Rigorous Numerical Enclosures for Positive Solutions of Lane–Emden's Equation with Sub-Square Exponents. (17th February 2022)

• Record Type: Journal Article
• Authors: Tanaka, Kazuaki; Plum, Michael; Sekine, Kouta; Kashiwagi, Masahide; Oishi, Shin'ichi
• Is Part Of: Numerical functional analysis and optimization. Volume 43:Number 3(2022)
• Page Start: 322
• Page End: 349
• Publication Date: 2022-02-17
• Subjects: Computer-assisted proofs -- elliptic boundary value problems -- Lane–Emden's equation -- numerical verification -- positive solutions -- rigorous enclosures -- sub-square exponent; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2022.2029485
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: The purpose of this paper is to obtain rigorous numerical enclosures for solutions of Lane–Emden's equation − Δ u = | u | p − 1 u with homogeneous Dirichlet boundary conditions. We prove the existence of a nondegenerate solution u nearby a numerically computed approximation u ̂ together with an explicit error bound, i.e., a bound for the difference between u and u ̂ . In particular, we focus on the sub-square case in which 1 < p < 2 so that the derivative p | u | p − 1 of the nonlinearity | u | p − 1 u 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 p = 3 / 2 on the unit square domain Ω = ( 0, 1 ) 2 .