TY - JOUR

T1 - Rigorous Numerical Enclosures for Positive Solutions of Lane–Emden’s Equation with Sub-Square Exponents

AU - Tanaka, Kazuaki

AU - Plum, Michael

AU - Sekine, Kouta

AU - Kashiwagi, Masahide

AU - Oishi, Shin’ichi

N1 - Funding Information:
This work was supported by CREST, JST Grant Number JPMJCR14D4 and by JSPS KAKENHI Grant Number JP19K14601.
Publisher Copyright:
© 2022 Taylor & Francis Group, LLC.

PY - 2022

Y1 - 2022

N2 - The purpose of this paper is to obtain rigorous numerical enclosures for solutions of Lane–Emden’s equation (Formula presented.) with homogeneous Dirichlet boundary conditions. We prove the existence of a nondegenerate solution u nearby a numerically computed approximation (Formula presented.) together with an explicit error bound, i.e., a bound for the difference between u and (Formula presented.) In particular, we focus on the sub-square case in which (Formula presented.) so that the derivative (Formula presented.) of the nonlinearity (Formula presented.) is not Lipschitz continuous. In this case, it is problematic to apply the classical Newton-Kantorovich theorem for obtaining the existence proof, and moreover several difficulties arise in the procedures to obtain numerical integrations rigorously. We design a method for enclosing the required integrations explicitly, proving the existence of a desired solution based on a generalized Newton-Kantorovich theorem. A numerical example is presented where an explicit solution-enclosure is obtained for (Formula presented.) on the unit square domain (Formula presented.).

AB - The purpose of this paper is to obtain rigorous numerical enclosures for solutions of Lane–Emden’s equation (Formula presented.) with homogeneous Dirichlet boundary conditions. We prove the existence of a nondegenerate solution u nearby a numerically computed approximation (Formula presented.) together with an explicit error bound, i.e., a bound for the difference between u and (Formula presented.) In particular, we focus on the sub-square case in which (Formula presented.) so that the derivative (Formula presented.) of the nonlinearity (Formula presented.) is not Lipschitz continuous. In this case, it is problematic to apply the classical Newton-Kantorovich theorem for obtaining the existence proof, and moreover several difficulties arise in the procedures to obtain numerical integrations rigorously. We design a method for enclosing the required integrations explicitly, proving the existence of a desired solution based on a generalized Newton-Kantorovich theorem. A numerical example is presented where an explicit solution-enclosure is obtained for (Formula presented.) on the unit square domain (Formula presented.).

KW - Computer-assisted proofs

KW - elliptic boundary value problems

KW - Lane–Emden’s equation

KW - numerical verification

KW - positive solutions

KW - rigorous enclosures

KW - sub-square exponent

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U2 - 10.1080/01630563.2022.2029485

DO - 10.1080/01630563.2022.2029485

M3 - Article

AN - SCOPUS:85129252708

VL - 43

SP - 322

EP - 349

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

SN - 0163-0563

IS - 3

ER -