The Department (now the Laboratory) of Mathematical Physics was established in 1989 on the basis of the Laboratory of Nonclassical Problems of Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine

The department specializes in the study of the correct solvability of ill-posed boundary value problems for partial differential equations.

From the time of its establishment until 2017, the department was headed by Corresponding Member of the National Academy of Sciences of Ukraine, Doctor of Physical and Mathematical Sciences, Professor Bohdan Yosypovych Ptashnyk. Since 2017 and until now the laboratory is managed by Ph.D. Mykhailo Mykhailovych Symotiuk.

During the existence of the department about 30 scientists were its employees. They defended 19 candidate dissertations and 6 (Vasyl Ilkiv, O. Verbitsky, Volodymur Korzhik, Irina Kmit, Natalia Protsakh, Igor Medynskyi) doctoral dissertations.

Since 1996, the department has included the Chernivtsi branch, which employs a group of researches led by Doctor of Physical and Mathematical Sciences, Professor Stepan Dmytrovych Ivasyshen. Since 2021, the branch is managed by Dr. Igor Mikhailovich Cherevko.

In 2022, the Department of Mathematical Physics was merged with the Department of Differential Equations and Theory of Functions as a Laboratory of mathematical physics.

The main research topics of the Team:

study of correctness of problems with nonclassical (nonlocal, including integral, multipoint, Dirichlet type) boundary conditions for partial differential equations or differential-operator equations and systems of such equations, the solvability of which is related to the problem of small denominators;
study of spectral properties of boundary value problems for differential and differential-operator equations, including multiparameter problems that arise when using the generalized method of separation of variables in cases where the classical Fourier method is not applicable;
study of the solvability of the Cauchy problem, mixed problems and problems without initial conditions for ultraparabolic equations with variable indices of nonlinearities, parabolic equations and systems of equations with different degeneracies;
research of solvability of inverse problems for linear parabolic equations;
study of solvability of local and nonlocal boundary value problems for linear and nonlinear hyperbolic systems with discontinuous and strongly singular initial data;
investigation of solvability of boundary value problems for multifrequency systems of ordinary differential equations with delay in fast and slow variables.

Main scientific results:

Methods for studying the correctness and construction of solutions of multipoint problems, Dirichlet problems, periodic, nonlocal and integral problems for hyperbolic, parabolic and typeless (including unsolved for the highest derivative with respect to time variable) partial differential equations and systems of equations (linear, weakly nonlinear, loaded) of finite and infinite orders, as well as for differential-operator equations. In addition, nonlocal problems for partial differential equations over fields of p-adic numbers are considered. These methods are based on a metric approach to the problem of small denominators, which arise in many cases when constructing solutions of these problems. (Bohdan Ptashnyk, Valentyna Polishchuk, B. Salyha, Volodymyr Ilkiv, Petro Shtabaliuk, V. Fihol, Lesia Komarnytska, Liudmyla Syliuha, N. Zadorozhna, Taras Hoi, P. Vasylyshyn, Iryna Klius, Natalia Bilusiak, Mykhailo Symotiuk , Oksana. Medvid, Ivan Savka, Anton Kuz, Sofiia Repetylo).
A generalized method for separating variables has been developed and applied to construct the solutions of a number of problems for linear equations and systems of partial differential equations with constant and variable coefficients and for differential-operator equations in different functional spaces. The spectral problems that arise during the application of the method are investigated (Petro Kaleniuk, Yaroslav Baranetskyi, Zinovii Nytrebych).
A fundamental matrix of solutions of the Cauchy problem for linear parabolic systems with degeneracies is constructed, its properties are investigated and applied to establish the correct solvability of the Cauchy problem and the problem without initial conditions, depending on the type of degeneracy, for linear parabolic systems. Local and global solutions of the Cauchy problem for quasilinear parabolic systems with degeneracy is investigated (Stepan Ivasyshen, Vitalii Dron, H.alyna Pasichnyk, T. Balabushenko, V. Laiuk, V. Litovchenko, Ihor Medynskyi).
The unique solvability of mixed problems for degenerate equations and systems of high-order equations with the second derivative over the time variable (linear and nonlinear), as well as mixed problems and Fourier problems (problems without initial conditions) for nonlinear ultraparabolic equations is investigated (Natalia Protsakh).
Conditions for solvability of inverse coefficient problems for parabolic equations in domains with known and unknown boundaries are established(Mykola Ivanchov, Halyna Snitko).
The apparatus of generalized Colombo algebras is used to study the correctness of boundary value problems with local and nonlocal conditions for linear and nonlinear hyperbolic systems of equations with discontinuous initial data (Iryna Kmit).
Using the averaging method, new theorems on the existence and uniqueness of solutions of problems with integral conditions for multifrequency systems of ordinary differential equations with delay are proved; accurate (relative to the order of a small parameter) estimates of the deviation of the solutions of the original and averaged problems are established (R. Petryshyn).