Abstract:
The differential operator of the ordinary
differential equation (ODE) is represented as the sum of two operators: basic
and supplementing operators. The order of the higher derivatives of a basic
operator and ODE operator should coincide. If the basic operator has explicit
system of fundamental solutions it is possible to make integral equation
Volterra of II kind. For linear equations the approximate solutions of the
integral equation are system of the approximate fundamental solutions of the
initial ODE.

Abstract:
Variations in level of fluid in wells caused by
Earth tides and earthquakes are governed by two leading processes: deformation
of medium and filtration. These processes are described by identical systems of
equations and are united by the term “hydrodynamical phenomena”. The exact
solution is obtained for the problem of level variations in wells due to Earth
tides. The similar problem is solved for short-term precursors of a tectonic
earthquake. Results are consistent with data of field observations.

Abstract:
The homogeneous system of the equations of the linear theory of
elasticity for the isotropic environment with one-dimensional continuous
heterogeneity is considered. Bidimensional transformation Fourier is applied
and the problem for images is led to the ordinary differential equations.
Generally, the differential equations are transformed in integro-differential
and the algorithm of such transformation is resulted. Solutions of specific
problems are resulted.

Abstract:
The mode of definition of the error at polynomial Richardson’s
extrapolation is described. Along with the table of extrapolations the new
magnitudes reflecting expediency and efficiency of extrapolation are entered. On
concrete examples it is shown that application of Richardson’s extrapolation to
a solution of integral equations has appeared rather effective and gives a
solution with a high exactitude. Application of formulas of interpolation leads
to a solution in the analytical aspect.

The scheme of creation of systems of the
integro-differential equations for evaluation of Green’s function in
non-uniform elastic boundless medium is described. The summand with singularity
is allocated. The isotropic medium with constant coefficient of Poisson and unidimensional
inhomogeneous isotropic medium are considered.

The calculation
of the precursor of the tectonic earthquake in the telluric current is carried
out on the basis of the main equations of the telluric current and the theory
of preparation of the tectonic earthquake. Results do not contradict data of
field observations. Methodical significance of work is especially important.

The analog of the quadrature solution of the equation of Fredholm of the second kind is considered. Fundamental difference from classical quadrature formulas is as follows. On segments of the chosen grid not values of functions, but their integral average values are used. Computing examples show expediency of such approach in appropriate cases.

The analog of the quadrature solution of the equation of the second kind is considered. Fundamental difference from classical quadrature formulas is as follows. On segments of the chosen grid not values of functions, but their integral average values are used. Equations of Fredholm and Volterra are considered. The graphical representation of the solution is discussed. Computing examples show expediency of such approach in appropriate cases.

Abstract:
We will find a constant of motion with energy units for a relativistic particle moving in a quadratic dissipative medium subjected to a force which depends on the position. Then, we will find the Lagrangian and the Hamiltonian of the equation of motion in a time interval such that the velocity does not change its sign. Finally, we will see that the Lagrangian and Hamiltonian have some problems.

Abstract:
This paper has two parts, in this occasion we will present the first one. Until today, there are two formulations of classical mechanics. The first one is based on the Newton’s laws and the second one is based on the principle of least action. In this paper, we will find a third formulation that is totally different and has some advantages in comparison with the other two formulations.