ACADEMICIAN MAHMUD SALAKHITDINOV

Mahmud Salakhitdinovich Salakhitdinov is a prominent mathematician, Doctor of physical and mathematical sciences, Professor, Academician of the Academy of Sciences of the Republic of Uzbekistan, Adyghe Circassian Academy of Sciences and the Islamic Academy, honored scientist of Uzbekistan, laureate of the State Prize of Uzbekistan named after Abu Rayhan Beruni.

Makhmud Salakhitdinov is the founder and leader of the modern scientific school of Uzbekistan in the field of differential equations, a prominent public and state servant who has made a great contribution to the development and organization of science and higher education.

As the author of fundamental research in the field of partial differential equations, integral equations, and the theory of functions, Makhmud Salakhitdinov left a deep imprint in modern mathematical science. His name is associated with a huge contribution to the development of mathematical research and the preparation of promising scientific personnel both in Uzbekistan and far beyond.

Makhmud Salakhitdinov was born on November 23, 1933 in the city of Namangan. In 1950 he entered the Physics and Mathematics Department of the Central Asian State University (now the National University of Uzbekistan) and in 1955 graduated with honors from the Department of Mathematical Analysis.

Starting from the student's bench, Mahmud Salakhitdinovich Salakhitdinov actively conducts scientific research work. After graduation, he was recommended to post-graduate school and in 1958 he graduated with successful completion of the thesis for the degree of candidate of physical and mathematical sciences. The first papers of Makhmud Salakhitdinov relate to the investigation of the Cauchy problem for systems of nonlinear partial differential equations in the presence of constraints, which themselves are relations between partial derivatives.

In the 60-s a new stage in the scientific activity of Makhmud Salakhitdinov began, directed to conduct research work at the Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan. He enthusiastically, with great energy, is working in a new direction - the study of boundary value problems for partial differential equations of mixed and mixed-compound types. A beneficial influence on the formation of M. S. Salakhitdinov as a scientist in those years was his stay at the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences under the guidance of Academician A.V.Bitsadze.

To this period belong his work on the complete classification of third-order equations, the indication of the canonical forms of these equations when they belong to a composite or mixed-composite type, as well as to study a number of new boundary-value problems for the model and general equations of a mixed-compound third-order type. The results of these studies were the content of a doctoral dissertation "On the theory of third-order equations of mixed-composite type which he successfully completed in May 1967.

The core direction of Makhmud Salakhitdinov's research was the discovery of new classes of partial differential equations, the definition of statements of correctly posed local and nonlocal boundary value problems for second and higher order differential equations, the study of the influence of the coefficients and the dimensionality of the space on the correctness of boundary value problems.

For equations of mixed type of the first and second kinds, as well as equations with discontinuous coefficients, he studied various boundary value problems, both with continuous and with discontinuous conditions on the degeneration line of type. Many of the problems formulated and considered by him are reduced to functional and integral equations. It should be noted that the results obtained by Makhmud Salakhitdinov in studying integral, especially singular integral equations, are of particular interest. His papers devoted to boundary value problems for a general equation of a composite third-order type degenerating on the boundary of a domain deserve much attention. In these papers he developed a theory of loaded integro-differential equations, and an alternative of Fredholm type was obtained for boundary value problems.

Since the 1970s, the scientific interests of Mahmud Salakhitdinovich have been associated with the study of boundary value problems with nonlocal conditions, namely, conditions with displacement and conditions of Bitsadze-Samarskii type for partial differential equations. Here, for a wide class of equations of mixed type with nonsmooth lines of degeneracy, he posed and investigated a number of boundary-value problems. Subsequently, Makhmud Salakhitdinov developed three-dimensional analogues of the Tricomi and Gellerstedt problem for equations of mixed type with two inner degeneracy planes in infinite cylindrical and prismatic regions. Of great interest are the studies of Mahmud Salakhitdinovich and his followers on the theory of operators of integro-differentiation of fractional order. Various properties and laws of compositions of generalized integro-differentiation of fractional order with identical and different beginnings are studied, the nuclei of which contain the functions of Gauss, Bessel, and Meyer. These results were applied to the study of boundary value problems for mixed parabolic-hyperbolic and elliptic-hyperbolic equations with two lines and different orders of degeneracy.

Significant in depth are the works of Makhmud Salakhitdinov on nonlocal boundary value problems for linear equations of mixed type with a spectral parameter. This includes the establishment of the absence of a point spectrum in certain sectors of the plane of the spectral parameter, the determination of the eigenvalues and eigenfunctions of local and nonlocal boundary value problems, the proof of the unique solvability of nonlocal boundary value problems in simply connected and doubly-connected domains, and the discovery of a new method for proving the uniqueness of the solution of boundary value problems with displacement.

One of the scientific directions of Makhmud Salakhitdinov's activity is the study of boundary value problems for mixed parabolic-hyperbolic equations in regions with a departure from the characteristic. In these studies, the correctness of the generalized Tricomi problem was proved and a conjugate boundary value problem was found, and the Volterra property (the absence of a spectrum) of the generalized Bitsadze problem was established. He has established the completeness and Riesz basis property of the root functions of a class of non-local problems for model and mixed parabolic-hyperbolic equations of the second and third orders, and also for a single mixed-composite equation.

Makhmud Salakhitdinov also developed a research of boundary value problemsfor loaded equations of hyperbolic and parabolic-hyperbolic types of the third order. It is especially noted that the study of the analogue of the Bitsadze-Samarskii problem for equations of mixed type with one and two degeneration lines, where the nonlocal condition connects the values of the desired solution to parts of the boundary and to several internal open curves, is a major contribution to the development of a new direction of the theory of equations of mixed type of the first and the second kind.

In recent years, Makhmud Salakhitdinov devoted special attention to the study of special functions of the hypergeometric type, as well as to the study of boundary value problems with the Franklle condition for a mixed type equation with two internal lines and different degeneracy orders. He found various properties of hypergeometric functions of several variables that are applied to the study of boundary value problems for degenerate equations of mixed type.

Makhmud Salakhitdinov's research is a major contribution to the development of the theory of differential and integral equations, as well as the theory of special functions and integro-differentiated operators. In 1974 Makhmud Salakhitdinov was awarded the State Prize of Uzbekistan named after Abu Rayhan Beruni for the cycle of research in the theory of differential equations. The same year he was elected a full member of the Academy of Sciences of Uzbekistan (Academician), and in 1984 he was awarded the honorary title "Honored Scientist of Uzbekistan".

The bibliography of M.Salakhitdinov's published works is 369 titles, including 5 monographs, 6 textbooks and 6 teaching aids, 247 scientific articles, as well as 74 popular science and journalistic articles in leading domestic and international publications.

Makhmud Salakhitdinov was a talented organizer of science and higher education. He headed for many years the Institute of Mathematics (1967-1985), where he significantly strengthened the institute is scientific positions and potential, expanded contacts with leading mathematical centers, actively promoted the fulfillment of the basic functions of the Institute as the main organization for the development of theoretical and applied mathematics in Uzbekistan. Being Vice-president (1984-1985), and then President (1988-1994) of the Academy of Sciences of the Republic of Uzbekistan, he contributed to the full development of fundamental research in various fields of science, accelerating scientific and technological progress and socio-economic development in country.

Makhmud Salakhitdinov constantly combined fruitful scientific work with great pedagogical and public-state activity. As a Minister of Higher and Secondary Special Education (1985-1988), he made a significant contribution to the development of higher education, strengthening its human resources. In the years 1980-1986, 1993-1995, and 2004-2011, he headed the Department of Differential Equations of the National University of Uzbekistan, created on his initiative. M.Salakhitdinov was the Head of the republican Seminar on differential equations organized by him in 1962, which played a huge role in the developments in this field.

M.S.Salakhitdinov was a caring and benevolent teacher, and at the same time a very fair, demanding and principled mentor of young researchers. Among the immediate students and successors of Academician Salakhitdinov's scientific school there are 9 doctors and 38 candidates of physical and mathematical sciences who work fruitfully both in Uzbekistan and abroad. In addition, he is made a tremendous impact on formation and professional growth of a whole galaxy of modern mathematicians, forming the trends of modern mathematical science.

As Chairman of the specialized council for dissertations for the degree of Doctor of Science at the Institute of Mathematics of the Uzbekistan Academy of Sciences. (1989-1997), Chief editor of journals  Reports of Uzbekistan Academy of Sciences. (1989-1997) and Izvestiya AN RUz. (1965-1988), the collection of differential equations (1965-1991), a member of the editorial boards journal "Uzbek Mathematical Journal"(1991-2013) and other scientific journals, Makhmud Salakhitdinov has made a great contribution to raise of the authority and prestige of science in Uzbekistan.

Makhmud Salakhitdinov took part in the work of major international scientific and mathematical congresses in Moscow (1966), Nice (France, 1970), Warsaw (Poland, 1983), Beijing (China, 1989), Kyoto (Japan, 1990), Tehran (Iran, 1992), Istanbul (Turkey, 1992, 1994). Makhmud Salahitdinov regularly participated in conferences and symposia on differential equations in Sofia (Bulgaria, 1976, 1987), Ashgabat (Turkmenistan, 1978) Samara (Russia, 1984), Nalchik (Russia, 1986, 1996, 2003, 2012), Novosibirsk (Russia, 1992, 2000, 2007), Alma-Ata (Kazakhstan, 1991, 2006, 2009), Tbilisi Georgia, 1986, 1991).

As a member of the Organizing Committee, Makhmud Salahitdinov actively participated in the preparation and holding of the First World Congress of the Mathematical Society after Bernoulli (Tashkent, 1986). He was also chairman and member of organizing committees of many conferences and seminars on differential equations (1970-2017). He was repeatedly lectured on differential equations in leading international scientific centers, and in higher educational institutions in the regions of Uzbekistan.

Along with state and research delegations, Makhmud Salahitdinov visited Jordan, Syria (1973), China (1989), India (1987, 1990, 1991, 1992), Japan (1990), Turkey (1991, 1992, 1994), the USA (1992), Iran (1992), France (1993), Chile (1993), Russia (1994, 2000, 2003, 2007, 2012).

For outstanding achievements in the development of mathematical science and higher education in Uzbekistan, fruitful pedagogical activity and training of scientific personnel, Makhmud Salahitdinov was awarded with lots of state orders and medals. He was repeatedly elected as a People's Deputy of the Republic of Uzbekistan (1985-2000).

Academician M.S.Salakhitdinov passed away on April 27, 2018. On  November 23, 2018, he would be 85 years old.

His research and work is continued by his colleagues and students. The  hole multifaceted life of a remarkable scientist and talented organizer of science and education, Makhmud Salakhitdinov, is an example of selfless service to science and people.