Let A(U) be the set of analytic functions f in a unit disk U := {z : |z| < 1}; B^(r)_2,γ and B^(r)_2,γ^β,α (r ∈ Z₊) be classes of functions f ∈ A(U), where f^(r) (r ∈ N) belongs to weighted Bergman space correspondingly with a weight γ and Jacobi weight γ_β,α = |z|^α(1 − |z|)^β (α,β > −1); H^(r)_2,γ^β,α (α ≥ 0, β > −1) be the class of functions B^(r)_2,γ^β,α, satisfying the condition ||f^(r)||_2,γ^β,α ≤ 1. In this paper we study the problem on finding the sharp values of mean squared approximation of functions f ∈ B^(r)_2,γ^β,α and their simultaneous derivatives f^(s) (1 ≤ s ≤ r − 1, r ≥ 2) in the metric of space B_2,γ^β,α. We prove the sharp Jackson-Stechkin type of inequality connecting the best mean squared approximation of f ∈ B^(r)_2,γ^β,α and Peetre functional.

Stochastic counterparts of the Cauchy problem for systems of nonlinear second order parabolic equations written in terms of forward SDEs and FBSDEs are derived and studied. There are selected two types of nonlinear PDE systems arising in applications and there are developed two different stochastic approaches to study them. The stochastic approach to systems of the first type allows to develop a numerical algorithm to construct the Cauchy problem solutions based on the suitable FBSDE and the neural network technique. The stochastic approach to systems of the second type extends the results obtained in the theory of McKean-Vlasov type equations.

The solution of the direct problem of geothermia under sedimentation conditions for geothermal reservoirs is considered. The main factors forming the thermal field of sedimentary basins are taken into account in the most complete way — it is the expenditure of heat flow energy on the base for heating of cold sedimentary material, partial shielding of heat flow due to the difference of thermophysical sediments and base rocks, heat generation in accumulating sediments, different rate of sedimentation. The problem of calculating the value of heat flux from the foundation based on temperature observations in wells — the inverse problem of geothermy in sedimentation conditions — has also been solved.

In this paper, the exact periodic solutions for some generalized Boussinesq equations with constant deviating arguments are found by the method of decomposition by the elliptic Jacobi function. In the case when not all deviating arguments are multiples of the solution periods, the exact solutions are found using the delta amplitude function dnξ, and the modulus of the function is calculated.