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UDC 517.948 DOI: 10.14529/ctcr150207 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.ruA.A. Ershova, South Ural State University, Chelyabinsk, Russian Federation, anya.erygina@yandex.ruAbstractIn 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. Keywordsregularization, integral equation, evaluation of inaccuracy, ill-posed problem References1. Goncharsky A.V., Leonov A.S., Yagola A.G [Linear Finite-difference Approximation of Improperly-posed Problems]. Journal of Calculus Mathematics and Mathematical Physics, 1974, vol. 14, no. 1, pp. 15–24. (in Russ.) 2. Tanana V.P., Sidikova A.I. [About Error Estimation of a Regularizing Algorithm Based of the Generalized Residual Principle at the Solution of Integral Equations]. Numerical Methods and Programming, 2015, vol. 16, no. 1, pp. 1–9. (in Russ.) 3. Tanana V.P. [A projective Method and Finite-difference Approximation of Linear Ill-posed Problems]. Siberian Mathematical Journal, 1975, vol. 16, no. 6, pp. 1301–1307. (in Russ.) 4. Vasin V.V. [Discrete Finite-dimensional Approximation and Convergence of Regularizing Algorithms]. Journal of Calculus Mathematics and Mathematical Physics, 1979, vol. 19, no. 1, pp. 11–21. (in Russ.) 5. Danilin A.R. [On Conditions for Convergence of Finite Dimensional Approximations of the Residual Method]. News of Higher Education Institutions: Mathematics, 1980, no. 11, pp. 38–40. (in Russ.) 6. Leonov A.S. [On the Relationship between the Generalized Residual Method and the Generalized Principle Residual for Nonlinear Problems]. Journal of Calculus Mathematics and Mathematical Physics, 1982, vol. 22, no. 4, pp. 783–790. (in Russ.) 7. Danilin A.R. [About Order-optimal Estimates of the Finite-dimensional Approximation of the Up-solving Ill-posed Problems]. Journal of Calculus Mathematics and Mathematical Physics, 1982, vol. 22, no. 4, pp. 1123–1129. (in Russ.) 8. Tanana V.P. [On a Projection-iterative Algorithm for Operator-tory Equations of the First Kind with a Perturbed Operator]. Reports of the Academy of Sciences, 1975, vol. 224, no. 5, pp. 1028–1029. (in Russ.) 9. Tikhonov A.N. [On the Solution of Ill-posed Problems Regularization Method]. Reports of the Academy of Sciences, 1963, vol. 151, no. 3, pp. 501–504. (in Russ.) 10. Goncharsky A.V., Leonov A.S., Yagola A.G. [Generalized Discrepancy Principle]. Journal of Calculus Mathematics and Mathematical Physics,1973, vol. 13, no. 2, pp. 294–302. (in Russ.) 11. Tanana V.P. Metody resheniya operatornykh uravneniy [Methods for Solving of Operator Equations]. Moscow, Nauka Publ., 1981, 156 p. 12. Tanana V.P., Erygina A.A. [An error estimate for the regularization method of A.N. Tikhonov for solving an inverse problem of solid state physics]. Siberian Journal of Industrial Mathematics, 2014, no. 2, pp. 125–136. (in Russ.) SourceBulletin 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) |