Vol 64, No 11 (2024)

The question of the effective solution of the basic initial boundary value problems for spatially onedimensional nonlinear equations of the parabolic type describing reaction-diffusion processes is considered. Such equations include the Kolmogorov–Petrovsky–Piskunov and Burgers equations. For these problems, a numerical analytical method based on implicit discretization of the differential operator in combination with explicit Adams–Bashforth extrapolation for the nonlinear term of the equation is proposed. At the same time, a new efficient algorithm based on analytical representations using an explicit form of a fundamental solution system has been developed to solve a sequence of emerging linear problems. The effectiveness of the developed method and its advantages over some traditional algorithms have been demonstrated for a number of complex model examples.

The classical problem of stationary waves on the surface of an ideal incompressible homogeneous liquid of finite depth is considered. The approach to solving the problem is related to the second Stokes method, but has the following differences: due to the obtained one-dimensional integrodifferential equation with cubic nonlinearity for the profile of a stationary wave on the surface of a liquid of finite depth, the original problem is reduced to a one-dimensional one. The solution is obtained up to the seventh approximation.

In classical works, equations for the fields of gravity and electromagnetism are proposed without subtracting the right-hand sides. Here we give the subtraction of the right–hand sides and the analysis of the momentum energy tensor in the framework of the Vlasov–Maxwell–Einstein equations and consider cosmological models of the Milne-McCree and Friedman types.

The paper is a continuation of the work of the same name by Belkarian, Belkarian (2024). The proof of the theorem of the existence and uniqueness of soliton solutions and their corresponding solutions of the functionally differential equation from the dual pair “function-operator”, which was formulated in the noted work in the form of a hypothesis, is presented. This made it possible, in particular, to study soliton solutions with more complex characteristics in the model of traffic flow on the Manhattan lattice than the characteristics given by the additive cyclic subgroup of the group R.

Approximations of random functions are studied, numerically modeled to study the stochastic process of particle transport, including problems of fluctuations in the criticality of the process in random breeding media. The simplest grid model of an isotropic random field is formulated, reproducing the effective average correlation length, which ensures high accuracy in solving stochastic transfer problems at a small correlation scale. The proposed algorithms have been tested in solving the test problem of estimating the overexponential average particle flux in a random multiplying medium.

The problems of unilateral discrete contact of an elastic half-space and a rigid punch of finite dimensions with a surface microrelief are considered. A variational formulation of the problems is obtained in the form of a boundary variational inequality using the Poincare-Steklov operator, which maps normal stresses into normal displacements on a part of the boundary of an elastic half-space. The minimization problem equivalent to the variational inequality is presented, as a result of approximation of which a quadratic programming problem with constraints in the form of equalities and inequalities is obtained. To solve this problem, a new computational algorithm based on the conjugate gradient method is proposed, which includes three equations of punch equilibrium in the calculation. The algorithm belongs to the class of active set methods and takes into account the specifics of the set of constraints. Some patterns of contact interaction of surfaces with regular microrelief have been established.