Rigorous numerics for nonlinear heat equations in the complex plane of time
Rigorous numerics for nonlinear heat equations in the complex plane of time
复制标题
复杂时间平面中非线性热方程的严格数值计算
DOI：
10.1007/s00211-022-01291-2
复制
发表时间：
2022
影响因子：
2.1
通讯作者：
Okamoto Hisashi
中科院分区：
文献类型：
--
作者：
Takayasu Akitoshi;Lessard Jean-Philippe;Jaquette Jonathan;Okamoto Hisashi
本文介绍了一种计算复值非线性热方程柯西问题解的严格局部包含的方法。该证明是构造性的，并且为包含柯西问题的解提供了明确的界，该问题被重写为某一Banach空间上的零点寻找问题。利用解映射算子，我们构造了一个简化的牛顿算子，并证明了它存在唯一的不动点。不动点及其严格的界提供了柯西问题解的局部包含。然后，迭代地应用局部包含技术来计算长时间间隔上的解。利用这一技巧证明了非线性热方程中分支奇点的存在性。最后，我们介绍了一种基于Lyapunov-Perron方法的计算中心稳定流形部分解的方法，并证明了Cauchy问题的开解集收敛到零，从而证明了解在复时间平面上的整体存在性。
In this paper, we introduce a method for computing rigorous local inclusions of solutions of Cauchy problems for nonlinear heat equations for complex time values. The proof is constructive and provides explicit bounds for the inclusion of the solution of the Cauchy problem, which is rewritten as a zero-finding problem on a certain Banach space. Using a solution map operator, we construct a simplified Newton operator and show that it has a unique fixed point. The fixed point together with its rigorous bounds provides the local inclusion of the solution of the Cauchy problem. The local inclusion technique is then applied iteratively to compute solutions over long time intervals. This technique is used to prove the existence of a branching singularity in the nonlinear heat equation. Finally, we introduce an approach based on the Lyapunov–Perron method for calculating part of a center-stable manifold and prove that an open set of solutions of the Cauchy problem converge to zero, hence yielding the global existence of the solutions in the complex plane of time.