**1. Introduction**

Methods for solving problems of mathematical physics can be divided into the following four classes [1–7].

Analytical methods (the method of separation of variables, the method of characteristics, the method of Green's functions [8], etc.) have a relatively low degree of universality, i.e. focused on solving rather narrow classes of problems. As a result of applying these methods, a solution is obtained in the form of analytical formulas. The use of these formulas for the implementation of the calculation may require the solution of auxiliary computational problems (solution of nonlinear equations, calculation of special functions, numerical integration, summation of an infinite series). Nevertheless, in a number of cases, the application of these methods makes it possible to quickly and with high accuracy calculate the desired solution.

Approximate analytical methods (projection, variational methods, small parameter methods, operational methods, and various iterative methods [4, 9]) are more universal than analytical ones. The use of such methods involves modifying the original problem or changing the problem statement in such a way that the new problem can be solved by the analytical method, and its solution itself differs little enough from the solution of the original problem.

Numerical methods (finite difference method, method of lines, control volume method, finite element method, etc. [1, 2, 5–7, 10–34]) are very universal methods. Often used to solve nonlinear problems of mathematical physics, as well as linear problems with variable operator coefficients.

Probabilistic methods (Monte Carlo methods) are highly versatile. It can be used to calculate discontinuous solutions. However, they require large amounts of calculations and, as a rule, lose with the computational complexity of the above methods when solving such problems to which these methods are applicable.

Comparing methods for solving problems of mathematical physics, it is impossible to give unconditional superiority to any of them. Any of them may be the best for solving problems of a certain class. At the same time, when characterizing a specific method, it is advisable to highlight those features that often determine its advantages or disadvantages in practical application compared to an alternative method.

The advantages of the finite difference method include its high universality, for example, much higher than that of analytical methods. The application of this method is often characterized by the relative simplicity of constructing a decision algorithm and its software implementation. Often it is possible to parallelize the decision algorithm.

The shortcomings of the method include: the problematic nature of its use on irregular grids; a very rapid increase in computational complexity with an increase in the dimension of the problem (an increase in the number of unknown variables); the complexity of the analytical study of the properties of the difference scheme.

The proposed method of moving nodes combines numerical and analytical methods [7, 8, 13, 35–38]. In this case, we can obtain, on the one hand, an approximate analytical solution to the problem, which is not related to the methods listed above. On the other hand, this method allows one to obtain compact discrete approximations of the original problem. Note that obtaining an approximate analytical solution to differential equations is based on numerical methods. The nature of numerical methods also makes it possible to obtain an approximate analytical expression for solving differential equations. For this, a so-called "movable node" is introduced.

The aim of the study is to develop a computing technology based on the proposed method of moving nodes, develop a two-point convective-diffusion problem an analytical method generated by numerical methods based on the method of moving nodes, and give test examples.
