ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Based on Interval Methods Solving a Singular Integral Fredholm Equation with Perturbed Operator


Alexander Anatolievich Rogoza

Bauman Moscow State Technical University (BMSTU),
Kaluga, Bazenova st, 2.


In this paper we consider some projection methods for solving a certain class of singular Fredholm integral equations of the second kind ( including a perturbed integral operator), which are relevant for robust control problems. An investigation the properties of these methods and implementation of algorithms for their solution were performed. Estimates of the method error are obtained in the norms of Lebesgue and continuous functions spaces for various smoothness classes of right-hand side functions of the singular integral equation. Estimates are obtained both for an arbitrary nonuniform grid and for equations, including equations with nondifferentiable kernel. The approximate model of the initial problem is a system of interval algebraic equations. The existence and uniqueness of this system solution is proved within the framework of standard assumptions of interval analysis. An effective computer-aided algorithm for numerical realization of the projection methods under consideration was developed. The proposed method allows parallelization and may be rather simply implemented on network platforms of the Grid system.



  1. Egupov N. D., Pupkov K. A., Rogoza A. A., Trofimov M. A. Algoritmicheskaya teoria system upravlenia, osnovannaya na spektralnih methodah. V dvuh tomah. Tom 2. Matrichno-vichislitelnie tehnologii na baze integralnih uravnenii [Algorithmic theory of control systems based on spectral methods. In two volumes. Volume 2. Matrix-computing technology based on integral equations]. Мoscow, MGTU Bauman, 2014. 464 p
  2. Sarii S. P. On the characterization of the joint set of solutions of interval linear algebraic systems, 1990; (3): 20 p. (In Russ. )
  3. Kreinovich V., Lakeyev A., Rohn J., Kahi p. Computational Complexity and Feasibility of Data Processing and Interval Computations. Dordrecht: Kluwer, 1997
  4. Li B., Markys L. Osnovi teorii optimalnogo upravlenia [Foundations of optimal control theory]. Moscow. Nauka publ, 1972. 270 p
  5. Kalman R., Falb P., Arbib M. Ocherki po matematichescoi teorii system [Essays on mathematical systems theory]. Moscow: Mir publ, 1971
  6. Alefeld G., Herchberger U. Vvedenie v intervalnie vichislenia [Introduction to interval computation]. Moscow: Mir publ, 1987. 230 p
  7. Oettli W. On the solution set of a linear system with inaccurate coefficients. SIAM Jornal on Numerical Analysis. 1965 Vol. 2, №1 pp. 115-118
  8. Lakeev А. V. The computational complexity of estimation of generalized sets of solutions of interval linear systems. Trudi XI mezdunarodnoi Baikalskoi skoli-seminara «Metodi optimizacii I ih prilozenia», Irkutsk, Baikal, 5 - 12 iulia 1998 г., sekcia 4. Irkutsk: ISEM, 1998. p. 115-118
  9. Geri М., Dzonson D. Vichislitelnie mashini i trudnoresaemie zadachi [Computers and trudnoreshaemyh tasks]. Moscow: Mir publ, 1982
  10. Kaucher E. Interval analysis in the extended interval space IR. Computing Supplement. 1980, Vol. 2. p. 33-49
  11. Kollatc L. Fynctionalnii analiz I vichislitelnai matematika [Functional analysis and computational mathematics]. Moscow: Mir publ, 1969
  12. Ortega Dz., Reinbolt В. Iteracionnie methodi reshenia nelineinih system yravnenii co mhogimi neizvestnimi [Iterative methods for solving nonlinear systems of equations with many unknowns]. Moscow: Mir publ, 1975
  13. Neumaer A. Interval methods for systems of equations. - Cambridge: Cambridge University Press, 1990
  14. Sarii S. P. External evaluation of generalized sets of solutions of interval linear system. Vichislitelnie tehnologii. 1999. vol. 4, №4. p. 82 - 110
  15. Fadeev D. К., Fadeeva D. K. Vichislitelnie methodi lineinoi algebri [Computational methods of linear algebra]. М. : Physmathlit, 1960. 656 p
  16. Rutily B., Chevallier L. Why is so difficult to solve the radiative transfer equation? EAS Publications Series, 2006. Vol. 18, pp. 1-23
  17. Ahues M., Largillier A., Titaud O. The roles of a week singularity and the grid uniformity in relative error bounds. Numer. Funct. Anal. and Optimiz. 2001. Vol. 22, 7-8, pp. 789-814
  18. Ahues M., d’Almeida F. D., Largillier A., Titaud O., Vasconcelos P. An !!!! ERROR!!! IMAGE IS NOT ALLOWERD! refined projection approximate solution of the radiation transfer equation in stellar atmospheres. Journal of Computational and Applied Mathematics, 2002, Vol. 140, 1-2, pp. 13-26
  19. Panasenko G., Rutily B. Titaud O. Asymptotic analysis of integral equations for a great interval and its application to stellar radiative transfer. C. R. Acad. Sci. Paris. Ser. Mecanique. 2002, Vol. 330, pp. 735-740
  20. Amosov A., Panasenko G., Rutily B. An approximate solution to the integral radiative transfer equation in an optically thick slab. C. R. Acad. Sci. Paris. Ser. Mecanique. 2003. Vol. 331, pp. 823-828
  21. Rutily B. Multiple scattering theory and integral equations. Integral Methods in Science and Engineering (C. Constanda, M. Ahues, and A. Largillier, eds. ). Birkhauser, Boston, pp. 211-232, 2004
  22. Rutily B., Chevallier L. The finite Laplace transform for solving a weakly singular integral equation occurring in transfer theory. Journal of Integral Equations and Applications. 2004, Vol. 16, 4, pp. 389 409
  23. Ahues M., Amosov A., Largillier A., Titaud O. L p error estimates for projection approximations. Applied Mathematics Letters. 2005. Vol. 18, pp. 381-386
  24. Amosov A., Panasenko G. Asymptotic analysis and asymptotic domain decomposition for an integral equation of the radiative transfer type. J. Math. Pure Appl. 2005. Vol. 84, pp. 1813-1831
  25. d’Almeida F., Titaud O., Vasconcelos P. B. A numerical study of iterative renement schemes for weakly singular integral equations. Applied Mathematics Letters. 2005, Vol. 18, 5, pp. 571 - 576
  26. Amosov A., Panasenko G. An approximate solution to the integral radiative transfer equation in an optically thick slab. Mathematical Methods in the Applied Sciences. 2007. Vol. 30, pp. 1593-1608
  27. Amosov A., Ahues M., Largillier A. Superconvergence of projection methods for weakly singular integral operators. Integral Methods in Science and Engineering: Techniques and Applications (Constanda C., Potapenko S. eds). Birthauser, Boston. 2008, pp. 17
  28. Amosov A., Ahues M., Largillier A. Supercovergence of some projection approximations for weakly singular integral equations using general grids. SIAM Journal on Numerical Analysis, 2009, Vol. 47, Issue 1, pp. 646-674
  29. Ahues M., d’ Almeida F., Fernandes R. Piecewise constant Galerkin approximations of weakly singular integral equations. Int. J. Pure Appl. Math. 2009. Vol. 55, 4, pp. 569-580
  30. Nunes A. L., Vasconcelos P. B., Ahues M. Error Bounds for Low-Rank Approximations of the First Exponential Integral Kernel. Numerical Functional Analysis and Optimization. 2013. Vol. 34, 1, pp. 74 - 93
  31. d’Almeida F. D., Ahues M., Fernandes R. Errors and grids for projected weakly singular integral equations. Int. J. Pure Appl. Math. 2013. Vol. 89, 2, pp. 203-213
  32. Marchyk G. I., Agoshkov V. I. Vvedenie v proekcionno-setochnie methodi [Introduction to projection-grid methods]. Moscow: Nauka Publ., 1981. 416 p
  33. Kantorovich L. V. Functional analysis and applied mathematics. YMN. 1948. Vol. 3, vip. 6 (28), p. 89 - 185. (In Russ)
  34. Kantorovich L. V., Krilov V. I. Priblizennie metodi vicshego analiza [Approximate methods of higher analysis]. Physmatlit. М.; 1962

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