disclaimer

This post contains explicit math. In fact, there’s so much math, your head just might explode. If you don’t like math, you’re better off doing something else.

Like having a tea party with the girlfriends.

Wuss.

This post is going to be a little bit different.

For starters, there aren’t any pictures of cats.

Instead, I’m going to spend a little bit of time writing about what happens when the solution to a partial differential equation has a jump discontinuity.

Before we get started, let’s look at two cases.

Case A: If the above didn’t make any sense whatsoever…

You should read my last post. If you’re still bored, you can drool over my choco-licious lime biscotti. And if that doesn’t do it, you can laugh at how I looked as a kid.

Otherwise, come back tomorrow.

Case B: If the above gets your blood flowing and produces a strong tingle in your special place…

Read on.

Actually, this topic is pretty easy to understand. What’s odd is that most references I’ve managed to dig up on the problem explain it horrendously at best. But as long as you have a clear understanding of how to solve PDEs using the method of characteristics, you should be well on your way to bigger and better things after this lil’ talk.

Let’s begin with a generalized first-order quasi-linear equation,

We’re going to still be interested in finding solutions u(x,y), but now we’d like to relax the condition that solutions have to be differentiable everywhere.

Weak Discontinuity: Let u(x,y) be defined in some domain D. Let C be a curve parametrized by x = x(t), and y = y(t). Assume that u(x,y) is continuous and differentiable everywhere except on C, across which a first-order derivative of the solution may be discontinuous. This is a weak discontinuity.

Now we can look at the limtiing value of the functions as the curve is approached from either side, so from the chain rule,

(1)
(2)

But what does that mean? Be sure to distinguish the two types of derivatives above. The dot, as in , is the derivative along the curve (with respect to t). This is in contrast to the partial derivatives, which measure the rates of change of u(x,y) in the x and y directions.

But actually, because u is continuous,

Why? Imagine it this way: Cars are travelling along the road C. An observer to the right of the road measures the speed of the car as . But on the other side, his buddy measures the speed of the car as . Assuming that both observers are sufficiently close (continuity), they’ll obviously measure the same speed.

And so, subtracting (1) from (2) we have,

(3)

where the square bracket denotes the ‘jump’ in the expression across C.

Remember that u(x, y) should satisfy the quasi-linear PDE except on C, so by subtraction we have,

(4)

Notice the c disappears because of the continuity of u, and this also explains why a and b can be evaluated on C without ambiguity from any direction.

(3) and (4) provide us with a system of equations for the size of the jump discontinuities. And of course, since we’re interested in non-zero jump discontinuities, we’ll require the determinant,

But this is the precise equation for the characteristic curves of the PDE!

Wowsers!

Solutions with Weak Discontinuities: If the solution of a quasi-linear PDE has a weak discontinuity across some curve C, then C must be a characteristic projection.

Actually, we can squeeze a little bit more information out of this theory. It turns out the size of the jumps across C can be calculated from a first order ordinary differential equation. For example, in the case of a linear PDE,

where we follow the characteristic curves and ,

And so if the jump is non-zero at some point in space (x,y), it will continue to be non-zero along a characteristic. More importantly, if there is initially a weak discontinuity at t = 0, then this jump will propogate along the characteristic.

Propogation of Weak Discontinuities: Suppose that there is initially a jump in the derivatives of the initial boundary data (Cauchy problem). Then this jump will propogate along its characteristic curve. Furthermore, if there is no jump, then the solution will be well behaved (around the characteristic).

Let’s finish with a bang.

Let’s solve the following PDE

subject to the Cauchy conditions:

using the method of characteristics, it’s quite easy to show that the solution is given by,

Take a look at the graph.

Notice once you’ve pieced everything back together, the solution is continuous for t > 0. The important observation is of course that the solution is smooth everywhere, except along the two special characteristics , where the discontinuity from the Cauchy data propogates along the characteristic.

And that about completes our look into what happens when there’s a jump discontinuity in the solutions of a PDE.

But we’ve only gotten started!

The more interesting and deeper question is what happens when the solution itself is discontinuous. These solutions contain so-called shock waves and are termed weak solutions. Does this happen in real life? Sure. Take a look at this picture.

Notice that a description of the surface waves would be discontinuous in the region where the waves are about to break, or the region where the wave is about to ‘bend’ over itself.

The basic ideas behind studying discontinuous solutions remains the same as for the above problem. Except now we can’t really use expressions like,

Right? Because since u(x,y) is itself discontinuous at certain points, we can’t be so cavalier about taking derivatives! The key to approach these types of problems is to consider equivalent integral formulations, as we know that integration is more robust with respect to discontinuous solutions.

But that’s all for another day.

Phew. I’m going to go lie down now.

math is hard

Make no mistake about it.

Math is hard.

It’ll pull down your pants, kick you where it hurts, then tell all your friends about how you cried like a little bitch.

For my next-next post, as in the post following this one, you’re going to see some weird and unusual things.

And some of you — most of you — are going to dislike it.

Allow me to explain.

Phil’s Proof is not a math blog. This, for example, is a math blog.

And let’s be honest. Who reads this shit?

Probably like, eight people on the Internet.

Let’s first establish two preliminary facts.

Fact 1: Math is hard. Math will kick you when you’re down.

Make no mistake about it. Math is hard. It’ll pull down your pants, kick you where it hurts, and then tell all your friends about how you cried like a little bitch.

It’s a whole new language. Don’t believe me? Read this.

Have you ever asked a math researcher what he does? Notice that casual, well practiced, nothin’-much shrug? You know why he does that?

It’s because you wouldn’t understand it.

He’d need three textbooks, several appendices, a blackboard, and a donkey just to give you the necessary background before you get to where the action’s at.

Try not to get offended. It’s just the nature of the beast.

Fact 2: There are two types of mathematical writing.

Good and bad. Technical and expository.

90% of the stuff out there is technical writing. That’s the stuff you see in textbooks, journal articles, math databases, and so on. Some of the stuff is highly unreadable, but there are the rare readable ones.

Here’s a random page from one of my books.

Imagine reading Harry Potter. But instead, the entire book consists of theorems, definitions, proofs, lemmas, and the odd footnote. Imagine having to do exercises at the end of each chapter just so you can understand what happened. Oh, and did I mention Hermione’s gone? Yeah, that’s right. You don’t get girls looking like that in math.

So then there’s expository writing. This is the kind of writing that’s slanted towards a broader audience.

Math is technical. It’s brutal. But it’s also beautiful. And if a writer can somehow remove the fog of technical jargon and doowacky, and show just how beautiful math can be — well, that’s wondrous writing indeed.

I’ve got a lot of respect for these kinds of mathematicians. Because to be a great expository writer, you have to have both a complete understanding of the nitty and gritty, as well as a tremendous writing ability. You need to have pizazz and flair.

Baby, you need a personality.

Phil’s Proof is not a math blog.

Not really.

I know most of you aren’t serious math buffs. And you know what? I love that!

In the rare instances that I do write about math, I need it to be fun. I need it to be light and frothy. And the fact that so many of you have written to tell me that I’ve managed to make it all comprehensible and interesting, well, that just warms my little geeky heart.

But that kind of expository writing takes energy. Hooo boy, does it ever! I need find creative ways to translate hoogie-boogie jibberish to everyday English. And that’s always a big effort.

Which brings me to my next post.

Sometimes, I don’t want to go through all that effort. Sometimes, I need to write a more technically slanted post. For most math gurus, the writing is still going to be pretty informal. But for most of my usual readers, you’ve been warned.

Unlike my usual chatter, it’s not going to go down well with a nice cup o’ tea and sugary pastry. You better have a family-sized bottle of Pepto-Bismol ready.

In any case, my usual programming of inane randomness and silliness will resume shortly after.

I have a friend who can tell you exactly what she wants in a guy.

She can give you a detailed list of qualities and attributes.

She can provide distributions, fancy pie charts, and precision rankings.

She can tell you how important looks are, and exactly what ratio of humour to physical attractiveness is an acceptable tradeoff.

Yeah, it’s pretty darn amazing.

Me? I have no idea what I want.

That is, until recently.

With summer in full swing, and blistering hot weather here in Ottawa, there’s been a definite pleasure in watching how fashion changes. You know, from the typical Canadian parka, snowshoes, and tuque, to something a little bit more festive.

And then I realized the other day: I do know what I want.

I want a girl with a skirt.

Really, it’s that simple. I just think there’s something so incredibly sexy about women in skirts. Whether it’s a long one, a short one, one of those frilly white skirts, or even denim — anything goes. I don’t know what it is.

Maybe it has to do with how a skirt accentuates the female form. Or maybe it has to do with the fact I hang around math geeks all day and none of them are qualified to wear skirts.

Some of them aren’t even qualified to wear shorts.

In any case, there it is. I put it out there, folks.

See? I’m not picky at all.

Edit: See? Even Mr. David agrees with me.

To: Phil
From: Dave
Date: Right after he read my post
Subject: Skirts

skirts are the bomb!

so sexy, i just get riled up whenever i see a skirt… yes, even ones at the wall, on mannequins…

I concur. Even headless mannequins wearing skirts are hella hot.

faith

How can you base your actions, your beliefs, your thoughts, your hopes, on something that is — for all intents and purposes — nonexistent?

I wrote this three years ago.

I remember it pretty vividly. It was the summer after high school and I was a bit heartbroken. Spittin’ angry, too. It was on my old blog, which has long been archived and safely tucked away for future mocking. So don’t bother looking.

Why would you want to look, anyways?

It’s just the incoherent ramblings of an angry kid.

You get almost the same experience watching re-runs of The O.C.

Except during one of my intense brooding sessions, I never have mournful indie-pop music playing in the background.

That’s right, Ryan Atwood. You sissy.

1. Being Alone

I’m starting to realize I have these anti-social, misanthropic feelings. My parents started warning me about it years ago, but I always told them it was because I was too busy to socialize.

It’s not that I’m bad at it. Or that I get the shakes when I’m at a party.

It’s just that things have changed. I’m not the same carefree bachelor free-spirited youth I was a few years back. I don’t feel the same.

2. The Box

I started pawing through the box.

If you’ve been in a busted relationship, you’ll know exactly what I mean.

You’ll know what’s in this box and why words like ‘the box’ need to be italicized, in contrast with other words which don’t strike as much fear into the hearts of singletons like us.

Take the word ‘cat’, for example. Can you say ‘cat’? I bet you can. And I bet you don’t break down into spasmic convulsions when you do. See, that’s why ‘cat’ isn’t italicized. Idiot.

And why did I do it? Just why did I dust off the box? Because I’m a melancholic retard, that’s why.

3. Being Tagged

I have shitload of Memes I need to get through.

Shit.

4. Leaving on a Jet Plane

I’m leaving on a jet plane. Soon. As a volunteer.

So I can bring the wonderful world of mathematics to poor children who may not otherwise have the chance to integrate a function. Ever.

Damn, can you imagine that? The horror!

Anyways, I’ll of course have to let you guys know exactly where I’m going and what I’ll be doing. Because there’s a very real chance I’ll be eaten by like, a lion or a gazelle.

5. Be a Hero (Or a Hiro)

Heroes Season Finale. Tonight. Prepare to be amazed.