Rigorous numerical enclosures for positive solutions of Lane-Emden's equation with sub-square exponents

TL;DR

This paper develops rigorous numerical methods to enclose positive solutions of Lane-Emden's equation with sub-square exponents, overcoming non-Lipschitz challenges and providing explicit error bounds for solutions near numerical approximations.

Contribution

It introduces a novel approach for obtaining explicit enclosures of solutions for nonlinear PDEs with non-Lipschitz nonlinearities, especially in the sub-square exponent case.

Findings

Successfully enclosed solutions for p=3/2 on the unit square

Proved existence of solutions near numerical approximations with explicit bounds

Extended Newton-Kantorovich theorem to non-Lipschitz nonlinearities

Abstract

The purpose of this paper is to obtain rigorous numerical enclosures for solutions of Lane-Emden's equation $-\Delta u=|u|^{p-1} u$ with homogeneous Dirichlet boundary conditions. We prove the existence of a nondegenerate solution $u$ nearby a numerically computed approximation $\hat{u}$ together with an explicit error bound, i.e., a bound for the difference between $ u $ and $\hat{u}$. 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 $\Omega=(0,1)^2$.