Sentence examples for autonomous problem from inspiring English sources

For the simplicity of the analysis, we assume that IVP (1) is an autonomous problem.

Whenever we want to solve more complicated initial value problems by means of the linear autonomous problem (20), we have to rely on a method of approximation.

In this subsection, we give some results for the autonomous problem varepsilon^{2s} -Delta)^{s} -Delta(u),quad uin H^{s}u+au=bh uhbb{R}^{N}bigr),quad1) which are uinful for tH^{s}biglof the nonlocal critical problemathbb{R}^{

In addition, the existing systems are lack of flexible and autonomous problem-solving capability, rich interactions, and pro-active and reactive features.

The main technical tools are evolution semigroups allowing the direct application of existing results on autonomous problems.

end{aligned}This slight extension of Lemma 33 to piecewise autonomous problems implies directly that the set ( big { |f(t)|^p big | t in [0,T], ; f cdot ) in mathcal{N}_{L^p}big } subset L^1({{mathbb R}^{N}}) ) is tight.

We present a simple analysis which, in many relevant cases, allows one to choose the most appropriate split to retain the high performance the methods show on the autonomous problems.

In order to use the method of normal form, the non-autonomous problem is transformed into an autonomous one by a generalized co-ordinate transformation.

Furthermore, by means of symmetric quadrature formulae of appropriate order, we propose a different but still equivalent formulation of the original non-autonomous problem that treats the time as two added coordinates of an enlarged differential system.

Splitting methods have also been used for separable non-autonomous problems either by solving each non-autonomous part separately or after each vector field is frozen properly.

In combination with a new way of handling non-autonomous problems, the schemes we present are based on Magnus expansions and they show very promising results when applied to Hamiltonian ODEs and PDEs.