The scientific lab was named the department of “Differential equations” for the first time. Academician M.S. Salohiddinov was the head of the department during the years 1964-1966 and  1974-2004. 

The scientific laboratory "Differential Equations and Their Applications" of the Institute of Mathematics named after VI Romanovsky of the Academy of Sciences of the Republic of Uzbekistan was established in accordance with paragraph 10 of the Decree of the President of the Republic of Uzbekistan dated July 9, 2019 No PP-4387    .  According to Annex 5 to this decision, the scientific laboratory has 7 staff units.

At present time, the scientific laboratory is headed by Doctor of Physical and Mathematical Sciences, Professor R. R. Ashurov. Also in this scientific laboratory the chief researchers Sh.A.Alimov, S.Umarov, A.Khasanov, S.Z.Djamolov, the leading scientists M.Kh.Ruziev, B.J.Kadirkulov, the senior scientists A.V.Yuldasheva, Yu. Fayziev, K. Buvaev and junior researcher F. Dehkanov are engaged in scientific work. J. Safarov, R. T. Zunnunov, O. H. Abdullaev are studying for a doctorate, J. Khojakulov, O. Kilichev, O. Muhiddinova, B. Toshtemirov, H. Turakulov are studying for a doctorate.

1.  Spectral theory of differential operators

2.  Harmonic analysis

3. Theory of pseudo differential operators

4. Initial-boundary value problems for the partial differential equations

5. Direct and inverse problems for the fractional differential equations and their applications

6. Investigating the boundary-value problems for mixed-type equations

7. Conducting research on the practical application of research results.

An unique-valued solution to the inverse problem of determining the fractional order for a sub-diffusion equation in two and multidimensional fields has been proved.

For the mixed-type linear equations of the first kind of three-dimensional (especially the Trikomi and Chapligin equations), the existence, uniqueness, and smoothness of generalized solutions of some non-local boundary value problems in infinite domains are proved.

For mixed-type loaded linear and nonlinear equations, the existence and uniqueness of the solution of  boundary value problems with special conjugating conditions have been proved.            

It has been proved the existence of the unique solution of the problem with non-local condition for the  fractional-order, decomposable mixed-type equation.         

Some properties of two-variable Mittag-Leffler and four-variable hypergeometric functions which are mostly used in the study of direct and inverse problems for partial differential equations  with integer and fractional order derivative are investigated.     

The properties of the Appel and Laurichella hypergeometric functions used in the study of single- and double-layer potentials were studied.

            For a mixed integral-differential equation with a spectral parameter involving Caputo fractional order differential operator, uniquely solvability of the boundary value problem is proved   at regular values of the spectral parameter.

The problem which was open since 1976, generalized localization problem for spherical partial sums of multiple Fourier series has been completely solved.

A uniquely solvability of local and nolocal boundary value problems for mixed-type equations has been proved.

Academician M. S. Salaxiddinov

•          In 1974, he was awarded the Beruni State Prize of Uzbekistan.

•          Anniversary  medal "For glorious labor" in 1970

•          In 1976 he was awarded the Medal of the Badge of Honor.

•          1993 Medal of Labor Glory

•          In 2007, he was awarded the Order of Merit.

•          In 1984, academician MS Salahiddinov was awarded the honorary title of "Honored Scientist of Uzbekistan."

•          In 1976, he was awarded the badge of Excellence in Public Education of Uzbekistan

•          In 1992 he was awarded the "Badge of Independence", and in 1992 - the "For Meritorious Service".

Academician Sh. O. Alimov.

•          In 1985, academician Sh. O. Alimov was awarded the Beruni State Prize of Uzbekistan.

•          In 2019, he was awarded the Order of Labor Glory for his achievements in science and education.

The laboratory cooperates with many research centers, including:

1. Moscow State University.

2. Institute of Mathematics and Mathematical Modeling of the Academy of Sciences of the Republic of Kazakhstan

3. Kabardino-Balkaria Scientific Center of the Russian Academy of Sciences, Institute of Applied Mathematics and Automation.

4. Novosibirsk State University

5. The Vanderbilt university (USA)

6. New Haven University (USA)

7. Tufts University, Medford, MA;

8. Oulu University (Finland)

9. Belgorod State National Research University NRU BelGU.

10. Ghent University (Belgium)

11. University of California, Merced, CA;

12. Fordham University, New York, NY;

13. CBPF, Rio de Janeiro, Brazil;

14. Free University, Berlin, Germany;

15. Berliner Hochschule fur Technik, Berlin, Germany;

16. University of Bologna, Bologna, Italy

17. Freiberg Technical University (Germany)

18. Malaysia International Islamic University (Malaysia)

19. Kazakh-Turkish University named after Ahmad Yassavi (Kazakhstan)