Refined least squares approach to the initial-value problems unstable in the Lyapunov sense1

Abstract

The paper presents application of the Refined Least Squares method to the initial value problems that are instable in the Lyapunov sense. There is shown that the method is not sensitive to this kind of instability. The method is especially useful in search of particular integral of the considered problem. The method has an additional tool to evaluate quality of approximation. The approach is based on minimization of the functional, which square root can is generalized norm L₂and can be used to estimate global error of approximation. The expected value of the functional is equal to zero. The approximation is satisfactory if both results converge and functional reaches value close to zero. The consideration is illustrated with examples. There are shown initial-value problems which have physical sense and are applicable in mechanics. Whereas numerical approach may fail for these tasks, Refined Least Squares approach returns reliable approximation. The last example presents application of the special feature of the method, which allows neglecting influence of general integral on the solution. The method may be used in sensitivity analysis, search of the problem parameters, verification of numerical methods and an antonymous method in computational physics and mechanics.