Abstract

We study a regularization of linear Volterra integral equation of the first kind with differentiable kernel degenerated at the initial point or at the end point of the interval on the diagonal. Accordingly, in the first case, the kernel on the diagonal is a non-decreasing function, in the second case, a non-increasing function. Regularizing Lavrentiev’s type operator is constructed that preserves the property of Volterra equation. We proved the uniform convergence of the regularized solution to the exact solution, and defined the conditions of uniqueness of the solution in the Holder space. We also considered the case of Volterra’s first kind linear integral equation with the approximated right hand side of equation. We established conditions under which a regularized solution to an equation with approximately given right-hand side can serve as an approximate solution to the original equation. The results can be used for numerical solution of the problem.

中文翻译：

---

具有光滑数据的第一类Volterra线性积分方程的正则化。

摘要

我们研究了第一类线性Volterra积分方程的正则化问题，该线性Volterra积分方程的对角线上的区间的起点或终点均具有可分解的可分解核。因此，在第一种情况下，对角线上的核是非递减函数，在第二种情况下是非递增的函数。构造正则化Lavrentiev的类型运算符，该运算符保留了Volterra方程的属性。我们证明了正则化解与精确解的一致收敛性，并定义了Holder空间中解唯一性的条件。我们还考虑了Volterra第一种线性积分方程的情况，该方程近似为方程的右侧。我们建立了条件，在该条件下，具有近似给定右侧的方程式的正则解可以用作原始方程式的近似解。结果可用于数值求解问题。