Regularization of the solution of integral equations of the first kind using quadrature formulas

Ill-conditioned systems of linear algebraic equations (SLAEs) and integral equations of the first kind belonging to the class of ill-posed problems are considered. This also includes the problem of inverting the integral Laplace transform, which is used to solve a wide class of mathematical problems. Integral equations are reduced to SLAEs with special matrices. To obtain a reliable solution, regularization methods are used. The general strategy is to use the Tikhonov stabilizer or its modifications, or to represent the desired solution in the form of an orthogonal the sum of two vectors, one of which is determined stably, and to search for the second requires some kind of stabilization procedure. In this article methods for the numerical solution of SLAEs with positive a certain symmetric matrix or with an oscillatory type matrix using regularization, leading to a SLAE with a reduced condition number. A method of reducing the problem of inversion of the integral Laplace transform to a SLAE with generalized Vandermonde matrices of oscillation type, the regularization of which reduces the ill-conditioning of the system, is indicated.