Journal of Applied Mathematics and Mechanics

This paper considers a third-order nonlinear differential equation with moving singular points and critical poles. This type of equation is not solvable by quadratures. The author's method, implemented in the study of the Van der Pol equation, was developed further in the equation under study. A proof of the existence theorem for moving singular points and a solution in the neighborhood of a moving singular point is given, based on a modified Cauchy majorant method. An analytical approximate solution in the neighborhood of a moving singular point is constructed, and a priori error estimates are obtained. A numerical experiment confirming the obtained theoretical results is presented.

We consider a spatial restricted elliptic problem of three bodies (material points) attracting each other according to Newton's law. The problem parameters (eccentricity of the orbits of the main attracting bodies and their mass ratios) are assumed to lie within the first approximation stability region of the Lagrangian libration points and do not fall on the third- and fourth-order resonance curves corresponding to the planar elliptic problem. The eccentricity is assumed to be small. The normal form of the Hamiltonian function is obtained up to and including terms of the fourth degree relative to deviations from the libration point. The normal form coefficients are written out with an accuracy of up to the second degree of eccentricity inclusive.

The problem of calculating the equilibrium shape of a capillary surface in a tube of circular cross-section, vertically lowered into a reservoir with a weighty incompressible liquid, is formulated. An equation for the equilibrium of a liquid column in a tube in the vertical direction is obtained, which closes the formulation of the problem under consideration without using Young's formula (and its modifications) and allows one to determine a free parameter – excess pressure at the pole point of the capillary surface. The results of natural experiments are presented, refuting the truth of Young's formula and its corrected variants. A numerical method for solving the formulated problem has been developed. The dependence of the height of rise (lowering) of liquid in a capillary tube and wetting angles on the variation of the input data of the problem: temperature, chemical composition of the liquid and the size of the inner diameter of the tube is investigated. For small diameter tubes, the possibility of the existence of two pairs of solutions has been demonstrated. The two solutions of the first pair correspond to the rise of the liquid to different heights, the solutions of the second pair correspond to the lowering of the liquid to different depths. In the first pair of solutions, the capillary surface is convex downwards, and in the second pair, upwards. The wetting angles in each type of solution of each pair are different. It is shown that for any tube diameters there is a trivial solution in which the liquid level in the tube is equal to its level in the main reservoir. There are maximum permissible values of tube diameters, beyond which non-trivial solutions cannot be implemented. These critical diameter values depend on the chemical composition of the liquid and its temperature. A graphical method for solving the problem under consideration has been developed for capillary tubes of variable length and circular cross-section. Simple formulas have been obtained that allow one to approximately determine the height of liquid rise in a capillary tube (with an accuracy of 1÷2%) and the value of the critical diameter (with an accuracy of 0.5%).

For three families of axisymmetrical relatively undistorted waves, that are Bateman-Hillion solutions, quasispherical waves and “complex source” solutions, differential conservation law for the L₂-norm, having a form of continuity equation, is obtained. For the first two families, simple formulae for the norm are presented. Conditions ensuring independence of L₂-norm of wave equation solutions from time are found.

This paper presents exact analytical solutions to the integro-differential equation describing fluid flow in a vertical finite-length hydraulic fracture. The study investigates the regime of constant wellbore pressure and impermeable/open fracture boundaries. For the cases of a closed fracture boundary and an open boundary (constant pressure at the crack end), formulas are derived to determine pressure dynamics and production rate as functions of time and fracture geometric parameters. Numerical calculations demonstrate the influence of fracture length, conductivity, and formation permeability on well productivity. It is shown that for fractures longer than 100 m, the results align closely with the infinite fracture model, simplifying practical calculations.

В связи с непредвиденными обстоятельствами произведена замена DOI статей с префиксом Российской академии наук за 2025 год. В первых выпусках журналов РАН за 2026 год размещена информация о замене цифрового идентификатора на действующий DOI.