Approximate Methods of Higher Analysis

Control and Stability Analysis of Spatially Dependent Nuclear Reactor Systems.

“…Theorem 3 gives a sufficient condition for asymptotic stability of system' (38), and condition c dictates the restriction on the nature of the nonlinearities in the system. The inequality (40) indicates that for a nonlinear system with an asymptotically linear part, the stability of the system depends strongly on the relative stability of the linear part of the system.…”

Orbit Computation in Celestial Mechanics by Urabe's Method

“…The unknown coefficients a v are determined by a balance procedure applied to the following equation This method to determine the coefficients of the approximate solution (2.2) of the system (2.1) is the Galerkin method (see also [13][14][15][16][17] …”

Convergence acceleration of modified Fourier series in one or more dimensions

“…Since the function f − g k satisfies the first k derivative conditions, then the new approximation F N [f − g k ] + g k converges at a faster rate to f . This is the principle of the polynomial subtraction process [20,22].…”

“…For univariate Fourier expansions, provided the values of these jumps are known, there is an effective tool to accelerate convergence: namely the polynomial subtraction device [20,22]. This idea was first considered by Krylov [21], and has been widely studied since then (see [5,19,23] and the references therein).…”

“…In standard implementations [4,10,20,22] such a function is constructed from a certain set of polynomials (the possibility of using other functions was suggested in [10]). Although this is the most convenient choice, it leads to poor numerical stability.…”