet boundary value problem of the Lotka-Volterra system with diffusion terms and confirm the efficacy of our method.",

keywords = "Computer-assisted proofs, Eigenvalue evaluation, Norm of inverse operators, Rigorous numerical computations, System of partial differential equations",

author = "Kouta Sekine and Kazuaki Tanaka and Shin'ichi Oishi",

note = "Publisher Copyright: {\textcopyright} 2021 The Authors",

year = "2022",

month = jan,

day = "15",

doi = "10.1016/j.camwa.2021.12.002",

language = "English",

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TY - JOUR

T1 - Inverse norm estimation of perturbed Laplace operators and corresponding eigenvalue problems

AU - Sekine, Kouta

AU - Tanaka, Kazuaki

AU - Oishi, Shin'ichi

PY - 2022/1/15

Y1 - 2022/1/15

N2 - In numerical existence proofs for solutions of the semi-linear elliptic system, evaluating the norm of the inverse of a perturbed Laplace operator plays an important role. We reveal an eigenvalue problem to design a method for verifying the invertibility of the operator and evaluating the norm of its inverse based on Liu's method and the Temple-Lehmann-Goerisch method. We apply the inverse-norm's estimation to the Dirichlet boundary value problem of the Lotka-Volterra system with diffusion terms and confirm the efficacy of our method.

AB - In numerical existence proofs for solutions of the semi-linear elliptic system, evaluating the norm of the inverse of a perturbed Laplace operator plays an important role. We reveal an eigenvalue problem to design a method for verifying the invertibility of the operator and evaluating the norm of its inverse based on Liu's method and the Temple-Lehmann-Goerisch method. We apply the inverse-norm's estimation to the Dirichlet boundary value problem of the Lotka-Volterra system with diffusion terms and confirm the efficacy of our method.

KW - Computer-assisted proofs

KW - Eigenvalue evaluation

KW - Norm of inverse operators

KW - Rigorous numerical computations

KW - System of partial differential equations

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U2 - 10.1016/j.camwa.2021.12.002

DO - 10.1016/j.camwa.2021.12.002

M3 - Article

AN - SCOPUS:85120953550

SN - 0898-1221

VL - 106

SP - 18

EP - 26

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

ER -

Inverse norm estimation of perturbed Laplace operators and corresponding eigenvalue problems

Abstract

Keywords

ASJC Scopus subject areas

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