One of the winning grant proposals of “Basic Grant Competition” of the Science Development Foundation (SDF) Under the President of the Republic of Azerbaijan is titled “Investigation of mathematical modeling of some class of stationary and non-stationary problems of mechanics and physics”, one of which participants is prof. Nazim Kerimov from the Department of Mathematics of Khazar University. In the project, along with Nazim Kerimov, there participate 7 researchers from Baku State University and Institute of Mathematics and Mechanics of the National Academy of Sciences. The project period is 12 months.

Within the framework of the project it will be investigated both the longitudinal and torsional oscillations of the Euler-Bernoulli beam with loads at the ends, as well as the bending oscillations of the Euler-Bernoulli beam with inertial load at both ends and the force acting on the axis at the cross sections. Such types of vibrations of the Euler-Bernoulli beams are encountered during flights of spacecrafts and aircrafts, the extraction of residual oil from oil wells, and many other processes in physics, mechanics, and engineering.

Mathematical models of the mentioned problems will be expressed by boundary value problems for the second and fourth order partial differential equations of hyperbolic type. These boundary value problems will be reduced to the eigenvalue problems for linear and nonlinear ordinary differential equations of the second or fourth order, which contain the spectral parameter in the boundary conditions by applying the method of separation of variables. The project will give a general description of the location of the eigenvalues of linear spectral problems on the real axis, determine the order of multiplicity of all eigenvalues, study the oscillation properties of the eigenfunctions, obtain asymptotic formulas for eigenvalues and eigenfunctions, and select two, three or four other systems, as will be shown to form bases in Lebesgue spaces.

Using these results we will prove the existence and uniqueness of classical solutions of direct and inverse problems for the corresponding partial differential equations, as well as the existence of an unbounded continuum of solutions of nonlinear problems containing spectral parameters in boundary conditions. The existence of a global solution for stationary and non-stationary nonlinear equations or systems of equations obtained as a result of mathematical modeling of processes such as heat transfer, air filtration in a certain environment, and gas separation in a closed medium will be studied, also sufficient conditions will be found for the absence of a global solution, and the accuracy of the sufficient conditions.

A package of recommendations and suggestions will be prepared for estimation of non-stationary oscillations of a vertical liquid column stratified by density and their stabilization for wave propagation in the elastodynamic system, wave frequency evaluation and practical application. Mathematical analysis of processes will be carried out in concrete tests, the corresponding analytical expressions of hydrodynamic quantities and physical parameters and the range of concrete values will be specified. The discovered solution of the mathematical model of the physical event and posses will allow to understand the essence of the considered model, to compare and evaluate the effectiveness of asymptotic and approximate numerical methods.