Self-validated computation of singular integral and integral equations

本研究有两个目的。利用DE变换建立奇异积分的自验证计算，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变换的自验证计算上，我们构造了一套保证奇异积分精度的计算机程序库的基本技术。对于程序的自变量，我们假定被积函数由初等函数组成。DE变换的类别是根据被积函数在积分区间末端的奇异性来选择的。为了估计误差界，我们需要在复杂区域的扩展区域上验证被积函数的正则性。为此，Sugiura等人的方法。是被收养的。这些结果在研究结果的报告中有详细的描述。在积分方程组的数值验证方面，我们开展了以下研究。建立了包含积分方程解的局部唯一性的数值验证。发展了一种由积分方程导出的具有初值的有序微分方程组的数值验证方法。1到2的实现。对于1，我们得到了一个重要的结果，该结果将对广泛的自我验证有用。对于2和3，我们已经开发了一种新的方法，现在正在改进以用于实际应用。

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.