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Different approximate crack opening models in a porous stratum are derived, based on a priori representations of the crack size, accurate solutions of motion equations of viscous fluid, approximate expressions of the filtering model in the stratum, and approximate expressions of the theory of elasticity of the stratum. The law of conservation of momentum of the fluid in the crack is used to obtain quasilinear parabolic equations, describing the crack opening.

A generalized phenomenological Leith model of wave turbulence in a medium with nonstationary viscosity is under study. Group analysis methods are used to obtain the main models possessing nontrivial symmetries. All invariant submodels are determined for each model. Invariant solutions describing these submodels are either determined in explicit form or satisfy the integral equations obtained. The main models are used to study turbulent processes. At an initial instance and with a fixed value of the wave number modulus, either turbulence energy spectrum and its gradient or turbulence energy spectrum and the rate of its variation are specified for the above-mentioned models. It is determined that solutions of the problems describing these processes exist and are unique under certain conditions.

Nonlocal equations of the coagulation theory are studied by methods of group analysis. In addition to the integrodifferential Smoluchowski equation, equivalent models are also considered, including the equation for the Laplace transform of the original equation, an infinite system of equations for the power moments of its solution, and the equation for the generating function of the power moments. Admissible Lie groups for the considered equations are found, their relationships are studied, and the corresponding invariant solutions are analyzed.

Horizontal shear motion of a homogeneous fluid in an open channel is considered in the approximation of the shallow water theory. The main attention is paid to studying the mixing process induced by the development of the Kelvin-Helmholtz instability and by the action of bottom friction. Based on a three-layer flow pattern, an averaged one-dimensional model of formation and evolution of the horizontal mixing layer is derived with allowance for friction. Steady solutions of the equations of motion are constructed, and the problem of the mixing layer structure is solved. If bottom friction is taken into account, the mixing process becomes slower and the width of the intermediate mixing layer does not increase. Verification of the proposed one-dimensional model is performed through comparisons with available experimental data and with the numerical solution of the two-dimensional equations of the shallow water theory.

The classical Boussinesq equation describing gravity waves in shallow waters is under consideration. Hirota's bilinear representation is used to construct exact solutions describing wave packets, waves on solitons, and «dancing» waves. The principle of multiplying the solutions of the Hirota equation is formulated, which helps constructing more complex structures made of solitons, wave packets, and other types of waves.

A fundamentally new unsaturated technique for the numerical solution of the Dirichlet-Neumann problem for the Laplace equation was designed. This technique makes it possible, due to the smoothness of the sought solution of the problem, to take into account the axisymmetric specificity of the problem, which prevents the use of any saturated numerical methods, i. e., methods with a leading error term.

Variational approach and sensitivity theory methods are used to construct algorithms for solving the problems of environmental forecast and design. The behavior of the model in a parameter space is studied by calculating sensitivity functions as partial derivatives of target functions with respect to the model, used to investigate the properties of mathematical models and solutions of inverse problems. Not only the proposed approach implies the atmospheric quality model in a Novosibirsk agglomeration and an algorithm based on an ensemble of sensitivity functions, but is also used to solve the inverse problem of position estimation and pollution source intensity.

Numerical modeling results on growth of axially symmetric hydraulic fracture new free surface in isotropic poroelastic medium are presented. The problem was solved using extended finite element method based on phantom nodes and cohesion model of failure. Trajectories of the fracture are calculated for different distances from free surface under injection of certain volume of working fluid with regard to the fluid leakage. The influence of impermeable boundary on the hydraulic fracture growth is analyzed.