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UDC 517.948
About One Numerical Algorithm for Solving Integral Equations of the First Kind in Space L2 Based on the Generalized Discrepancy Principle
A.I. Sidikova, South Ural State University, Chelyabinsk, Russian Federation, 7413604@mail.ru
A.A. Ershova, South Ural State University, Chelyabinsk, Russian Federation, anya.erygina@yandex.ru
Abstract
In this paper we consider a one-dimensional Fredholm integral equation of type I closed with a kernel having a solution in the class W12 [a,b] with homogeneous boundary conditions of the first kind at the point a. The problem is reduced to a new integral equation for the derivative of the desired solution. The resulting integral equations is subjected to finite-dimensional approximation of a special form, that allows to use the variational regularization Tikhonov’s method with the choice of regularization parameter according to the generalized discrepancy principle to reduce the problem to a special system of linear algebraic equations. A priori estimation of the accuracy of the resulting finite-stable approximate solution that takes into account the accuracy of the finite-dimensional approximation of the problem is also carried out. Using of this approach is made on the example of the problem of determining the phonon spectrum on its heat capacity, depending on the temperature, which is known to be reduced to integral equations of the first kind.
Keywords
regularization, integral equation, evaluation of inaccuracy, ill-posed problem
References
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Source
Bulletin of the South Ural State University. Ser. Computer Technologies, Automatic Control, Radio Electronics, 2015, vol. 15, no. 2, pp. 66-74. (in Russ.) (Modeling and Computer Technologies)