The results of the scientific research work conducted by the department members are discussed at scientific seminars organized by the department for students, postgraduates, and faculty members. They are also presented at regional, national, and international scientific symposiums and conferences. Based on the results of these studies, more than 200 scientific articles and 2 monographs have been published in scientific collections and journals.

The faculty members of the department conduct research in two main scientific directions:

I. Under the supervision of Professor S.K. Karimov, research is being conducted in the field of “Singularly Perturbed Differential Equations”. In this direction, research work is continued by Associate Professor G.M. Anarbaeva, Professor K.S. Alybaev, and senior lecturers: M.M. Mamatkulova, A. Akmatov, and A. Abdilazizova.

In this field, from 1994 to 2014, research was conducted on the project titled: "Investigation of solutions for singularly perturbed differential equations and their application in applied problems," registered by the Ministry of Education and Science of the Kyrgyz Republic under No. 020005725. Head: Doctor of Physical and Mathematical Sciences, Professor Karimov S.K.; researchers: Candidate of Physical and Mathematical Sciences, Associate Professors Anarbaeva G.M., Toigonbaeva A.K., senior lecturers Mamatkulova M.M., Kambarova A.D., Akmatov A.A., Abdilazizova A.A., and senior laboratory assistant Murzabekova A.S.

The solution of many physical problems is reduced to differential equations. In particular, the study of physical processes such as gas motion in a pipe in an ordinary medium, problems of quantum physics, and other similar phenomena leads to the solution of singularly perturbed problems. Boundary value problems consist of two parts: the boundary part and the initial part. The second part includes initial value problems for singularly perturbed differential equations. The research conducted in these works is relevant for the theoretical development of mathematics. Therefore, the problem of studying the asymptotics of solutions to singularly perturbed differential equations is considered one of the current urgent problems.

In the asymptotic evaluation of the solution of a system of singularly perturbed ordinary differential equations, the stability of the solution of the initial value problem for linear and nonlinear singularly perturbed ordinary differential equations and systems of equations was investigated using level line methods, transition to polar coordinates (the coordinate method), successive approximations, and the induction method.

As a result of the research work, an asymptotic evaluation of the solution for a system of singularly perturbed equations in generalized cases of eigenvalues was obtained, and an asymptotic evaluation of the solution for a system of singularly perturbed differential equations in special critical cases was conducted. Furthermore, the stability of the initial value problem for linear and nonlinear singularly perturbed ordinary differential equations was investigated; in cases where stability conditions were violated, the asymptotics of the solution for a system of ordinary differential equations containing a small parameter were constructed; the asymptotics of the solution for a system of singularly perturbed linear differential equations in the instability region were investigated.

II. Associate Professor A.K. Toigonbaeva and senior lecturer A.D. Kambarova, under the supervision of Doctor of Physical and Mathematical Sciences, Professor A. Asanov, are conducting research in the field of "Fredholm-Stieltjes Integral Equations of the First Kind and Volterra Equations of the Third Kind" .

As a result of conducting scientific research while working at the department, the following members successfully defended their candidate and doctoral dissertations: S.K. Karimov (1984, Kyiv), K.S. Alybaev (1995, Bishkek), G.M. Anarbaeva (1993, Bishkek), D.A. Tursunov (2005, Bishkek), M.A. Azimbaev (2010, Bishkek), and A.K. Toigonbaeva (2015, Osh).

The results of the scientific research work of the department members are discussed at scientific seminars organized by the department for students, postgraduates, and teachers, reported at regional, national, and international scientific symposiums and conferences, and more than 200 scientific articles and 2 monographs have been published in scientific collections and journals.

In 2009, Professor S. Karimov and A. Abdilazizova participated in the III World Congress of Mathematicians of Turkic-Speaking Countries in Almaty (Kazakhstan). In 2010, department staff participated in the anniversary of Academician A. Borubaev, and in 2011, in the anniversary dedicated to the 80th birthday of Academician M. Imanaliev. In 2012, S. Karimov, M. Mamatkulova, M. Azimbaev, and A. Toigonbaeva paid a scientific visit to the international conference "Functional Analysis and Its Applications" in Astana (Kazakhstan). From June 5 to 7, 2014, S. Karimov participated in the V World Congress of Mathematicians of Turkic-Speaking Countries at Issyk-Kul, where he presented reports on: "Uniform approximations of the solution of a singularly perturbed system of differential equations in a special critical case" and "Behavior of solutions of a singularly perturbed system of differential equations in a special critical case." In 2017, the abstracts of A. Asanov and A.K. Toigonbaeva on the topic: "The choice of the regularization parameter for solutions of linear Fredholm-Stieltjes integral equations of the first kind" were published in the proceedings of the VI World Congress of Mathematicians of Turkic-Speaking Countries held in Astana (Kazakhstan).

In addition, the teachers of the department successfully carry out scientific supervision of qualification (thesis) and master′s works of students and graduate students on the topics of the above-mentioned scientific research areas.