On Regularization of the Solution of Integral Equations of the First Kind Using Quadrature Formulas

• Published in: Vestnik, St. Petersburg University. Mathematics Vol. 54; no. 4; pp. 361 - 365
• Main Authors: Lebedeva, A. V., Ryabov, V. M.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.10.2021; Springer Nature B.V
• Subjects: Analysis; Ill posed problems; Ill-conditioned problems (mathematics); Integral equations; Laplace transforms; Linear algebra; Mathematical analysis; Mathematical problems; Mathematics; Mathematics and Statistics; Matrices (mathematics); Matrix algebra; Numerical methods; Quadratures; Regularization; Regularization methods; Vectors (mathematics); ill-posed problems; ill-conditioned problems; regularization method; system of linear algebraic equations; integral equations of the first kind; condition number
• ISSN: 1063-4541; 1934-7855
• DOI: 10.1134/S1063454121040129

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 class includes also the problem of inversion of 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 the orthogonal sum of two vectors, one of which is determined stably, while, to search for the second vector, it is necessary to use some kind of stabilization procedure. In this paper, the methods of numerical solving an SLAE with a symmetric positive definite matrix or with an oscillatory-type matrix with the use of regularization leading to an SLAE with a reduced condition number are considered. A method of reducing the problem of inversion of the integral Laplace transform to an SLAE with generalized Vandermonde oscillatory-type matrices, the regularization of which reduces the ill-conditioning of the system, is indicated.