Abstracts

Siberian participants

The method of the solution of common differential second-degree equations with boundary conditions of the third kind is offered. In many practical problems of this type descripting mathematical models of diffusion-convective processes or the related physical phenomena, the factor at second derivative is small on a comparison with a factor at the first derivative and/or by other addends of an equation. The examples can be served by problems about carry of heat with large numbers the Peclet, about currents of the Navier-Stokes with large Reynold's numbers and problem of magnetic hydrodynamics with large numbers the Hartman. The solution of these problems can fast be changed near to boundary points, i.e. we have a boundary layer. At the solution of such problems the standard numerical methods arise large difficulties because of absence of approximation of an input equation [1,2].

Characteristics of an offered method is the absence of limitations on factors of an equation (at preservation positive of a factor at a higher derivative) and stability on small parameter at a higher derivative.

For construction of the incremental scheme on a range of definition of a problem the grid is set. On each cell of a partition of a function included in an equation are substituted by constants close to assumed functions. The functions of an equation are not necessary for substituting by constants, but at replacement them more composite functions, will arise complexities with finding of the analytical solution of an obtained equation. After replacement on each section the linear second-kind equation with constants by factors is received which has the analytical solution. The kind of this solution depends on the radicals of a secular equation. As the scheme allows to have change of branches of the solutions on adjacent sections of a grid, it can be used for any common equation of the 2-nd order. In an outcome of the requirement of a continuity of a derivative from the obtained solutions in units of a grid, the system of linear algebraic equations with a three-diagonal matrix allowed a method scalar прогонки is received.

The theoretical error of approximation of this scheme generally has the second order, that will be agreeed practical outcomes [3].

The literature [1] Бахвалов Н.С. Численные методы. Т.1. М.:Наука, 1973.

[2] Дулан Э., Миллер Дж., Шилдерс У. Равномерные численные методы решения задач с пограничным слоем. М.:Мир, 1983.

[3] Агафонцев С.В. Об одной модификации итерационно – интерполяционного метода.//Сопряженные задачи механики и экологии: Избранные доклады международной конференции, Томск :Изд-во ТГУ, 2000. С.5-28.

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