Moving Node Method for the Approximate Analytical Solution One-dimensional Convection-Diffusion Problems

Methods for solving problems of mathematical physics can be divided into the following four classes.
Analytical methods (the method of separation of variables, the method of characteristics, the method of Green's functions, etc.) methods have a relatively low degree of universality, i.e. focused on solving rather narrow classes of problems.
Approximate analytical methods (projection, variational methods, small parameter method, operational methods, various iterative methods) are more universal than analytical ones.
Numerical methods (finite difference method, method of lines, control volume method, finite element method, etc.) are very universal methods.
Probabilistic methods (Monte Carlo methods) are highly versatile. 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.
This chapter contains information about new approaches to solving boundary value problems for differential equations. It introduces a new method of moving nodes. Based on the approximation of differential equations (by the finite difference method or the control volume method), introducing the concept of a moving node, approximately analytical solutions are obtained. To increase the accuracy of the obtained analytical solutions, multipoint moving nodes are used. The moving node method is used to construct compact circuits. The moving node method allows you to investigate the diskette equation for monotonicity, as well as the approximation error of the differential equation. Various test problems are considered.
Subject Areas: Mathematics.