The eigenvalues and eigenfunctions of two boundary value problems for three-dimensional equations of elliptic types with singular coefficients with lower terms are found.
In this work a composition of Erd´elyi-Kober operator with differential operators of the high order is proved. Applying the proved theorems, explicit formulas of a solution of the analogue of the singular Cauchy problem for the iterated Klein-Gordon-Fock equation.
Lie group method is applicable to both linear and non-linear partial differential equations, which leads to find new solutions for partial differential equations. Lie symmetry group method is applied to study Newtonian incompressible fluid’s equations flow in turbulent boundary layers. The symmetry group and its optimal system are given, and group invariant solutions associated to the symmetries are obtained. Finally the structure of the Lie algebra such as Levi decomposition, radical subalgebra, solvability and simplicity of symmetries is given.
The model integro-differential Bernlulli equation is considered in the paper. This equation was reduced to a differential equation with derivatives of fractional orders and solved numerically with the help of Newton’s iteration method. Depending on the values of the control parameters, calculated curves were constructed.
The paper presents a new mathematical model of the van der Pol oscillator, Duffing with external periodic influence given hereditarity. An algorithm for finding the numerical solution of the original model equation, which is based on the finite-difference scheme. A computer program implementing this algorithm. With this program built waveform and the phase trajectories for hereditarity oscillator Van der Pol-Duffing.
The paper proposes and implements, in the form of a mathematical model and software package, the concept of using value coefficients, for ranging answers in universal hybrid question-answer systems, which will improve the efficiency of their work.
The article considers nonlinear dynamic systems with impulse perturbation. The described method of generation of impulse random processes on a computer. On the example of the Lorenz system are considered a software module.
A web interface was developed to visualize the calculation of the spatial distribution of correlation links between the whistler flux intensity and planetary thunderstorm activity.
The change in the magnitude of the atmospheric electric field (AEF) before the earthquake and immediately after it, according to our model, is due to the fact that the protons of water atmospheric complexes turn out to be quantum-entangled to protons of hydrogen bonds (HB) in lithosphere material. After the establishment of the quantum entanglement regime in the system, a decoherence follows. It manifests itself in AEF change, and then there is a recoherence, during which the quantum entanglement of the largest possible number of elements in the system with HB which ends with a new decoherence, is again tuned in the system. It is during this process that a shock wave or an earthquake entailing the change of AEF magnitude is generated.
The paper deals with the problem of constructing a straight line perpendicular to a given line, and on constructing a line perpendicular to a given plane, as a rule, reduce to constructing the height of some triangle. In the general case this problem is a metric problem.
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