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Alan
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Re: About solving a transport equation

February 1st, 2020, 4:04 pm

Sure. 

BTW, looking again at my Mathematica session, there's still a small glitch. I told Mathematica that [$]y_1 < y_2[$]. But, indeed since [$]\dot{\xi} < 0[$] for this problem, actually I should have said to Mathematica that [$]y_2 < y_1[$]. That makes both the l.h.s. and r.h.s. positive for [$]t>0[$] (since [$]b(x)<0[$]). However, after making this change in the Assumptions, and rerunning: Mathematica gets the same expression for integral[y1,y2]. In other words, the answer is still the same, confirming Paul's answer. 
 
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Paul
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Re: About solving a transport equation

February 1st, 2020, 5:45 pm

Easy ones:

[$]u_t+u_x=0[$]

[$]u_t+\sgn(x)u_x=0[$]

[$]u_t-\sgn(x)u_x=0[$]

All to be solved for t>0 with initial data on t=0 (for all x or just bits).
 
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Cuchulainn
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Re: About solving a transport equation

February 1st, 2020, 8:46 pm

1... [$]u_t+u_x=0[$] .

2 ...[$]u_t+ sgn(x) u_x=0[$] 

3...[$]u_t- sgn(x) u_x=0[$] 

I suspect classification of characteristics. May need to do a bit of reading up.

1, unique solution
2. multiple solutions
3. non-unique solution (after a while)

Is [$]x = 0[$] a pathological case?
 
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Paul
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Re: About solving a transport equation

February 1st, 2020, 10:58 pm

I would urge you to draw the pictures!
 
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Cuchulainn
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Re: About solving a transport equation

February 2nd, 2020, 9:59 am

I would urge you to draw the pictures!
Exactly. I did. They tell a story. BTW how do I get a jpeg from my computer directly into a post?

BTW How do we take the discussion from here in general? 

TBD the outstanding questions posed by Alan.
Last edited by Cuchulainn on February 2nd, 2020, 10:51 am, edited 1 time in total.
 
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Cuchulainn
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Re: About solving a transport equation

February 2nd, 2020, 10:42 am

A side step: PW in his book describes non-probabilistic model of short interest rates:

[$]\frac{\partial V}{\partial t} + c(\frac{\partial V}{\partial r})\frac{\partial V}{\partial r}  - rV = 0 [$] 

This is an answer to many questions I reckon. It's a nonlinear pde.
 
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Paul
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Re: About solving a transport equation

February 2nd, 2020, 6:19 pm

I would urge you to draw the pictures!
Exactly. I did. They tell a story. BTW how do I get a jpeg from my computer directly into a post?
Below where you write it says attachments. Upload there. Then you will see something like "insert inline." It's a bit small but it's there.

Can you do the plots for the sgn problems?

Then maybe we find some exam questions!
 
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Alan
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Re: About solving a transport equation

February 2nd, 2020, 6:24 pm

I would urge you to draw the pictures!
BTW how do I get a jpeg from my computer directly into a post?
To display it in the post, just google "free image hosting" or some such, pick a site, upload, get a link ending in a supported picture format (.jpg, .png, .gif, others?), and insert the link into "Insert an image".

Haven't ever tried "insert inline". Maybe do both and see which is better.
Last edited by Alan on February 2nd, 2020, 6:26 pm, edited 1 time in total.
 
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Cuchulainn
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Re: About solving a transport equation

February 2nd, 2020, 6:26 pm

I use imgur, it's a pain in the assets. I'll check the small print.
 
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Cuchulainn
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Re: About solving a transport equation

February 2nd, 2020, 6:34 pm

I don't know much about shocks. But, 
This set of lecture notes by the late Piet Hemker (Amsterdam) discuss some of the issues, especially chapter 6. (one of the note takers, R. Mirani did some Asian FDM coding for me a while back).

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.24.565&rep=rep1&type=pdf

Might be useful.

And ... the challenges using FDM for these pdes.
Last edited by Cuchulainn on February 2nd, 2020, 7:07 pm, edited 2 times in total.
 
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Cuchulainn
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Re: About solving a transport equation

February 2nd, 2020, 6:57 pm

I would urge you to draw the pictures!
The pictures are just geometrical representations of plane curves (just like a circle is a point and a radius)?

The wave diagram in fig 5 is just the hyperbola [$]x^2 - y^2[$] = const

http://www.robertus.staff.shef.ac.uk/gian/chapter5.pdf
 
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Paul
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Re: About solving a transport equation

February 2nd, 2020, 7:07 pm

For my three simple problems it looks like this:
chars3.jpg
 
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Alan
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Re: About solving a transport equation

February 2nd, 2020, 7:15 pm

1519870931505.png
Had to test "place inline". This is great!
 
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Paul
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Joined: July 20th, 2001, 3:28 pm

Re: About solving a transport equation

February 2nd, 2020, 7:16 pm

Cuch is easily distracted at the best of times!
 
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Cuchulainn
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Re: About solving a transport equation

February 2nd, 2020, 7:29 pm

For my three simple problems it looks like this:

chars3.jpg
This is what I knocked up last night. BTW why does \sgn not work?
Paul, please can you write on blank high-quality DIN A4 paper and on one side only! With a black pen, not blue.

Image
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