Numerical verification for nonlinear equations including a principal value integral

不用说，在工程和科学领域，求解数学模型都需要数值计算。我们试图通过数值计算来获得近似解。因此，误差估计是非常重要的，特别是对于非线性问题。本项目的主要内容是发展数值验证方法，即利用计算机对数值解中包含的误差进行严格估计。这项工作考虑了在弹性体理论、流体力学、声学、电磁学等二维势问题中常见的非线性、周期和奇异积分方程解的数值验证方法。其基本思想是将原始方程转化为不动点形式，设置适当的函数空间，并应用Schauder不动点定理。近似解的邻域由有限维部分和截断部分的乘积给出。不动点定理保证了在一定条件下邻域内存在精确解。作为例子，研究了水波的Nekrasov积分方程。结果表明，对于中等波高的情况，该方程的解可以用本文提出的方法进行验证。当波高较大时，由于计算复杂，该方法不起作用。这一问题还有待于进一步研究。此外，本文还考虑了分叉点的数值验证，特别是双转折点的数值验证。给出了分岔点存在的充要条件及其验证方法。

It is needless to say that numerical calculation is required to solve mathematical models in the field of engineering and science. We try to obtain approximate solutions using numerical calculation. Then error estimate is very important, in particular for nonlinear problems. The main subject of this project is development of the numerical verification method which is rigorous estimation of errors included in numerical solutions using computers.This work considered the numerical verification method of solutions of nonlinear, periodic and singular integral equation which can be often found in the two-dimensional potential problems in elastic body theory, fluid mechanics, acoustics, electromagnetism, and so on. The basic idea is to transform an original equation into a fixed point form, to set a suitable function space, and to apply Schauder's fixed point theorem. The neighbourhood of approximate solutions is given by product of the finite dimensional part and the truncated part. The fixed point theorem guarantees that exact solutions exist in the neighbourhood under some conditions. As an example, Nekrasov's integral equation for water waves is investigated. It was found that solutions of this equation can be verified using the proposed method for the case of moderate wave height. When the wave height is large, this method did not work due to computational complexity. This problem remains as a future work.Also this work considered numerical verification of bifurcation points, in particular double turning points. The necessary and sufficient conditions for existence of the bifurcation points and the verification method using them are shown.