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## Bulletin of TUIT: Management and Communication Technologies

#### Abstract

In this article, the interval expansion of the structure of solving basic types of boundary value problems for partial differential equations of the second order of making the basic operations that compose interval arithmetic is developed. For the differential equation (1) of the type, when constructing the interval expansion of the structure of the formula, structural formulas were used to construct with the R-function method and 4 problems were studied — the Dirichlet problem, the Neumann problem, the third type problem, the mixed boundary conditions problem. For the Dirichlet problem, the solution is an interval expansion of the structure in the form (5), where �={�� ,��̅,�̅�,�̅�̅} и [ �,�̅]is an indefinite interval function. For the Neumann problem, a solution is solved in the interval extension of the structure (10), (11), [ �1,�1̅̅̅̅], [ �2,�2̅̅̅̅] is an indefinite interval function and �1 is a differential operator of the form (9). For the problem of the third type, the solution is solved in the interval extension of the structure (16), (17), [ �1,�1̅̅̅̅], [ �2,�2̅̅̅̅] - indefinite, interval function, �1- differential operator of the form (9). For the problem, mixed boundary conditions are treated. The solution In the interval extension of the structure (22), (23), (24),[ �1,�1̅̅̅̅], [ �2,�2̅̅̅̅] is an indefinite interval function and �1is a differential operator of the form (9).

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#### References

[1] V.L. Rvachev, Theory of R-functions and some applications / . - Kiev g Sciences. Dumka, 1982. -552 p. [2] S.P. Shary, Finite- interval analysis . Novosibirsk :Publishing house «XYZ» Institute of Computational Technologies , 2009 . - 580 p. [3] V.I. Faddeev, Computational methods of linear algebra. - M.: Gosteh " publish , 1950. -240 With [4] L.V. Kantorovich and V.I. Krylov, Approximate methods of higher analysis. - M. ; L. Gostekhizdat , 1950 . - 695 p. [5] B.S. Dobronets, Interval Mathematics. - Krasnoyarsk: Publisher KSU, 2004. Course of lectures , read by the author repeatedly in various universities of Krasnoyarsk. [6] S.A. Kalmyikov, Y.A. Shokin, Z.K. Yuldashev, Methods of interval analysis. - Novosibirsk : Nauka, 1986. -223 P. [7] Sh. A. Nazirov, The method of interval- valued R-functions in mathematical modeling / / Proceedings of the International Scientific Conference " Actual problems of applied mathematics and information technology al-Khwarizmi 2012", Volume number 2, 19-22 December, 2012 Tashkent. Pp. 65-70. [8] Sh. A. Nazirov, Two interval -valued R- functions of argument . Republican Scientific and Methodological Conference "Modern Information Technologies in Telecommunications." Tashkent. 2011. P.15 -20. [9] Sh. A. Nazirov, Calculating the values of n- tuples of differential multivariate interval-valued functions. Computational technologies. Novosibirsk. 2014. Issue. 2. [10] Sh. A. Nazirov, Multidimensional interval - valued R-function. // Problems of Computational and Applied Mathematics., Tashkent , 2011. - № 126. -P. 23-59. [11] Walster, G. William; Hansen, Eldon Robert (2004). Global Optimization using Interval Analysis (2nd ed.). New York, USA: Marcel Dekker. ISBN 0-8247-4059-9. [12] Jaulin, Luc; Kieffer, Michel; Didrit, Olivier; Walter, Eric (2001). Applied Interval Analysis. Berlin: Springer. ISBN 1-85233-219-0. [13] Application of Fuzzy Arithmetic to Quantifying the Effects of Uncertain Model Parameters, Michael Hanss, University of Stuttgart [14] Tucker, W. (1999). The Lorenz attractor exists. Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 328(12), 1197-1202. [15] Young, R. C. (1931). The algebra of many-valued quantities. Mathematische Annalen, 104(1), 260-290. [16] Dwyer, P. S. (1951). Linear computations. Oxford, England: Wiley. [17] Sunaga, Teruo (1958). Theory of interval algebra and its application to numerical analysis. RAAG Memoirs. pp. 29–46. [18] Moore, R. E. (1966). Interval Analysis. Englewood Cliff, New Jersey, USA: Prentice-Hall. ISBN 0-13-476853-1. [19] Cloud, Michael J.; Moore, Ramon E.; Kearfott, R. Baker (2009). Introduction to Interval Analysis. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). ISBN 0-89871-669-1. [20] "IEEE Std 1788.1-2017 - IEEE Standard for Interval Arithmetic (Simplified)". IEEE Standard. IEEE Standards Association. Retrieved 2018-02-06.

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