Rigorous Numerical Enclosures for Positive Solutions of Lane?Emden’s Equation with Sub-Square Exponents
Rigorous Numerical Enclosures for Positive Solutions of Lane?Emden’s Equation with Sub-Square Exponents
复制标题
带次方指数的 Lane?Emden 方程正解的严格数值封闭式
DOI：
10.1080/01630563.2022.2029485
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发表时间：
2022
影响因子：
1.2
通讯作者：
Oishi Shin’ichi
中科院分区：
文献类型：
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作者：
Tanaka Kazuaki;Plum Michael;Sekine Kouta;Kashiwagi Masahide;Oishi Shin’ichi
本文的目的是得到齐次Dirichlet边界条件下Lane-Emden方程解的严格数值封闭。我们证明了一个非退化解的存在unearby数值计算的近似连同一个明确的误差界，即，特别地，我们关注次平方情形，使得非线性项的导数不是Lipschitz连续的。在这种情况下，它是有问题的应用经典的牛顿Kantorovich定理获得存在性证明，而且在程序中出现的几个困难，以获得严格的数值积分。我们设计了一种显式封闭所需积分的方法，基于广义牛顿-康托洛维奇定理证明所需解的存在性。给出了一个数值算例，在单位平方区域上得到了一个显式的封闭解
The purpose of this paper is to obtain rigorous numerical enclosures for solutions of Lane–Emden’s equationwith homogeneous Dirichlet boundary conditions. We prove the existence of a nondegenerate solutionunearby a numerically computed approximationtogether with an explicit error bound, i.e., a bound for the difference betweenuandIn particular, we focus on the sub-square case in whichso that the derivativeof the nonlinearityis 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 foron the unit square domain