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

Abstract

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.

Keywords

• Sphere
• stiff equation
• multi-layer method
• best parameter singularly perturbed equation