http://group22x.narod.ru/grenkin/music_new/DancePhD.mp3
The blackbody is emitting
At all positive wavelengths.
We consider a gray medium
Where radiation propagates.
The radiative intensity is governed by equation
And it depends of the direction of propagation,
So this equation can’t be solved in an effective way,
That’s why we have to use approximations if we may.
Assuming that the medium is optically thick,
We average intensity and, after doing trick,
The radiative transfer equation is reduced
To the P1 approximation which we gonna use.
If you can hear and you can see,
Stand up and dance, dance, ..., PhD
Complex heat transfer is modelled by to PDE
The first is heat equation and second is the RTE.
The medium is surrounded by the boundary
That has some heat transfer coefficient and the emissivity
And these parameters are going to be optimized
And the heat transfer model is to be analyzed.
So, we have a system of two PDE,
Existence of its solution is that we want to see.
Also our question is if it’s unique.
Existence and uniqueness are this chapter’s topic.
Fixed points exist, they exist, they exist,
Schauder principle states this, it states this, it states this.
Fixed points exist, they exist, they exist,
And contracting mapping principle also states this.
There’s a compact mapping from a set to itself
Which is complex, bounded, closed in Banach space.
Or let us consider a closed set in a Banach space,
And let a mapping be contracting.
If we find a boundary coefficient
Such that temperature is the closest to desired field,
Then the optimality conditions
Must be regular to not be killed.
To obtain the regular conditions
We must prove that there will be unique
A solution of linearized system,
And to prove that, we apply some trick.
Multiply the each of the equations
By the sign of function in this one
After adding the just obtained relations
We derive that the solution is only one.
Then we add the just obtained relations
The solution is only one.
If you can hear and you can see,
Stand up and dance, dance, ..., PhD
Complex heat transfer is modelled by to PDE
The first is heat equation and second is the RTE.