Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Homework Help
Calculus and Beyond Homework Help
What does it mean for a linear approximation to be reliable?
Reply to thread
Message
[QUOTE="soarce, post: 5480827, member: 116188"] Linearized equations are "reliable" when the equations are linearly stable, i.e. the time dependent solution of the linearized system do not diverge from the nonlinear solution. The linearized solution won't capture all features of the nonlinear solution but at least it gives you a rough idea about the time evolution. This is equivalent to saying that the equations are linearly stable. To study the linear stability you replace, roughly speaking, the nonlinear solution as following [tex]\Phi(x)\rightarrow\Phi_0(x) + \delta(x)e^{\lambda t}[/tex] where ##\Phi_0(x)## is a time independent solution of the nonlinear equation. After plugging ##\Phi_0(x)+\delta(x)e^{\lambda t}## into the nonlinear equation one has to determine the eigenvalues ##\lambda##. If ##Re\{\lambda\}>0## then perturbation ##\delta## will grow with time and the solution ##\Phi_0(x)## it is said to be linearly unstable. Very important: keep in mind that the linear stability depends on the (is associated with a) time independent solution of the nonlinear equation. It may happened that a solution may be linearly stable while others not. See for instance the wikipedia [URL='https://en.wikipedia.org/wiki/Linear_stability#Example_2:_NLS']page[/URL]. However, the linear stability is a weak criteria when deciding whether a system is stable or not. This means that even if solution is linearly stable don't imply that it will follow the long time behavior of the nonlinear equation. Aside from the linear perturbations there are other types of perturbations which may set in and affect the time development. The stability chain is as following [tex]Energetic\: stability \Rightarrow Dynamical\: stability\Rightarrow Linear\: stability[/tex] The linear stability is used to rule out the stability, is the system is not linearly stable then it won't be neither dynamical nor energetic stable. The energetic and dynamical stabilities are in general cumbersome to undertake, one should study the Hamiltonian structure and, something like, the Lyapunov stability (related directly to the time dependent evolution of the solution). They are performed only for simple nonlinear systems and solutions. LE: You can follow, for instance, this notes as guide on [URL='http://people.uleth.ca/~roussel/nld/stability.pdf']linear stability analysis[/URL]. [/QUOTE]
Insert quotes…
Post reply
Forums
Homework Help
Calculus and Beyond Homework Help
What does it mean for a linear approximation to be reliable?
Back
Top