SCIENTIFIC, PEDAGOGICAL AND PUBLIC ACTIVITY OF PROFESSOR ASHUROV R.R.

Professor Ravshan Radjabovich Ashurov is a renowned mathematician recognized for his expertise in differential equations and functional analysis. He has made significant contributions to the development of spectral theory of differential operators and multidimensional harmonic analysis. Additionally, he is acknowledged worldwide for his scientific work in the field of fractional-order equations.

Ravshan Radjabovich Ashurov was born on March 19, 1955, in Tashkent. In 1972, he graduated with a gold medal from School No. 90. He entered the Faculty of Applied Mathematics and Mechanics at Tashkent State University (now the National University of Uzbekistan named after Mirzo Ulugbek). From 1976, R.R. Ashurov continued his studies at the Faculty of Computational Mathematics and Cybernetics of Moscow State University (MSU), graduating with honors in 1978. In the same year, he was admitted to the graduate program at the same faculty, where he began his research on the spectral theory of elliptic differential operators under the supervision of Professor Alimov Shavkat Arifjanovich (now an academician). In October 1981, he defended his candidate dissertation in the specialty of differential equations and mathematical physics (specialty code: 01.01.02) under the guidance of Academician A.N. Tikhonov at the Faculty of Computational Mathematics and Cybernetics of Moscow State University (MSU).

In October 1981, after returning to Tashkent State University, he began his pedagogical career as an assistant at the Department of Differential Equations in the Faculty of Mathematics. He later served as a senior lecturer and associate professor. In 1984, R.R. Ashurov was awarded a research fellowship in the United Kingdom, where he worked at the University of Birmingham under the supervision of the renowned scientist Professor V.N. Everitt. During this period, he published several articles on quasi-differential operators. Additionally, during his time in the UK, he delivered lectures at various universities, including Oxford University (seminar led by Professor J.B. McLeod), King’s College London (Professor E.B. Davies), Cardiff University (Professor W.D. Evans), Edinburgh University (Professor J.M. Ball), and the University of Birmingham (Professor V.N. Everitt). During these seminars, he presented his results on the spectral theory of differential operators.

In 1989, he was admitted to the doctoral program at Tashkent State University and was sent to MSU for further study. His scientific advisor during his doctoral research was Professor Sh.A. Alimov. In December 1992, he defended his doctoral dissertation in the specialty of differential equations at the Faculty of Computational Mathematics and Cybernetics of Moscow State University (MSU) under the guidance of Academician A.N. Tikhonov.

Upon returning to Tashkent State University in December 1992, R.R. Ashurov continued his work as an associate professor, professor, and head of the Department of Mathematical Physics. From 2001 until February 2003, he served as the dean of the Faculty of Mechanics and Mathematics. From September to December 2002, he served as a visiting professor at Bowling Green University in the United States, supported by an IREX grant, where he examined the organizational structures of U.S. universities.

From February 2003 to October 2004, R.R. Ashurov served as the rector of the Tashkent Regional State Pedagogical Institute. From 2004 to 2006, he served as the rector of the National University of Uzbekistan, named after Mirzo Ulugbek. From July 2006 to February 2009, he worked as a senior scientific researcher at the Mathematics Institute of the Academy of Sciences of Uzbekistan. Simultaneously, from 2007 to 2009, he held the position of professor at the Tashkent branch of MSU.

In February 2009, R.R. Ashurov received an invitation to work as a senior scientific researcher at the Institute of Advanced Technology (ITMA) at Putra University, Malaysia, where he conducted research until February 2012. During his time at Putra University, several graduate and postgraduate students defended their theses under his supervision. Putra University awarded the research conducted with his students gold, silver, and bronze medals. He also led two scientific research grants at the university and a grant from the Ministry of Higher Education of Malaysia. He was a member of the organizing committee for several international conferences held in Malaysia.

From 2012 to 2018, R.R. Ashurov continued his scientific career as a senior researcher at the Mathematics Institute of the National University of Uzbekistan, now known as the Mathematics Institute of the Academy of Sciences of Uzbekistan. Since 2018, he has served as the head of the Laboratory of Differential Equations and Their Applications at the Mathematics Institute of the Academy of Sciences of Uzbekistan, a position he continues to hold.

R.R. Ashurov, as the rector of the National University of Uzbekistan, visited China, Iran, Germany, and Russia. Currently, he regularly travels to Russia, the United Kingdom, the USA, Japan, Bulgaria, Germany, Italy, Hong Kong, Malaysia, the United Arab Emirates, Brazil, Finland, Turkey, China, Belgium, and other countries for scientific research and lectures, contributing to the development of science.

In the 1980s, R.R. Ashurov conducted scientific research on the convergence of multiple Fourier integrals and series in regions bounded by the surface of elliptic polynomials. During these studies, it was shown that the conditions for the localization of these expansions are intrinsically related to the geometry of the surface of the elliptic polynomial. R.R. Ashurov derived specific conditions for the localization of these expansions, proving that these conditions depend on the number of nonzero principal curvatures of the corresponding surface. Additionally, he demonstrated that, unlike classical localization, the convergence of these expansions at almost all points does not depend on the geometry of the surface of the elliptic polynomial; these conditions apply to all elliptic polynomials similarly to spherical expansions.

Furthermore, R.R. Ashurov researched the asymptotic estimates of spectral functions for elliptic operators of arbitrary order with smooth coefficients in the space of square-integrable functions. During this study, he proved a remarkable Tauberian theorem, applicable in the spectral theory of operators. Using this Tauberian theorem, R.R. Ashurov obtained an exact asymptotic estimate for the spectral function of an elliptic operator with a fixed principal part in the space of bounded functions.

During his time in the United Kingdom, R.R. Ashurov, in collaboration with V.N. Everitt, laid the theoretical foundations for quasi-differential operators in fully metrizable locally convex topological spaces. The results obtained for these operators were generalized to the case of quasi-differential vector operators.

R.R. Ashurov also studied the quadratic and cubic partial sums of multiple Fourier series and integrals. He derived the necessary and sufficient conditions for the localization of these expansions in Sobolev spaces, significantly strengthening the previously known results of S. Hoffman and Liu Feng-Chi. Later, together with his student A. Metwali, he obtained similar results for triangular sums.

In the 1990s, R.R. Ashurov studied the spectral expansions of general elliptic pseudo differential operators. He found the necessary and sufficient conditions for the convergence of such expansions at almost all points in the class of functions integrable with respect to a certain degree of their derivatives. It is essential to note that the differential operator class is too small to provide a complete solution to this problem, and the solution was obtained precisely for the class of pseudodifferential operators. In a special case, where the spectral expansions of differential operators coincide with multiple Fourier integrals, R.R. Ashurov, together with his student K.T. Buvaev, succeeded in deriving the necessary and sufficient conditions for the convergence of differential operators at almost all points.

R.R. Ashurov together with his student Y.E. Fayziev, investigated the convergence and summability of spectral expansions associated with the Schrödinger operator featuring a singular potential. The authors successfully derived specific conditions for the uniform convergence of these spectral expansions. It is worth mentioning that the potential considered in this work belongs to the Stummel class, which includes all previously studied potential courses for this purpose. Furthermore, R.R. Ashurov proved that if the potential has stronger singularities, then all eigenfunctions vanish at the point of potential singularity, and it makes no sense to study the convergence of the corresponding spectral expansion at this point.

R.R. Ashurov, together with his graduate student M. Sokolov, investigated the spectral expansions of self-adjoint differential vector operators in Hilbert space.

In 2006, R.R. Ashurov began studying the convergence and summability of multiple Fourier series and integrals of piecewise-smooth functions. In particular, he proved that the Riesz means of such series and integrals converge similarly to spherical summation in regions bounded by the surface of an elliptic polynomial. Later, together with his student A. Butaev, he proved that the well-known Pinsky effect for spherical partial sums also holds in the general case. This result significantly strengthened and clarified the previously known results of M. Taylor and M. Pinsky. Subsequently, R.R. Ashurov and A. Butaev proved that the well-known Cahn effect for spherical partial sums also holds in the general case. This latest result was published in the proceedings of the French Academy of Sciences at the recommendation of academician J. Cahan.

During his tenure at Putra University in Malaysia, R.R. Ashurov focused on the theory and applications of wavelet expansions. His early results in this field were related to one-dimensional continuous wavelet expansions. It is well known that the result concerning the convergence of wavelet series expansions of integrable functions at all Lebesgue points belongs to Terence Tao. American mathematicians M. Rao, H. Sikic, and R. Song proved that for continuous wavelet expansions, convergence almost everywhere is valid only for functions integrable to some  degree. R.R. Ashurov demonstrated that T. Tao’s result is also valid for continuous wavelet expansions. Later, together with his student A. Butaev, R.R. Ashurov obtained results on the convergence and spherical summability of multivariate continuous symmetric wavelet expansions at all Lebesgue points. It is worth noting that before these results, no convergence results had been obtained for multivariate continuous wavelet expansions at individual points. These authors introduced a new class of multivariate wavelets, termed “quadratically symmetric,” and studied the convergence of expansions based on these wavelets. It is important to note that applying these new wavelets in image reconstruction provided significantly better results compared to using previously known wavelets.

R.R. Ashurov, in collaboration with Professor B. Turmetov has published several articles, particularly finding the conditions for the existence of solutions to polyharmonic equations for various Neumann problems. The obtained conditions were more straightforward to verify and significantly more uncomplicated compared to previous conditions. Later, these authors proposed a straightforward method for constructing solutions to fractional-order differential equations. The difference between this method and the previously known Mikusinski method is that there is no need to introduce new objects and verify their properties in this case. It is worth mentioning that the method proposed by R.R. Ashurov and B. Turmetov is a development of the method suggested by B.A. Bondarenko in the 1980s for solving partial differential equations.

R.R. Ashurov, together with his student A. Butaev, conducted research on the generalized localization of multiple Fourier series of compactly supported distributions. Necessary and sufficient conditions were derived for generalized localization to be valid in Sobolev classes of negative order. Additionally, they solved the generalized localization problem for spectral expansions of multi-dimensional symmetric wavelet expansions. Necessary and sufficient conditions were derived for generalized localization to be valid for distributions belonging to Sobolev classes of negative order.

Another contribution of R.R. Ashurov to the field of mathematics was his solution to the generalized localization problem of multiple Fourier trigonometric series, which had been posed by V.A. Il’in in 1968 and had remained unsolved for 50 years.

In recent years, R.R. Ashurov has achieved significant success in solving forward and inverse problems for fractional-order equations. He positively resolved some unsolved problems mentioned by the Japanese mathematician M. Yamamoto. In addition, he derived the conditions for the existence and uniqueness of solutions to fractional-order equations defined in any bounded domain with a smooth boundary, where the elliptic part is of arbitrary order, thus opening a new direction in this field. R.R. Ashurov also achieved great success in solving inverse problems for fractional-order equations. In collaboration with Academician Sh.A. Alimov, they studied the subdiffusion equation with a self-adjoint operator defined in a separable Hilbert space. They successfully solved the inverse problem of determining the order of the fractional derivative with respect to time.

In 2020, during the COVID-19 pandemic, a need arose to predict the intensity and spread rate of the virus. In this regard, under the directive of the Cabinet of Ministers of the Republic of Uzbekistan, Professor R.R. Ashurov was appointed as the head of the scientific working group formed at the Institute of Mathematics. The group was tasked with developing a mathematical model to predict the spread of the pandemic and forecast the number of people infected using this model. Professor R.R. Ashurov and his students established contact with scientists from leading universities in the USA, including Professor S.R. Umarov from the University of New Haven and Professor Yan Quan Chen from the University of California, to discuss this issue. The scientific research was conducted in direct collaboration with Johns Hopkins University, which was managing pandemic-related studies worldwide. As a result of this collaboration, a new adequate mathematical model of pandemic processes based on the theory of fractional-order differential operators was created, and the forecast results obtained using this model were sent to the Cabinet of Ministers from April 8, 2020, until the end of the year. These results enabled the advance planning of key pandemic measures, including the establishment of specialized departments in existing hospitals, the construction of new hospitals, the recruitment of medical personnel, and the determination of the necessary volume of medications.

R.R. Ashurov has won several individual scientific grants from leading foreign countries. Specifically, from September 1984 to July 1985, he undertook a scientific trip to the University of Birmingham in the UK under a British Council individual grant, and from September to December 2002, he visited Bowling Green University in the USA under an IREX individual grant. In May 1991, he participated in an international conference in Oberwolfach, Germany, with the support of a Volkswagen grant.

From 2001 to 2010, he led the scientific group working under the grant PRJ-24, “Functional Analysis and Applications,” at the International Centre for Theoretical Physics (ICTP) in Trieste, Italy. He also led scientific groups at Putra University under the Malaysian Ministry of Higher Education’s FRGS grant from 2009 to 2011 and RUGS grants from 2009 to 2012.

R.R. Ashurov has also led several local grants. He was the head of the fundamental project “Boundary value problems for differential equations with fractional derivatives and free boundary nonlinear problems” F4-FA-F010 from 2012 to 2016, and “Studies of direct and inverse problems for second-order and higher mixed-type equations” OT-F4-88 from 2017 to 2021 under the Uzbek Academy of Sciences. Since 2022, he has been heading the fundamental project “Differential Equations with Whole and Fractional Orders” F-FA-2021-424, funded by the Ministry of Innovative Development of Uzbekistan.

Additionally, R.R. Ashurov has established close scientific collaborations with foreign researchers, including V.N. Everitt (UK), M. Yamamoto (Japan), A. Pskhu (Russia), A. Ashyralyev (Turkey), S.R. Umarov (USA), and S.M. Sitnik (Russia), as well as T. Sh. Kalmenov, M.A. Sadybekov (Kazakhstan), V.S. Serov (Finland), and M. Ruzhansky (Belgium).

He regularly presents at international conferences and scientific seminars at leading universities. For instance, in 2023, he gave lectures at the “Modern Issues in Mathematical Physics” seminar at the University of New Haven (USA) and participated in a “WORKSHOP” at Ghent University (Belgium). In 2024, he delivered lectures at international conferences in Malta, Azerbaijan, the UAE, and Turkey, where he also chaired sessions. Moreover, he was invited to the “Fractional Calculus Seminars” at SISSA in Italy as a speaker.

R.R. Ashurov’s significant contributions to the advancement of science have been recognized worldwide. On November 15, 2022, he was awarded a certificate by the international publisher “Springer Nature” as the “Most Published Author.” In 2023, he was honored with a diploma and medal from the Turkic World Mathematical Society (TWMS) for his contributions to mathematics at its VII Congress, as well as with a commemorative badge for the 80th anniversary of the Uzbek Academy of Sciences.

He has published over 150 scientific articles, textbooks, and methodological manuals, with more than 100 appearing in prestigious international journals. One of his notable works, co-authored with academician Sh. A. Alimov, is the textbook “Mathematical Analysis,” which has been republished twice and comprises three volumes. This book is one of the first textbooks on mathematical analysis written in the Uzbek alphabet, utilizing the Latin script. In addition, R.R. Ashurov is the author of several monographs published in Uzbekistan and leading foreign countries, including “Nonlocal and Inverse Problems for Fractional Order Differential Equations of Higher-Order Elliptic Type” (2024) and “Initial-Boundary and Inverse Problems for Higher-Order Differential Equations” (2025), co-authored with his students.

R.R. Ashurov is also a member of the editorial boards of the following scientific journals: “Uzbek Mathematical Journal,” “Vestnik KRAUNC” (Russia), and “Fractional Differential Equations” (Zagreb, Croatia).

Among his students are 5 Doctors of Science and 8 Candidates of Science, who are currently working in Uzbekistan, Malaysia, Canada, and the Arab Republic of Egypt. Currently, 5 of his students are on the verge of defending their PhD dissertations. Additionally, R.R. Ashurov actively engages young people in science, regularly organizing scientific seminars and discussions with undergraduate and graduate students at the National University of Uzbekistan.

Currently, R.R. Ashurov leads four scientific seminars, one of which is an international seminar in collaboration with foreign scientists. He is also a member of the Specialized Scientific Councils at the V.I. Romanovskiy Institute of Mathematics and the National University of Uzbekistan. We wholeheartedly congratulate Ravshan Rajabovich on his anniversary and wish him robust health, great success in his scientific and teaching activities, family happiness, and long life!