Precise Asymptotics for Bifurcation Curve of Nonlinear Ordinary Differential Equation

Shibata, Tetsutaro

doi:10.3390/math8081272

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2020

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“…In what follows, we denote by C the various positive constants independent of α. We modify the time-map method used in [6]. By (1), we have…”

Section: Proof Of Theoremmentioning

confidence: 99%

Asymptotic Behavior of Solution to Nonlinear Eigenvalue Problem

Shibata

2020

Mathematics

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We study the following nonlinear eigenvalue problem: −u″(t)=λf(u(t)),u(t)>0,t∈I:=(−1,1),u(±1)=0, where f(u)=log(1+u) and λ>0 is a parameter. Then λ is a continuous function of α>0, where α is the maximum norm α=∥uλ∥∞ of the solution uλ associated with λ. We establish the precise asymptotic formula for L1-norm of the solution ∥uα∥1 as α→∞ up to the second term and propose a numerical approach to obtain the asymptotic expansion formula for ∥uα∥1.

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“…In what follows, we denote by C the various positive constants independent of α. We modify the time-map method used in [6]. By (1), we have…”

Section: Proof Of Theoremmentioning

confidence: 99%