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Alan
Posts: 10165
Joined: December 19th, 2001, 4:01 am
Location: California
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### Re: About solving a transport equation

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.

Paul
Posts: 10771
Joined: July 20th, 2001, 3:28 pm

### Re: About solving a transport equation

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).

Cuchulainn
Posts: 62113
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: About solving a transport equation

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?

Paul
Posts: 10771
Joined: July 20th, 2001, 3:28 pm

### Re: About solving a transport equation

I would urge you to draw the pictures!

Cuchulainn
Posts: 62113
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: About solving a transport equation

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.

Cuchulainn
Posts: 62113
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: About solving a transport equation

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.

Paul
Posts: 10771
Joined: July 20th, 2001, 3:28 pm

### Re: About solving a transport equation

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!

Alan
Posts: 10165
Joined: December 19th, 2001, 4:01 am
Location: California
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### Re: About solving a transport equation

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.

Cuchulainn
Posts: 62113
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: About solving a transport equation

I use imgur, it's a pain in the assets. I'll check the small print.

Cuchulainn
Posts: 62113
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: About solving a transport equation

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).

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.

Cuchulainn
Posts: 62113
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: About solving a transport equation

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

Paul
Posts: 10771
Joined: July 20th, 2001, 3:28 pm

### Re: About solving a transport equation

For my three simple problems it looks like this:

Alan
Posts: 10165
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: About solving a transport equation

Had to test "place inline". This is great!

Paul
Posts: 10771
Joined: July 20th, 2001, 3:28 pm

### Re: About solving a transport equation

Cuch is easily distracted at the best of times!

Cuchulainn
Posts: 62113
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: About solving a transport equation

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.