Self-validated computation of singular integral and integral equations

本研究主要有两个方面.利用DE变换建立了奇异积分的自验证计算方法，这是计算奇异积分近似值的最有效方法之一.发展了积分方程解的存在性和局部唯一性的数值验证方法，在DE变换的自验证计算基础上，建立了一套保证奇异积分计算精度的计算机程序库的基本技术。对于程序的参数，我们假设被积函数由初等函数组成。根据被积函数在积分区间两端的奇异性选择DE变换的类。为了估计误差界，我们需要验证被积函数在复区域中的扩展区域上的正则性。为此，采用Sugiura等人的方法。这些结果在研究成果报告中有详细的描述。在积分方程的数值验证方面，我们进行了如下研究.建立了函数方程（包括积分方程）解的局部唯一性的数值验证.发展由积分方程导出的具初值的常微分方程组的数值验证方法3.实现1到2。对于1，我们得到了一个重要的结果，这将是有用的广泛的自我验证。对于2和3，我们已经开发了一种新的方法，现在正在改进它以供实际使用。

The present research has two purposes.1. Establishment of self-validated computation for singular integral using DE transformation, which is one of most effective methods for calculating approximate values of singular integrals.2. Development of numerical verification methods for the existence and the local uniqueness of solutions to integral equations.On the self-validated computation of DE transformation, we constructed a set of basic techniques for a computer program library of computation with guaranteed accuracy of singular integrals. For the arguments of the programs, we suppose integrands composed of elementary functions. The class of the DE transformation is chosen corresponding to the singularity of the integrand at the ends of the interval of integral. In order to estimate the error bounds, we need to verify the regularity of the integrand on an expanded area in a complex region. For this purpose, the method by Sugiura et al. is adopted. These results are described in detail in the report of the research results.On the numerical verification of integral equations, we carried out our study as follows.1. Establish a numerical verification for the local uniqueness of solutions to function equations including integral equations.2. Develop a numerical verification methods for the systems of ordinal differential equations with initial values which are derived from integral equations3. Implementation of 1 to 2.For 1, we have got an important result which will be useful for a wide range of self-validation. For 2 and 3, we have developed a new method and are now improving it for practical use.