COMPARISON OF SOLVING A STIFF EQUATION ON A SPHERE BY THE MULTI-LAYER METHOD AND METHOD OF CONTINUING AT THE BEST PARAMETER

A stiff equation, linked with the solution of singularly perturbed differential equations with the use of standard methods of numeral solutions of simple differential equations often lead to major difficulties. First difficulty is the loss of stability of the counting process, when small errors on separate steps lead to an increase in the systematic errors in general. Another difficulty is, directly linked with the first one, consists of the need of decreasing the integrating step by a lot, which leads to a major decrease in the time taken for the calculations. On an example of a boundary value problem for a differential equation of second power on a sphere, comparison of our two approaches of constructing approximate values are held. The first approach is connected with the construction of an approximate multi-layer solution of the problem and is based on the use of recurrent equalities, that come out from classical numeral methods to the interval of a non-constant length. As a result, a numeral, approximated solution is replaced with an approximate solution in form of a function, which is easier to use for adaptation, building a graph and other needs. The second approach is linked with the continuation of the solutions by the best parameter. This method allows us to decrease majorly the number of steps and increase the stability of the computing process compared to standard methods

NUMERICAL METHODS FOR SOLVING PROBLEMS WITH CONTRAST STRUCTURES

In this paper, we investigate the features of the numerical solution of Cauchy problems for nonlinear differential equations with contrast structures (interior layers). Similar problems arise in the modeling of certain problems of hydrodynamics, chemical kinetics, combustion theory, computational geometry. Analytical solution of problems with contrast structures can be obtained only in particular cases. The numerical solution is also difficult to obtain. This is due to the ill conditionality of the equations in the neighborhood of the interior and boundary layers. To achieve an acceptable accuracy of the numerical solution, it is necessary to significantly reduce the step size, which leads to an increase of a computational complexity. The disadvantages of using the traditional explicit Euler method and fourth-order Runge-Kutta method, as well as the implicit Euler method with constant and variable step sizes are shown on the example of one test problem with two boundaries and one interior layers. Two approaches have been proposed to eliminate the computational disadvantages of traditional methods. As the first method, the best parametrization is applied. This method consists in passing to a new argument measured in the tangent direction along the integral curve of the considered Cauchy problem. The best parametrization allows obtaining the best conditioned Cauchy problem and eliminating the computational difficulties arising in the neighborhood of the interior and boundary layers. The second approach for solving the Cauchy problem is a semi-analytical method developed in the works of Alexander N. Vasilyev and Dmitry A. Tarkhov their apprentice and followers. This method allows obtaining a multilayered functional solution, which can be considered as a type of nonlinear asymptotic. Even at high rigidity, a semi-analytical method allows obtaining acceptable accuracy solution of problems with contrast structures. The analysis of the methods used is carried out. The obtained results are compared with the analytical solution of the considered test problem, as well as with the results of other authors

Fresh approaches to the construction of parameterized neural network solutions of a stiff differential equation

A number of new fundamental problems expanding Vasiliev's and Tarkhov's methodology worked out for neural network models constructed on the basis of differential equations and other data has been stated and solved in this paper. The possibility of extending the parameter range in the same neural network model without loss of accuracy was studied. The influence of the new approach to choosing test points and using heterogeneous complementary data on the solution accuracy was analyzed. The additional conditions in equation form derived from the asymptotic decomposition were used apart from the point data. The classical and non-classical definitions of the problem were compared by entering a parameter into the complementary data. A new sampling scheme of test point choice at different stages of minimization (the procedure of test point regeneration) under various initial conditions was investigated. A way of combining two approaches (classical and neural network) based on the Adams PECE method was considered