By the twentieth century, classical computation had evolved into distribution theory. The main central objects of modern mathematical analysis are integrals according to Lebesgue and derivatives according to Sobolev.

The Lebesgue and the Sobolevs went down in history with a whole and fresh look at their products. It is known that mathematical talent is passed from teacher to student. It is a chain, the loops of which are closely connected, and it connects the schools of mathematics. This can be seen in the famous school of Euler Sobolev.

Academician N.N. Luzin says: “The older the scientific school, the higher its value. In it, traditions, customs, customs, methods of work, which have been selected and developed over the centuries, reach even modern mathematicians.

It is known that S.L. Sobolev wrote his dissertation under the guidance of P.L. Chebyshev's student N.M. Gunther. S.L. Sobolev considered V.A.Steklov's student V.I.Smirnov to be his mentor

In 1935, S.L. Sobolev applied his methods of functional analysis to equations of mathematical physics and developed a new theory, namely generalized functions, generalized derivatives, and Sobolev spaces. Ten years after S.L. Sobolev, the French mathematician Laurent Schwartz introduced Fourier substitutions into distribution theory. S.L. Sobolev began work on the nuclear project in 1942. Here in Laboratory 2, Academicians I.V. Kurchatov, S.L. Sobolev, and Corresponding Academy Members I.K. Kikoin and Voznesensky enriched uranium-235 and plutonium-239 to create explosive weapons. For this work, Sobolev was awarded the title of Hero of Socialist Labor in 1952. Sobolev's main role in the development of the atomic bomb was to lead this huge calculation. During this time, more than 200 only girls worked for him.

Professor M.D. Ramazanov asked his teacher S.L. Sobolev, "Why you?" "There could have been others, but I did it very well," he said. The nuclear project greatly enriched Sobolev's scientific and personal potential. Therefore, until the end of his life, computational mathematics occupied a high place in his scientific research.

From 1952 to 1960, S.L. Sobolev headed the Department of Computational Mathematics at Moscow State University which he organized. In Siberia, he headed the Institute of Mathematics. It was during this time that he developed his theory of cubature formulas.

            It is well known that S.L Sobolev made a worthy contribution to the development of functional analysis in Uzbekistan.

            U 1967 yilda O‘zR FA “Kibirnetika” instiutida “Integrallarni taqribiy hisoblash” laboratoriyasiga asos solgan. Bu laboratoriyaning birinchi mudiri, S.L.Sobolevning shogirdi, professor G‘.N.Salixov bo‘lgan. Ushbu laboratoriyani 1976-1995 yillarda dunyoga mashhur olim N.P.Romanovning shogirdi, professor M.I.Isroilov boshqargan, hamda 1995-2019 yillarda S.L.Sobolevning shogirdi, professor X.M.Shadimetov boshqargan. Bu laboratoriyani 2019 yildan shu kungacha S.L.Sobolevning nevara shogirdi, professor A.R.Hayotov boshqarib kelmoqda. S.L.Sobolev rahbarligida doktorlik dissertasiyalarini G‘.N.Salixov va X.M.Shadimetovlar, nomzodlik dissertasiyalarini Z.J.Jamalov, T.X.Sharipov va I.I.Jalolovlar himoya qilganlar.

In 1967 he founded the Laboratory of Approximate Integrals at the Institute of Cybernetics of the Academy of Sciences of the Republic of Uzbekistan. The first director of this laboratory was Professor G.N. Salikhov, a student of S.L. Sobolev. From 1976 to 1995, this laboratory was headed by Professor M.I. Isroilov, a student of the world-famous scientist N.P Romanov, and from 1995 to 2019, it was headed by Professor Kh.M. Shadimetov, a student of S.L. Sobolev. The laboratory has been headed by Professor A.R. Hayotov, a grandson student of S.L. Sobolev, since 2019. G.N. Salikhov and Kh.M. Shadimetov defended their doctoral dissertations under the supervision of S.L. Sobolev, and Z.J. Jamalov, T.Kh. Sharipov, and I.I. Jalolov defended their dissertations.

In Uzbekistan, S.L. Sobolev's students continue his scientific work with dignity. They are doing research in the following areas.

-The theory of optimal quadrature and cubature formulas with grids.

-Optimal interpolation formulas and splines theories.

- Optimal differential methods for approximate solution of differential equations.

- Problems of construction of discrete analogs of differential operators.

-Optimal methods for approximate solution of integral equations.

The laboratory currently conducts research in the following areas:

1.      The theory of grid optimal quadrature and cubature formulas.

In this direction, the optimal quadratic and cubic formulas for the approximate calculation of regular, singular, and Fourier integrals in Sobolev spaces were constructed and their estimates were obtained. These results were obtained by Kh.M.Shadimetov, A.R.Hayotov, F.A.Nuraliev, O.X.Gulomov, N.H.Mamatova, I.O.Jalolov, D.M.Akhmedov, S.S.Azamov, N.D.Boltaev, A.K.Boltaev, S.Babaev, B.N.Abdukayumov, X.Quldoshev, G.Akhmadaliev, I.I.Jalolov, B.Daliev, B.Bozarov, R.Rasulov, U.Khayriev and J.Davronov.

2.      Optimal interpolation formulas and spline theories.

The problems of construction of optimal interpolation formulas and splines in different Gilbert spaces have been studied in the scientific researches of Kh.M.Shadimetov, A.R.Hayotov, F.A.Nuraliev, N.H.Mamatova, I.O.Jalolov, A.K.Boltaev, S.Babaev and J.J.Eshniyozov

3.      Optimal difference methods for approximate solution of differential equations.

Kh.M.Shadimetov, A.R.Hayotov, R.N.Mirzakabilov, R.Karimov, and Sh.Esanov are conducting research on the construction of optimal differential formulas for the approximate solution of differential equations in certain spaces based on the methods of functional analysis.

4.      Kh.M.Shadimetov, A.R.Hayotov, F.A.Nuraliev, S.S.Azamov, A.K.Boltaev, X.M.Quldoshev, I.I.Jalolov and Sh.Esanov are engaged in the construction of discrete analogs of differential operators.

Optimal methods for approximate solution of integral equations. Kh.M. Shadimetov, A.R. Hayotov, D.M. Akhmedov, B. Daliev are conducting research in this area

The quadrature and interpolation formulas in different functional spaces were found to be the extreme functions of the error functions and the corresponding error function norms were calculated.      

The problem of the existence and uniqueness of optimal quadrature and interpolation formulas in differentiated functions in the Hilbert spaces is solved, which proved that the solution gives a minimum to the norms of the error functionals in the corresponding spaces.

              Discrete analogs of known differential operators were constructed.

        Optimal quadrature formulas with positive coefficients were constructed in Sobolev space and the optimal error functional norm was calculated.

         Optimal quadrature formulas in the sense of Sard were constructed that were exact for polynomials and exponential functions not exceeding m-2, and for polynomials and trigonometric functions not exceeding m-3.

          Sobolev's grid optimal interpolation formulas were constructed in the space of periodic functions with n variables.

          In Sobolev space, optimal quadrature formulas were constructed for the approximate calculation of Fourier coefficients and singular integrals.

          The problem of constructing optimal Euler-Maclaren type quadrature formulas was completely solved in Sobolev's space, and the exact value of such quadrature formulas was found..

          An optimal quadrature formula was developed for the approximate calculation of Fourier integrals, which served as the basis for the development of a new algorithm for image reconstruction in the field of computed tomography.         

Researchers of the Laboratory of Computational Mathematics of the V.I. Romanovsky Institute of Mathematics of the Academy of Sciences of Uzbekistan and professors of the Department of Informatics and Computer Graphics of Tashkent State Transport University together with scientists from leading scientific centers and universities of the world study current problems of modern computational mathematics are conducting scientific collaborations on solutions. In particular, scientific cooperation with the following scientists and their scientific schools is very well established.

1) Professor V.L.Vaskevich, S.L.Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Russia.

2) Professor M.D.Ramazanov, Ufa Institute of Mathematics, Russia.

3) Professor M.V.Noskov, Siberian Federal University, Krasnoyarsk, Russia.

4) Academic G.V.Milovanovich, Institute of Mathematics, Serbian Academy of Sciences and Culture, Serbia.

5) Professor E.Novak, Friedrich-Schiller University of Jena, Germany.

6) Professor A.Kabada, Santiyago de Kompostela universiteti, Ispaniya.

7) Professor Ch.-O. Li, KAIST, South Korea.

8) Professor S.S.Sherbakov, Belarus State University, Republic of Belarus.

9) Professor Parovik R.I., V. Bering Kamchatka State University, Russia.