Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity"
Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity"
复制标题
指数非线性常微分方程爆炸解的严格数值计算"
DOI：
10.1016/j.cam.2019.112607
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发表时间：
2020
影响因子：
2.4
通讯作者：
K. Matsue and A. Takayasu
中科院分区：
文献类型：
--
作者：
サーバス・ローリー;島田悠彦;K. Matsue and A. Takayasu
本文从动力系统的角度研究具有指数非线性的微分方程的爆破解及其数值验证。作为一个例子，考虑了具有齐次Dirichlet边界条件的u t= u x x+ e u m的有限差分离散。我们的想法是基于相空间的紧化和时间尺度的去具体化。在目前的情况下，处理指数非线性是主要问题。幸运的是，在一种指数齐次的向量场下，我们可以用多项式向量场的方法来处理这个问题。特别地，对于具有指数非线性的微分方程，我们可以用与先前工作类似的方法来描述和验证爆破解及其爆破时间。给出了爆破问题中指数非线性的一系列技术处理方法，并给出了具体的验证实例。
Our concerns here are blow-up solutions for ODEs with exponential nonlinearity from the viewpoint of dynamical systems and their numerical validations. As an example, the finite difference discretization of u t= u x x+ e u m with the homogeneous Dirichlet boundary condition is considered. Our idea is based on compactification of phase spaces and time-scale desingularization as in previous works. In the present case, treatment of exponential nonlinearity is the main issue. Fortunately, under a kind of exponential homogeneity of vector field, we can treat the problem in the same way as polynomial vector fields. In particular, we can characterize and validate blow-up solutions with their blow-up times for differential equations with such exponential nonlinearity in the similar way to previous works. A series of technical treatments of exponential nonlinearity in blow-up problems is also shown with concrete validation examples.