Let me continue a little bit with what Alexander Martin was talking about in the early afternoon

and also the other speakers were talking about transmission problems, interface problems,

and let me go to the first picture which is in fact precisely out of the context that

Alexander presented earlier. So you have a gas network, in fact you have the gas network of

Germany and obviously this is a very complicated thing. Well to us maybe if you talk to people

from scientific computing they say well I mean come on this is a locally 1D problem. If you do

exascale computing you just put it in and fine right, but you have to be careful. I mean we are

talking about finally about optimization, optimal control problems, so you have lots of variables

and if you have a dynamic process, I'm talking not about the Watson network that Alex mentioned

where there is little maybe little dynamics, but in gas networks you do have tremendous dynamics

and this has in particular to do with changing markets, with changing providers and different

desires of the customers. So it is indeed a problem where you do have to discretize a lot and then you

get into millions or billions of variables and for that matter it is important to have something

at hand which is called domain decomposition. Domain decomposition can be in space and time

and Alex mentioned particular time domain decomposition or also space domain decomposition.

I will dwell a little bit on this but not stay for too long on the equations that you presented,

but there is one piece of equation in the model, there's a zoo of models and we discussed this with

Matthias after the lecture about the proper use of the word Euler's equation. There is a zoo of

equations and one is friction dominated flow and I would like to show you at least in one slide

this model, but not stay with that for a long time because I will also want to go into explaining

the main decomposition techniques. Okay so first of all clearly the pointer is maybe here, no this

is something else. What did I do? Okay so what you do is obvious, so you do decomposition of the

entire network into smaller networks. By that you also want to decompose the original goal which

I didn't define so far namely optimal control problems. So this is taken from from gas slip

and I don't have to go into details because Alex was so kind to mention this already. So you

decompose such a system into smaller systems and you want to do this with non-overlapping

domain decomposition because at interfaces, multiple interfaces not so clear what an

overlapping domain decomposition physically would be about. Okay so let me show you these equations.

The equations are the first line you see is like a doubly monlinear parabolic equation to deal

with and you see lots of symbols here. So my intention today is to put two things together.

First of all problems on graphs, problems on metric graphs, friction dominated flow and also

fractional derivatives and not because I like fractional derivatives per se because this is

very fashionable but because it has to do with the applications in a variety of ways. There is

a non-local in fact fractional continuity equation, there is non-local Darcy's law,

there is non-local, there is anomalous diffusion, sub diffusion, super diffusion. In a variety of

ways these non-local things I will show you in the next slide what this means at all are important

and I would like to take this up in order to put this into the context of processes on networks.

I hope I can use this pointer at some point without messing up. So obviously I cannot. So

let me dwell on this equation. The equation is as I said, I mean this is the master equation,

I will go away from this in a number of slides because I was talking about this earlier at other

occasions. This is a parabolic equation, these are fractional derivatives in time and space and I

would like to show you these definitions. So there is a lot of different fractional derivatives,

there is a so-called Caputo derivative and there is Riemann-Liouville derivative. These are hereditary

concepts and in fact that was in my early time when I was doing my PhD I was working on viscoelasticity

with fading memory and I somehow come back to this issue. So the first thing is the Caputo

derivative so you see what the connection between derivative and fraction is. You can take a classical

derivative and then you can take a convolution integral and integration over time or space

depending on what you mean. Or you can first take the convolution with a time kernel and then do

the derivative which is then Riemann-Liouville derivative. So these are classical concepts and

I don't want to go into the details here. So this is what a fraction derivative is. So let me go