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.
1. Denisov A.M. The approximate solution of a Volterra equation of the first kind. USSR Computational Mathematics and Mathematical Physics, 1975, vol. 15, no 4, pр. 237—239.
2. Lavrentiev M.M. Some Improperly Posed Problems in Mathematical Physics. Springer, Berlin, 1967.
3. Lusternik L.A., Sobolev V.J. Elements of Functional Analysis. Frederick Ungar Publishing Co., 1961.
4. Sidorov D.N. Methods of Analysis of Integral Dynamical Models: Theory and Applications. (in Russian) Irkutsk State Univ. Publ., 2013.
5. Sidorov D.N. Solvability of systems of integral Volterra equations of the first kind with piecewise continuous kernels. (in Russian) Izv. Vyssh. Uchebn. Zaved. Mat., 2013, no. 1, pp. 62-72.
6. Sidorov D. N., Tynda A. N., Muftahov I. R. Numerical Solution of Volterra Integral Equations of the First Kind with Piecewise Continuous Kernel. Bul. of the South Ural State University. Ser. “Math. Model., Programming and Comp. Software”, 2014, vol. 7, no 3, pp. 107-115.
7. Sidorov N.A., Trenogin V.A. Linear Equations Regularization using the Perturbation Theory. Diff. Eqs., 1980, vol. 16, no 11, рp. 2038-2049.
8. Sidorov N.A., Trenogin V.A. A Certain Approach the Problem of Regularization of the Basis of the Perturbation of Linear Operators. Mathematical notes of the Academy of Sciences of the USSR, 1976, vol. 20, no. 5, pр. 976-979.
9. Sidorov N.A., Sidorov D.N., Muftahov I.R. Perturbation Theory and the Banach–Steinhaus Theorem for Regularization of the Linear Equations of the First Kind. IIGU Ser. Matematika, 2015, vol. 14, pp. 82-99.
10. Tikhonov A. N. Solution of Ill-Posed Problems. Winston. New York, 1977.
11. Tikhonov A.N., Ivanov V.K., Lavrent’ev. Ill-posed Problems. In the book Partial Differential Eqs, Moscow, Nauka, 1970, pp. 224-239.
12. Trenogin V. A. Functional analysis. Nauka. Moscow, 1980, 496 p.
13. Tynda A.N., Malyakina E.N. Direct Numerical Methods for Solving Volterra Integral Equations of the I Kind with Discontinuous Kernels. Sbornik statey VIII Mezhd. nauch.-tekh. konferentsii molodykh spetsialistov, aspirantov i studentov (in Russian) [Proc. 8th Int. Sci. Tech. Conf. Young Specialists, Post-graduate students], PGU, Penza, 2014, pp. 84–89.
14. Tynda A.N., Boginskaya P.A. Numerical Analysis of Volterra Integral Equations of the I Kind with Discontinuous Kernels [Chislennyy analiz integral’nykh uravneniy Vol’terra I roda s razryvnymi yadrami]. Sbornik statey VIII Mezhd. nauch.-tekh. konferentsii molodykh spetsialistov, aspirantov i studentov (in Russian) [Proc. 8th Int. Sci. Tech. Conf. Young Specialists, Post-graduate students]. PGU, Penza, 2014, pp. 76-81.
15. Brunner H., Houwen P.J. The Numerical Solution of Volterra Equations. North-Holland, Amsterdam, 1986.
16. Kythe P.K., Puri P. Computational Methods for Linear Integral Equations. Birkh¨auser, Boston, 2002.
17. Sidorov D. Integral Dynamical Integral Dynamical Models: Singularities, Signals and Control Ed. by L. O. Chua, Singapore, London, World Scientific Publ., 2015, vol. 87 of World Scientific Series on Nonlinear Science, Series A. 258 p.
18. Sizikov V. S. Further Development of the New Version of a Posteriori Choosing Regularization Parameter in Ill-Posed Problems. Intl. J. of Artificial Intelligence, 2015, vol. 13, no 1, pp. 184-199.