Main research topics:
Finite-difference and spectral methods for numerical solving partial differential equations and singular integral equations. Mathematical modeling and numerical simulations in solving physical and engineering problems. Polynomial interpolation of operators and functions of matrix arguments together with associated computational algorithms.
Scientific interests of the department are analytical theory of ordinary and partial differential equations, theory of Painleve equations, isomonodromic deformation of linear differential systems, theory of nonlinear waves and solitons.
The main research directions of the Mathematical Cybernetics Department in graph theory are:
The main research directions in the Function Theory Department are boundary value problems of the theory of analytic functions on Riemannian surfaces, rational approximation, function spaces, analysis on metric measure spaces.
Theory of generalized random processes and differential equations with generalized coefficients.
Research area of our department includes stochastic analysis and stochastic differential equations. We investigate stochastic differential equations by using new generalized stochastic processes (mnemoprocesses). In parallel the stochastic calculus of variation or Malliavin calculus for processes with conditionally independent increments was developed. Similar methods of theory of new generalized functions are applied to deterministic differential equations with generalized coefficients.
Also the research interests of Functional Analysis Department members include spaces of functions, distributions and operators therein.
The research on the problem of multiplying distributions lead to the development of a general method to construct algebras of new generalized functions (a.k.a. mnemofunctions). With the use of this theory a detailed research on the equations with Dirac delta-like coefficients has been conducted.
An active research in going on in Analysis over Non-Archimedean metric spaces, in particular, over Non-Archimedean valued fields, and over the number theoretic rings of adeles.
The theory of operator algebras generated by dynamical systems has been developed. With the use of this theory invertibility conditions and one-sided invertibility conditions are obtained for functional operators. Symbols for non-local pseudo-differential operators has were introduced, necessary and sufficient conditions for their invertibility and one-sided invertibility were obtained.
The important research direction of the department is computer and mathematical modelling of the biological systems behavior under the action of the dynamic and static loads. Based on the obtained results the programs for the supporting of medical procedures, surgical operations and postoperative rehabilitation are developed in the field of dentistry, maxillofacial surgery, cardiology, sensor systems, in particular for the middle ear, as well as surgical cutting of tumor-like lesions of the long bones. These computer programs are used by health institutions and hospitals in Belarus and other countries of the world.