Lavrentiev Regularization of Integral Equations of the First Kind in the Space of Continuous Functions
The regularization method of linear integral Volterra equations of the ﬁrst kind is considered. The method is based on the perturbation theory. In order to derive the estimates of approximate solutions and regularizing operator norms we use the Banach- Steinhaus theorem, the concept of stabilising operator, as well as abstract scheme for construction of regularizing equations proposed in the monograph N.A. Sidorov (1982, MR87a: 58036). The known results (existence of the second derivatives of kernel and source) of A.M. Denisova (1974, MR337040) for the Volterra equations' regularization were strengthened. The approximate method was tested on the examples of numerical solutions of integral equations under various noise levels in the source function and in the kernel. The regularization method is tested on Volterra integral equations with piecewise continuous kernels suggested by D.N. Sidorov (2013, MR3187864). The desired numerical solution is sought in the form of a piecewise constant and piecewise linear functions using quadrature formulas of Gauss and midpoint rectangles. The numerical experiments have demonstrated the eﬃciency of Lavrentiev regularization applied to Volterra integral equations of the ﬁrst kind with discontinuous kernels.