Regards. I would like to ask how could we use statistics in a constructive way in a PDE?
This should be the simplest statistical usage in PDE. I already read some about stochastic PDE. In this question, we may classify into two :
PDE with statistics (non-stochastic)
Simplest stochastic PDEs
For example, in the linear advection equation we could put random generating numbers from a statistical distribution, let it be a normal one :
$$ u_{t} + c(t) \: u_{x} = S(x) $$
$ c(t) = N(\mu_{c}, \sigma_{c}) $ is the speed that generates from a normal distribution. $S(x) = N(\mu_{s},\sigma_{s})$ is the source function that also generates from another normal distribution. Numerically, this addition of statistics would produce a solution with noises.
The above is a non-stochastic example. I would like to know whether there are other constructive methods, especially the non-stochastic ones. I may also open for the stochastic ones, yet simplest. I am a beginner in this topic, I appreciate some inputs on this. Thanks. Regards, Arief.