“ … partial differential equations are the basis of all physical theorems. In the theory of sound in gases, liquid and solids, in the investigations of elasticity, in optics, everywhere partial differential equations formulate basic laws of nature which can be checked against experiments.”

Bernhard Riemann

And since “who said it” sometimes carries more importance than “what is said”, let me emphasize that this quote is attributed to Bernhard Riemann! The same Riemann of geometry, complex analysis, topology, number theory; the “pure” mathematician Riemann. This post is for those “pure” mathematicians who believe that applications of a mathematical result mar its aesthetic beauty. Once not very long ago, I too was of this opinion. “PDE is not mathematics”, a friend quipped and I agreed wholeheartedly. I now realize how shallow this notion was.

In what follows, I will describe a smooth PDE result that involves interesting real analysis results. The Cauchy problem asks for a solution to a PDE with initial data defined on a hypersurface. In the case we are discussing, we have a linear PDE of first order with initial data. The “transport equation” is the simplest PDE in the sense that any simplification will result in it being an ordinary differential equation (ODE). (If I were to rechristen the subject, I’d rather call PDE as EDE, extra-ordinary differential equations, not because they are NOT ODE but because when learnt properly, they can be extraordinarily amazing).

We shall take Fourier transform on both sides of both equations, convert the PDE into an ODE, use the initial data and finally prove that the solution is

The justification of taking Fourier transforms, doing some algebra with them and equating two functions whose Fourier transforms are themselves equal, was discussed in this post. There we saw that two square-integrable functions with the same FT are equal almost everywhere.

We fix the variable and take Fourier transforms on the given PDEwith respect to the variable which gives,

The Fourier transform converts an -th order derivative into a polynomial of degree , so here the PDE in and has been converted into an easy ODE in , which we know how to solve. (Say by Clairut’s method to solve the ODE – ). Its solution is given by

Here, denotes shifting by . (See the Wikipedia link for basic properties of Fourier transforms).

By taking inverse-Fourier transform, we see that

An assertion on uniqueness of the solution (which we shall not prove) shows that this solution is unique. Observe that the solution is just a shift (transport) of the initial wave with speed . If we started with a sine wave at , we would end up with a sine wave propagating in space-time. Isn’t it remarkable?

## 7 comments

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February 11, 2012 at 21:36

Ojas Sahasrabudheomg….. !! what a change !!

never expected a post on PDE from you (& that too you are describing it as EDE !!). how’s the change happened ??

I remember that, you became once the top scorer in our PDE Quiz & then you did not used to like it at all whenever reminded of it !! 🙂

couldn’t resist to comment !!

– Ojas.

February 11, 2012 at 22:50

Amar MainkarThis post should be emailed to the address akp@math.iitb.ac.in; I am sure an award for highest scorers in PDE in IITB will be instituted in the honour of PDE’s latest fan!

February 12, 2012 at 00:08

Abhishek ParabYes I know, I know. Its me, this is my new avatar. Guess, I have grown up now. Mathematically mature. Or may be some day the sleeping algebraist in me will wake up and denounce this meta-mathematics crap!

But guys, on a serious note, in India the lack of good analysis teachers has made research in such topics a taboo. Algebra and number-theory are over-hyped. And poor PDE is left to Mamta and akp. Did you know GKS did his PhD in PDE?

And Ojas, feel free to comment on the blog. Its no secret that you have been following my blog under various pseudonyms. 😀

May 13, 2015 at 08:09

Gopala KrishnaHi, This is GKS here 🙂 Not only was my PhD on PDEs, I use the representation theory of SL_2(C) in the space of binary forms – classical invariant theory

February 15, 2012 at 08:21

Ojas SahasrabudheYes, I do follow your blog, in fact I’ve subscribed it so that whenever you post something on your blog, I get a notification via e-mail. So you can count me as one of the fans of your blog !! 🙂

btw this semester, GKS is teaching a PhD course on PDE & SAM wanted to teach the PDE (M.Sc) last time but didn’t get a chance. Imagine SAM teaching a PDE (full of hardcore Analysis & maybe categories) !! 🙂

One of my friend told me that “PDE is the pinnacle of Analysis” !!

So maybe it’s not as bad as it seems…

May 13, 2015 at 08:15

Gopala KrishnaDear Abhishek, I recently taught a UG course MA 207 to sophomores on PDEs. Could send you the slides. You may find it interesting or amusing.

GKS

May 13, 2015 at 21:31

Abhishek ParabHi sir! Your comment came as a pleasant surprise to me. (And it makes me wonder what people might think when they read these posts I wrote eons ago, as a kid mathematically. Would they get offended? Probably not.)

The more Mathematics I learn, the more I’m convinced of its unity. It’s all inter-connected. You cannot understand one topic in isolation. Or can we, my algebraist friends?

Yes, I would be glad to take a look at the slides. My email is firstnamelastname[at]gmail. Although I don’t solve PDEs explicitly, the area of number theory I work needs me to know Lie groups, Lie algebras and their representations; so differential operators are all over the place.