"State of the Art" in nonlinear differential equations?

[Mr. Moose]

In my introductory ODE class we have focused mostly on linear differential equations. I know that nonlinear differential equations are much harder to solve, and I am wondering what exactly the "state of the art" methods are for dealing with them, or also what recent developments have been made in the field?

 

[jedishrfu]

Many are dealt with using numerical methods as often there is no exact solution. A coomon tool is Matlab where you can setup a program to integrate the ODE.

 

[epenguin]

A a dabbler I can't answer about the present state of the art, but at least be assured that a lot of what you have learned on lde's still useful for nlde's. Particularly the part about the nature of the solutions, stability of the stationary point, whether attractive, repulsive. You just linearise about the s.p.'s - the new thing is there may be more than one of them. You can almost never get analytical solutions (but then you realize that even when you could the interest was really qualitative anyway) but it is far from the case that there is nothing mathematical between analytical solutions and blind numerical computation - there is a lot of math in it, and there are whole fields like chaos.

 

[jedishrfu]

One interesting area of non-linear is with ocean waves. There was a NOVA documentary about rogue waves which were thought to be figments of the imagination of captains at sea until one was measured on an oil rig at 75 ft when swells of the time were only half that height. Later it was discovered thru satellite that they may be more frequent than that and could be the cause of many missing ships over time.

The rogue wave was a non-linear phenomena where neighboring wave contributed energy to the rogue to make it bigger. Ships that were hit by it would fall into a deep trough and then the wave would literally crash on top of them. A cruise ship lost power due to one such wave that hit, took out windows and flooded key electrical systems.

http://www.pbs.org/wnet/savageseas/neptune-side-waves.html

http://channel.nationalgeographic.com/channel/alien-deep/videos/rogue-waves/

One scientist explained them using a non-linear quantum mechanics equation.

 

[pasmith]

One interesting area of non-linear is with ocean waves. There was a NOVA documentary about rogue waves which were thought to be figments of the imagination of captains at sea until one was measured on an oil rig at 75 ft when swells of the time were only half that height. Later it was discovered thru satellite that they may be more frequent than that and could be the cause of many missing ships over time.

I recall a BBC Horizon documentary on that subject. In particular I recall some expert saying "This (points at straight line) is what the model used by the shipbuilding industry assumes, and this (points at set of points lying on a curve to which the previous line was tangent at some point) is the observed data". This was long before the program thought it worth mentioning the idea that "the linear model is inappropriate at large amplitudes".

 

[DivergentSpectrum]

In general, most numerical methods fall into the category of runge-kutta algorithms.

The first order runge kutta method is also known as Newtons method, and it is the simplest of all.
if you have dx/dt=f(x,t)
and have initial conditions x₀,t₀
then f(x₀,t₀)*dt=dx (note that here dt isn't actually dt, but it is a very small number called the stepsize- you probably know it as h)
the above equation can be verified using the identity (dx/dt)*dt=dx (also dx is not an actual dx, but it is the increment of x with respect to the stepsize of t)
then x₀+dx=x₁
and t₀+dt=t₁
the idea is to perform this many times setting dt as small as possible and provided the approximation is suitable it will converge to the correct answer.

There are higher order methods, The most popular(it is used by programs like mathematica by default) is the runge kutta forth order method, which is given by:
dx₁ = dt*f(x₀,t₀)
dx₂=dt*f(x₀+dx₁/2,t₀+dt/2)
dx₃=dt*f(x₀+dx₂/2,t₀+dt/2)
dx₄=dt*f(x₀+dx₃,t₀+dt)
then
x₁=x₀+(dx₁+2*dx₂+2*dx₃+dx₄)/6
and
t₁=t₀+dt
The basic idea of this is to estimate the x stepsize with respect to a t stepsize at varrying points and average them together, putting the most weight at the middle points. The real proof of the method is far more fascinating, but this is the basic idea.

One popular method is to use estimation methods of differing orders, for example find the estimate using a third order method and another estimate using the 4th order method, and as long as the estimations are close enough to being equal, that means the estimation is converging already, while if the estimations are a way off, it indicates it isn't converging well enough, and a smaller step size is used. this is called adaptive stepsize, and can be very powerful.

Its really a pretty interesting area of study.